Volume 25 Issue 2 August 2018 ISSN 2576-5299

University of Bahrain

Association of Arab Universities

Arab Journal of Basic and Applied Sciences

المجلة العربية للعلوم ا��ساسية

والتطبيقية

Arab Journal for Basic and Applied

Sciences

Published by Taylor and Francis on the Behalf of

the University of Bahrain

Volume 25 Issue 2 August 2018

Editorial board Editor-in-Chief Prof. Mohammad El-Hilo - General Secretary of Society of Colleges of Science in Arab Universities; and Dean of College of Science, University of Bahrain, Kingdom of Bahrain Managing Editor Prof. Waheeb E. Alnaser - Vice President for academic and postgraduate programs, University of Bahrain, Kingdom of Bahrain Editorial Secretary Dr. Ali Salman Bin Thani - University of Bahrain, Kingdom of Bahrain Editorial Board Prof. Yaseen Al-Soud (Chemistry) - Al al Bayt University, Jordan Prof. Dumitru Baleanu (Mathematics) - Cankayu University, Turkey Prof. Belkheir Hammouti (Environmental Chemistry) - University Mohammed Premier, Morocco Prof. Shoukry S. Hassan (Mathematics) - University of Bahrain, Kingdom of Bahrain Prof. Ji-Huan He (Mathematics) - Soochow University, China Prof. M. Ishaque Khan (Chemistry) - Illinois Institute of Technology, USA Prof. Badiadka Narayana (Environmental Chemistry) - Mangalore University, India Prof. Xiao-Jun Yang (Mathematics) - China University of Mining and Technology, China Prof. Haq Nawaz Bhatti (Physical and Environmental Chemistry) - University of Agriculture, Pakistan Prof. Cemil Tunç (Mathematics) - Van Yuzuncu Yil University, Turkey International Advisory Editorial Board Prof. Farouk El-Baz (Geology) - Centre for Remote Sensing, Boston University, USA Prof. Mustafa Amr El-Sayed (Chemistry) - Georgia Institute of Technology, USA Prof. Mourad E.H. Ismail (Mathematics) - University of Central Florida, USA Prof. Ahmed Sameh (Computer Science) - Minnesota University, USA Prof. Munir H. Nayfeh (Physics) - Illinois University, USA Prof. Sultan Abu Orabi (Chemistry) - Secretary General of AAU, Jubeyha, Amman, Jordan Prof. Riyad Y. Hamzah (Biochemistry) - President of University of Bahrain, Kingdom of Bahrain Distinguished Advisory Editorial Board Prof. Ahmed H. Zewail (26 February 1946 - 2 August 2016)

Arab Journal of Basic and Applied SciencesVolume 25 Issue 2 August 2018

CONTENTS

Original Articles

45 Evaluation of newly synthesized hydrazones as mild steelcorrosion inhibitors by adsorption, electrochemical,quantum chemical and morphological studiesTuruvekere K. Chaitra, Kikkeri N. Mohana, and HarmeshC. Tandon

56 Optimization of methyl esters production from non-edibleoils using activated carbon supported potassium hydroxideas a solid base catalystAbdelrahman B. Fadhil, Akram M. Aziz, and MarwaH. Altamer

66 New similarity solutions for the generalized variable-coefficients KdV equation by using symmetry group methodRehab M. El-Shiekh

71 New interaction solutions and nonlocal symmetries for the(2 + 1)-dimensional coupled Burgers equationHengchun Hu and Yueyue Li

77 Application of two different algorithms to the approximatelong water wave equation with conformable fractionalderivativeMelike Kaplan and Arzu Akbulut

85 Exact solutions of the classical Boussinesq systemHong-Qian Sun and Ai-Hua Chen

ORIGINAL ARTICLE

Evaluation of newly synthesized hydrazones as mild steel corrosioninhibitors by adsorption, electrochemical, quantum chemical andmorphological studies

Turuvekere K. Chaitraa, Kikkeri N. Mohanaa and Harmesh C. Tandonb

aDepartment of Studies in Chemistry, Manasagangotri, University of Mysore, Mysuru, India; bDepartment of Chemistry, SriVenkateswara College, New Delhi, India

ABSTRACTInhibition effect of newly synthesized heterocyclic aromatic compounds, 3-(cyano-dimethyl-methyl)-benzoic acid thiophen-2-ylmethylene-hydrazide (CBTH) and 3-(cyano-dimethyl-methyl)-benzoic acid furan-2-ylmethylene-hydrazide (CBFH) was studied on mild steelcorrosion in hydrochloric acid medium by gravimetric, electrochemical and morphologicaltechniques and correlated with quantum chemical indices of the respective molecules. Atoptimized concentration (2mM), CBTH and CBFH showed the highest inhibition efficiency of87.1% and 85.3%, respectively. Impedance study revealed that simultaneous increase inpolarization resistance and decrease in double layer capacitance with increasing inhibitorconcentration is due to adsorption phenomenon of hydrazones. Inhibitors shift the corrosionpotential to less negative value which hinders dissolution of mild steel and evolution ofhydrogen and it is more pronounced in anodic reaction. Corrosion current density decreasedfrom 0.2mA cm�2 (blank) to 0.0177 and 0.026mA cm�2 and polarization resistance increasedfrom 199 X cm2 to 1590 and 1552 X cm2 for CBTH and CBFH, respectively. Quantitativestructure activity relationship (QSAR) results showed good correlations between a number ofquantum chemical parameters and the experimentally determined inhibition efficiency.Scanning electron microscopy (SEM) and energy dispersive X-ray spectroscopy (EDX) analysesconfirmed the formation of protective inhibitory film.

ARTICLE HISTORYReceived 14 June 2017Accepted 18 December 2017

KEYWORDSMild steel; corrosion;hydrazone; adsorption;electrochemical techniques

1. Introduction

The suffering of mild steel (MS) surface whenexposed to various service environments containinghydrochloric acid and sulfuric acid due to corrosionis commonly observed in chemical processing, acidpickling, acid cleaning, ore production, petroleumproduction, oil well acidification and acid descaling.Mitigation of corrosion is usually done by eitheralloying the metal with other elements or protectivefilm forming through chemical compounds (Kumar,Danaee, Avei, & Vijayan, 2015). Organic inhibitors arepreferred for their high efficiency, ease of synthesis,less toxicity and cost-effective nature (Tourabi,Nohair, Traisnel, Jama, & Bentiss, 2013). Most com-monly used corrosion inhibitors are organic com-pounds containing polar groups including nitrogen,sulfur, and/or oxygen atoms and heterocyclic com-pounds with polar functional groups and conjugateddouble bonds (Ansari, Quraishi, & Singh, 2017; Verma& Quraishi, 2017).

Hydrazones are close relatives of imines whichexist as R1R2C¼NNH2. Many researchers studied com-pounds containing>C¼N– bond as corrosion inhibi-tors and obtained excellent results (Daoud, Douadi,Issaadi, & Chafaa, 2014; Saha, Dutta, Ghosh, Sukul, &Banerjee, 2015; Singh, 2012). Hydrazones have beenmuch investigated owing to their wide range ofproperties including antibacterial (Khalil, Berghot, &Gouda, 2009), antitubercular (Telvekar, Belubbi,Bairwa, & Satardekar, 2012), antifungal (Secci et al.,2012) and antimalarial (Fattorusso et al., 2008) activ-ities and recently reported corrosion inhibitingproperties. Inhibition efficiency study of somewater-soluble hydrazones for C-steel corrosion inhydrochloric acid has been done, and the maximuminhibition efficiency obtained was 95% (Moussa,El-Far, & El-Shafei, 2007). Corrosion inhibition effi-ciency of quinolinyl thipropano hydrazones was alsotested and obtained excellent inhibition of 98%(Saliyan & Adhikari, 2009). Negm, Morsya, and Said(2005) studied some novel hydrazones as corrosion

CONTACT Kikkeri N. Mohana drknmohana@gmail.com Department of Studies in Chemistry, Manasagangotri, University of Mysore, Mysuru,Karnataka, India

Supplemental data for this article can be accessed here.

� 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

University of BahrainARAB JOURNAL OF BASIC AND APPLIED SCIENCES2018, VOL. 25, NO. 2, 45–55https://doi.org/10.1080/25765299.2018.1449347

inhibitors in NaCl medium containing 5% H2S andobtained maximum inhibition efficiency up to 56%(Negm et al., 2005).

In continuation of our earlier work (Chaitra,Mohana, & Tandon, 2015; Gurudatt & Mohana, 2014;Kumar, Mohana, & Muralidhara, 2014), present paperreports the synthesis of 3-(cyano-dimethyl-methyl)-benzoic acid thiophen-2-ylmethylene-hydrazide(CBTH) and 3-(cyano-dimethyl-methyl)-benzoic acidfuran-2-ylmethylene-hydrazide (CBFH) along withtheir characterization using FTIR, 1H NMR and massspectral studies and testing their corrosion inhibitionproperties using weight loss, electrochemical imped-ance spectroscopy (EIS) and potentiodynamic polar-ization techniques. The protective inhibitory filmformed on the steel surface was studied by scanningelectron microscope and energy dispersive X-rayanalyses. In addition to this, quantum chemical cal-culations were done and obtained parameters werecorrelated with experimental findings.

2. Experimental

2.1. Materials and sample preparation

Mild steel specimen of dimension 2 cm�2 cm�0.1 cm and having composition (wt. %) of C – 0.051;Mn – 0.179; Si – 0.006; P – 0.005; S – 0.023; Cr –0.051; Ni – 0.05; Mo – 0.013; Ti – 0.004; Al – 0.103;Cu – 0.050; Sn – 0.004; B – 0.00105; Co – 0.017; Nb –0.012; Pb – 0.001, and the remainder being Fe wasused in all experiments. Preparation of specimensand solutions were as reported earlier (Chaitra,Mohana, Gurudatt, & Tandon, 2016).

2.2. Synthesis of inhibitors

Scheme for the synthesis of CBTH and CBFH is out-lined in Figure 1. Procedure for synthesis is given insection 1 of supporting data.

Synthesis of 3-(cyano-dimethyl-methyl)-benzoic acidhydrazide (compound 3): Compound 3 was synthe-sized from compound 1 via 3-(cyano-dimethyl-methyl)-benzoic acid ethyl ester according to thereported procedure (Harish, Mohana, Mallesha, &Veeresh, 2014). Product yield is 92% and meltingrange is 455–457 K.

Syntheses of 3-(cyano-dimethyl-methyl)-benzoic acidthiophen-2-ylmethylene-hydrazide (compound 4) and3-(cyano-dimethyl-methyl)-benzoic acid furan-2-ylmethylene-hydrazide (compound 5): Compounds 4and 5 were synthesized by reacting compound 3with thiophene-2-carbaldehyde and furan-2-carbalde-hyde respectively according to the reported proced-ure (Sherif & Ahmed, 2010). Product yield is 89% andmelting range is 459–462 K.

2.3. Weight loss measurements

Weight loss experiments were done as described byChaitra et al. (2016).

2.4. Electrochemical measurements

Potentiodynamic polarization and electrochemicalimpedance spectroscopy (EIS) experiments were car-ried out using CHI660D electrochemical workstation.Electrochemical experiment was conducted asreported earlier (Chaitra et al., 2016). The electro-chemical tests were performed using the synthesizedhydrazones for concentrations of 0.5–2.0mM at 303K. Potentiodynamic polarization measurements wererecorded by changing the electrode potential from�200mV toþ200mV, related to the open circuitpotential, with the scanning rate of 10mV s�1. EISdata were taken using AC sinusoidal signal in the fre-quency range of 0.1 to 100,000Hz with amplitude0.005 V. Simulation of EIS data and fitting of curveswere done using Zsimpwin software.

CN

1

N

O

OH SO

0

1

OCl2, EtOH

0–5°C

CN

O

O

2

OEt

C

ETOH,

0-5 o

CN

NN

O

HS

NH2NH2.H2O

oC

4

S

O

H

EtO

H

CN

O

3

tOH

, AcO

H, r

.t. EtO

H,

CN

O

NH

NH2

O

OH

, AcO

H, r.t.

NN

O

HO

5

O

H

Figure 1. Scheme for the synthesis of inhibitors.

46 T. K. CHAITRA ET AL.

2.5. Quantum chemical calculations

The geometrical optimization of the investigatedmolecules was done by Ab initio method at 6-31G��basis set for all atoms. The convergence limit at 1.0and rms gradient at 1.0 kcal/A mol were kept forenergy minimization. The Polak-Ribiere conjugategradient algorithm which is quite fast and precisewas used for optimization of geometry. TheHYPERCHEM 7.52 (Hypercube Inc., Gainesville, FL,2003) professional software was employed for allcalculations.

2.6. Scanning electron microscopy and energydispersive X-ray spectroscopy

The scanning electron microscopy (SEM) used wasZeiss electron microscope with the working voltageof 15 kV and the working distance of 10.5mm. SEMimages were taken of polished MS specimen andspecimen immersed in acid solution with and with-out optimum concentration of inhibitors after 4 h ofimmersion. Energy dispersive X-ray spectroscopy(EDX) experiments were performed using FESEMquanta-200 FEI instrument.

3. Results and discussion

3.1. Weight loss measurements

3.1.1. Effect of inhibitor concentration

Gravimetric experiments were performed on MS inthe temperature range of 303–333 K in 0.5 M HCl forboth the hydrazones for 24 h of immersion.Following formulae was used for the calculation ofcorrosion rate (CR) and inhibition efficiency (IE) andresults are tabulated (Table 1 of supporting data).

CR¼ DWSt

(1)

IE ð%Þ ¼ ðCRÞa�ðCRÞðCRÞa

� 100 (2)

where, Dw is the weight loss, s is the surface area ofthe specimen (cm2), t is the immersion time (h), and(CR)a, (CR)p are corrosion rates in the absence andpresence of the inhibitor, respectively.

Variations in CR with the change in temperatureand concentration are depicted in Figure 2.CR decreased considerably with increase in concen-tration of both the hydrazones, but the extent ofinhibition offered by CBTH was higher compared toCBFH. The increase in inhibition efficiencies withincrease in concentration is due to increased surfacecoverage caused by adsorption of inhibitors. Themaximum inhibition efficiencies exhibited were87.1% and 85.3% by CBTH and CBFH, respectively.The protective property of hydrazones may be due

to the interaction between p-electrons and lone pairof electrons present on oxygen, nitrogen and sulphurwith charged steel surface. Presence of nitrile groupattached to two electron donating methyl groupsalso contributes to the inhibition efficiency becausethe electron density is more towards nitrile group. Asa result, the net negative charge developed on thenitrile group resulting in good inhibition effect(Gurudatt & Mohana, 2014). In the presence ofhydrochloric acid, protonation can occur on carbonyloxygen atom or nitrogen atom. When the proton-ation occurs, solubility of the inhibitor increases. Thishas positive effect on inhibition efficiency. The gen-eral trend reported in the inhibition efficiencies ofmolecules containing heteroatoms is in the orderO<N< S< P (Obot & Obi-Egbedi, 2011). The higherinhibition efficiency shown by CBTH compared toCBFH is due to better electron donating capacity dueto the presence of sulphur instead of oxygen atom.

3.1.2. Activation and thermodynamic parameters

Corrosion rate for both CBTH and CBFH increasedwith increasing solution temperature and maximumefficiency was obtained at 303 K. This can be attrib-uted to the quick desorption of adsorbed inhibitormolecules from the surface in addition to thedecomposition of these molecules at elevatedtemperatures (Aljourani, Raeissi, & Golozar, 2008).Activation thermodynamic parameters were calcu-lated using Arrhenius equation and its alternativeform given as equations (3) and (4), respectively.

CR ¼ kexp � Ea�

RT

� �(3)

CR ¼ RTNh

expDSa�

R

� �exp

�DHa

RT

� �(4)

where, Ea� is activation energy, DSa� is the entropyof activation, DHa

� is the enthalpy of activation, k isArrhenius pre-exponential factor, h is Planck’s con-stant, N is Avogadro’s number, T is the absolute tem-perature and R is the universal gas constant. Plot ofCR vs 1/T gave the values Ea� and k whereas ln CR/Tvs 1/T gave DHa

� and DSa� (Figure 1 and Figure 2 of

30 30 30 40 40 40 50 50 50 60 60 60

-0.2

0.3

0.8

1.3

1.8

0.000 0.5 1 1.5 2

CR

(mg

cm-2

h- 1)

Concentration (mM)

CBTH

CBFH

Figure 2. Variation of corrosion rate with solution tempera-ture and concentration of inhibitors after 24 h of immersion.

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 47

supporting data). All parameters involved inArrhenius equation were calculated and listed inTable 2 of supporting data. It was found that Ea� val-ues for systems containing CBTH (27.6 to 33.3 kJmol�1) and CBFH (ranged from 33.2 to 34.9 kJ mol�1)were higher as compared to inhibitor free acid solu-tion (23.9 kJ mol�1). Increase in Ea� value after theaddition of hydrazones indicates the reduction in theease for charge and mass transfer due to the forma-tion of energy barrier. It is reported that if inhibitionefficiency decreases with an increase in temperature,then Ea (inhibited solution)> Ea (uninhibited solu-tion) (Singh, 2010). Results obtained in the presentstudy justify this statement. Positive values of DHa

�which varied between 24.9 and 32.2 kJ mol�1 showthat MS dissolution is endothermic. Negative valuesof DSa� which varied from �159.8 and �174.4 JK�1

mol�1 in the presence of inhibitors suggests that for-mation of the activated complex in the rate deter-mining step is association rather than dissociationleading to decrease in randomness (Ghareba &Omanovic, 2011). Movement from reactant to acti-vated complex state is accompanied with change inactivation entropy due to complex adsorption anddesorption between hydrazones and water moleculeson the metal surface which is referred to as quasisubstitution. Negative value of activation entropyindicates the decrease in randomness due toincrease in solute entropy due to adsorptionof hydrazones.

3.1.3. Adsorption isotherm

Adsorption of organic molecule occurs because theinteraction energy between the metal surface andthe inhibitor is higher than the interaction energybetween the metal surface and the water molecules(El-Maksoud, 2008). Attempts were made to fit thedata to various isotherms like Langmuir, Freundlich,Temkin and other isotherms among which Langmuirisotherm was formed to be most suitable. Theregression co-efficient (R2) for three of the studiedisotherms is included in the Table 3 of supportingdata. The equation for Langmuir isotherm is

C=h ¼ 1Kads

þ C (5)

where, h is the surface coverage and C is the concen-tration of the inhibitor. A plot of C/h versus C gavestraight line (Figure 3) using which different adsorp-tion parameters were calculated and listed inTable 1. Regression co-efficient and slope for CBTHand CBFH are around 0.99 and 1, respectively whichconfirms the validity of Langmuir isotherm model foradsorption. Free energy of adsorption can be calcu-lated from Kads using the equation,

DGoads ¼ �RT Inð55:5kadsÞ (6)

where, R is gas constant and T is the absolute tem-perature of the experiment and the constant value55.5 is the concentration of water in solution in moldm�3. The negative value obtained for DGo

ads indi-cates that the adsorption process onto the metal sur-face is spontaneous and the adsorbed layer is stable(Scendo & Trela, 2013). The interaction betweenhydrazones with the MS could be described by phys-isorption, chemisorption or by complexation whichare influenced by the nature and charge of themetal, the type of electrolyte and the chemical struc-ture of the inhibitor. (Yang, Li, & Mu, 2006). Thevalue of DGo

ads for studied inhibitors varies between�31.6 kJ mol�1 and �33.9 kJ mol�1, so the kind ofadsorption cannot be completely either physisorp-tion or chemisorption, but it is a complex and com-prehensive adsorption involving both (Singh, Gopal,Prakash, Ebenso, & Singh, 2013). Entropy of adsorp-tion and enthalpy of adsorption process is calculatedusing the following thermodynamic equation.

DGoads ¼ DHo

ads�TDSoads (7)

A graph of DGoads vs T gives straight line with

slope �DSoads and intercept DHoads (Figure 4).

The gain in entropy is attributed to the increase insolvent entropy. That is, hydrazones get adsorbed onto the MS surface as soon as water molecules getdesorbed. It is reported that endothermic adsorption

(a)

(b)

Figure 3. Langmuir isotherm for the adsorption of (a) CBTHand (b) CBFH on MS in 0.5 M HCl at different temperatures.

48 T. K. CHAITRA ET AL.

is chemisorption whereas exothermic adsorption ischemisorption when its magnitude is more than100 kJmol�1 and physisorption if it is less than40 kJmol�1 (Noor & Moubaraki, 2008). The DHo

ads

value obtained for CBFH and CBTH are �17.6 kJmol�1 is �8.68 kJ mol�1, respectively indicating thatboth the inhibitors undergo physisorption.

3.2. Electrochemical impedance spectroscopy

EIS determines different impedance parametersbetween MS-hydrochloric acid interface in theabsence and presence of optimum concentration oftwo hydrazones, and results are presented inTable 2. Detailed information about the EIS experi-ments is presented in Table 4, Figures 3, 4(a–c),5(a–c) of supporting data. Open circuit potential vstime, Nyquist, bode and phase angle plots for CBTHand CBFH are presented in Figure 5(a–d). Nyquistplots exhibit single semicircle whose diameterincreases with increase in the concentration whereasshapes of the curves remains the same after the add-ition of hydrazones. Nyquist plots are explainedbased on equivalent electrical circuit consisting ofpolarization resistance (Rp) parallel with constantphase element (CPE) both in series with solutionresistance (Rs) Figure 5(b). Inhibition efficiency canbe calculated by Rp using the following formula,

IE ð%Þ ¼ ðRpÞp � ðRpÞaðRctÞp

� 100 (8)

where, (Rp)a and (Rp)p are the polarization resistancein the absence and presence of inhibitor,

respectively. Rp increases with increase in the con-centration and reaches maximum value of 1590 and1522X cm2 at optimum concentration of 2mM forCBTH and CBFH, respectively. The increase in Rpcould raise the tendency of the current to passthrough the capacitance in the circuit.

CPE can be expressed as

ZCPE ¼ Yo�1ðixÞ�i (9)

where, Yo is the magnitude of CPE, x is the angularfrequency (in rad s�1), i2¼�1 is the imaginary num-ber. The double layer capacitance (Cdl) can be calcu-lated from CPE parameters using the equation,

Cdl ¼ ðYoRp1�nÞ1=n (10)

Here n¼ a/(p/2) in which a is the phase angle ofCPE. “n” is fractional exponent for solid electrodes/solution interfaces and its value is �1� n� 1(Chaitra et al., 2016). Depending on the value of n,CPE can represent resistance (n¼ 0, Y0¼ R), capaci-tance (n¼ 1, Y0¼ C), inductance (n¼�1, Y0¼ L) orWarburg impedance (n¼ 0.5, Y0¼W) (Murulana,Kabanda, & Ebenso, 2015). In the present study,value of n for blank 0.76 increases in case of bothhydrazones (varying between 0.78 and 0.85) repre-sents deviation from the ideal behaviour (wheren¼1). Generally, high value of n is associated withlow surface roughness and high surface coverage(Verma, Ebenso, Bahadur, Obot, & Quraishi, 2015).Increase in the value of n shows that after theadsorption of inhibitor the surface becomes morehomogeneous, also there is increase in surface cover-age of MS by CBTH and CBFH that leads to increasein capacitive behavior.

Double layer capacitance is calculated from thick-ness of the double layer (d) by the following equa-tion,

Cdl ¼ Aeeod

(11)

where, e is the dielectric constant of the protectivelayer, eo is the permittivity of the free space and A isthe area of the working electrode. In the presentstudy, decrease in Cdl after the addition of hydra-zones is due to increase in thickness of the protect-ive film which results in decrease in local di-electricconstant (Verma, Singh, & Quraishi, 2016). The single

Table 1. Adsorption thermodynamic parameters in the absence and presence of various temperatures.

Inhibitor T (K) R2 SlopeKads

(L mol�1)DGads

(kJ mol�1)DSads

(J mol�1 K�1)DHads

(kJ mol�1)

CBTH 303 0.9965 1.0679 5042 �31.6 76.1 �8.68313 0.9983 1.0975 5195 �32.7323 0.9931 1.0990 4243 �33.2333 0.9912 1.0961 3832 �33.9

CBFH 303 0.9995 1.0855 5449 �31.7 47.2 �17.6313 0.9966 1.0987 4677 �32.4323 0.9974 1.0903 3565 �32.7333 0.9998 1.0987 2978 �33.2

Figure 4. Plot of DGads vs. T for CBTH and CBFH.

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 49

peak obtained in Bode plots for both CBTH andCBTH indicates that the electrochemical impedancemeasurements fit well in the one-time constantequivalent model. In the present study, phase anglevalue close to 70� has been obtained whereas anideal capacitor is characterized by slope value of 90�.

3.3. Potentiodynamic polarization

Tafel curves for MS in the absence and presence ofoptimum concentrations of CBTH and CBFH in 0.5 MHCl at 303 K are presented in Figure 5(e). The corro-sion kinetic parameters such as corrosion potential

Table 2. Impedance and polarization parameters for the corrosion of MS in 0.5 M HCl in absence and presence of optimumconcentration of inhibitors at 303 K.

Inhibitor Rp (X cm2) Yo (m X�1 sn) nCdl

(mFcm�2) IE (%) Ecorr (mV)icorr

(mA cm�2)ba

(mV dec�1)bc IE

(mV dec�1) (%)

Blank 198.9 356.5 0.7631 132.8 – �502 0.2000 220 376 –CBTH 1590 68.7 0.8563 14.0 87.8 �498 0.0177 79.3 106 91.2CBFH 1552 71.2 0.8513 14.2 85.1 �440 0.0265 69.9 162 86.7

Figure 5. (a) Open circuit potential versus time (b) Nyquist (c) Bode (d) Phase angle (e) Tafel plots in the absence and pres-ence of different concentrations of CBTH and CBFH.

50 T. K. CHAITRA ET AL.

(Ecorr), corrosion current density (Icorr), cathodic Tafelslope (bc), anodic Tafel slope (ba) and percentageinhibition were derived from the curves and arerecorded in Table 2. Detailed information on polar-ization measurement is included in the Figure 6 andTable 5 of supporting data.

Inhibition efficiency (IE %) values were derivedfrom corrosion current density (icorr) were calculatedby the Tafel plot,

IE ð%Þ ¼ iocorr � icorriocorr

� 100 (12)

where, iocorr and icorr are the uninhibited and inhibitedcorrosion current densities, respectively. It is evidentfrom Figure 5(e) that both hydrogen evolution andmetal dissolution reactions were retarded after theaddition of hydrazones. Corrosion current densityvalues decreased on increasing concentration ofinhibitors indicating that inhibitors bring down elec-trochemical reaction. The corrosion current densitywhich was 0.2mAcm�2 for blank decreased to0.0177 mAcm�2 and 0.02654 mAcm�2

, respectivelyfor CBTH and CBFH at optimum concentration. Thevariations in the values of both ba and bc signify theinfluence of CBTH and CBFH on both anodic andcathodic reactions and its irregularities indicate thatsome mechanisms are involved. The corrosion poten-tials for inhibitors are slightly anodically displacedwith respect to the bare MS in solution without aninhibitor. This shows the predominant hindrance ofMS dissolution compared to hydrogen evolution.According to the literature, when Ecorr is displacedmore than ±85mV relating to potential of the blank,the inhibitor can be considered as either cathodic oranodic type (Chaubey, Savita, Singh, & Quraishi,2017). In the present investigation, the shift in Ecorr is47mV and 62mV towards anodic for CBTH andCBFH, respectively. Therefore, the studied inhibitorsare neither anodic nor cathodic, but are ofmixed type.

3.4. Quantum chemical calculations

Quantum chemical methods are useful in elucidatingthe electronic structure and reactivity which could

be helpful in designing high-efficiency inhibitors byquantitative structure–activity relationship (QSAR)method (Zhang & Musgrave, 2007). The structures ofCBTH and CBFH were first optimized and the simu-lated to calculate the electronic parameters. Variousquantum chemical parameters such as EHOMO Energyof highest occupied molecular orbital, ELUMO Energyof lowest unoccupied molecular orbital, Energy gap(DE), dipole moment (m), ionization potential (I), elec-tronegativity (v), hardness (g) and softness (r) werecalculated for inhibitors and their protonated formsand these are listed in Table 3. Following equationsare used for the calculation of quantum chemicalparameters:

I ¼ �EHOMO (13)

A ¼ �ELUMO (14)

x ¼ Iþ A2

(15)

g ¼ I� A2

(16)

r ¼ 1g

(17)

Difference in the energy between inhibitor mole-cules and their protonated forms (energy of proton-ation) is calculated and found to be �19,520 kJ mol�1

for CBTH and �14,048 kJmol�1 for CBFH. From elec-trostatic potential map of CBTH and CBFH, negativecharge is accumulated near oxygen atom (indicatedby red color lines) so protonation on oxygen atom ismost favorable. After protonation, nitrogen atombecomes negative therefore the adsorption of hydra-zones occurs mainly through interaction betweennitrogen atom and MS surface. From Table 4, HOMOand LUMO orbitals of CBTH is mainly distributedaround nitrogen atoms, CO–NH–NH2 group and thia-zole ring, whereas in CBFH they are distributedmainly around CO–NH–NH2, so these act as activeadsorption centers. A good correlation was foundbetween ELUMO and dipole moment with experimen-tally determined inhibition efficiency. Higher HOMOenergy results in more reactivity towards electro-philes and lower LUMO energy results in higherreactivity towards nucleophiles (Rauk, 2001). As theenergy required to remove the last occupied orbital

Table 3. List of quantum chemical parameters for CBTH and CBFH.Quantum chemical parameters CBTH CBFH CBTH (Protonated) CBFH (Protonated)

Total Energy (kJ mol�1) �3265108 �2421885 �3284628 �2435933Dipole (Debyes) 6.2875 5.6444 6.8650 6.5201EHOMO (eV) �9.0228 �8.8556 �9.3422 �7.4542ELUMO (eV) 1.6668 1.8809 2.9564 3.3280DE¼ ELUMO� EHOMO (eV) 10.689 10.736 12.2986 10.7822DE#¼ ELUMO of acid� EHOMO of 8.8228 8.6556 9.1422 7.2542base (eV)Ionisation potential, I¼�EHOMO 9.0228 8.8556 9.3422 7.4542Electron affinity, E¼�ELUMO �1.6668 �1.8809 �2.9564 �3.3280Electronegativity (v)¼ (Iþ E)/2 3.678 3.4873 3.6774 2.0631Hardness of the molecule (g)¼ (I� E)/2 5.3348 5.3682 6.1493 5.3911Softness (r) 0.1874 0.1862 0.1626 0.1854

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 51

will be low, lower values of energy gap will havegood inhibition efficiency (Obot, Obi-Egbedi, &Umoren, 2009). The present study showed lowerELUMO for CBTH, thereby implying better electronicinteraction with the metal surface. Hence it has bet-ter inhibitor property compared to CBFH.

Dipole moment is the measure of polarity in abond and is related to the distribution of electronsin a molecule (Gece, 2008). As the energy of deform-ability increases with increase in dipole moment,high value of dipole moment increases the adsorp-tion between chemical compound and metal surface.There is increase in dipole moment after protonationfor both CBTH and CBFH indicating ease of inter-action with the metal surface. Higher value of dipolemoment for CBTH as compared to that of CBFHhelps the former to strongly adhere on MS surface.Chemical hardness is defined as the resistanceoffered towards the deformation of the electron

cloud of the atoms, ions or molecules under smallperturbation of chemical reaction (Zarrok et al.,2012). For the simple transfer of electron, adsorptioncould occur on the part of the molecule where soft-ness (r), which is a local property, is maximum(Hasanov, Sadikoglu, & Bilgic, 2007). Since iron is asoft acid it is more likely to react with soft bases.The quantum chemical study results show goodcorrelation with experimental values.

3.5. Scanning electron microscopy

To confirm the presence of protective inhibitory film,SEM images were taken with and without CBTH andCBFH at optimum concentration (Figure 6(a–d)) at303 K. In the absence of inhibitors, MS surface exhib-ited increased pitting. But, in the presence of boththe inhibitors the surface showed great improvementwithout the formation of any pits.

Table 4. List of quantum chemical structures for CBTH and CBFH.Quantum chemical structure

CBTH CBTH (protonated) CBFH CBFH (Protonated)

Optimized geometry

Total Charge Density

HOMO

LUMO

Electrostatic potential

map

52 T. K. CHAITRA ET AL.

3.6. Energy dispersive X-ray analyses

Energy dispersive X-ray analysis was done to getinformation about the composition of MS in 0.5 MHCl in the absence and presence of inhibitors. EDXspectra of CBTH and CBFH are given in Figure 7 ofsupporting data. Table 5 presents weights of ele-ments in uninhibited and inhibited MS samples. Theweight percent of Fe considerably decreased by theaddition of inhibitors. The decrease in percentweight of Fe from 56.95 to 22.95 and 8.5, respect-ively after the addition of CBTH and CBFH. The peaksof other elements such as nitrogen, oxygen, carbonand sulphur were also appearing in inhibited EDXspectra. This clearly proves the adsorption of inhibi-tors on to the Fe surface.

3.7. Mechanism of inhibition

Inhibition effect of hydrazones on mild steel can beexplained on the basis of the mode of interactionwith the metal surface. Studied hydrazones contain

nitrogen atoms as well as the carbonyl both of whichcan undergo protonation. From electrostatic poten-tial map (Table 4), carbonyl group has more negativecharge compared to nitrogen atom so carbonylgroup predominantly undergoes protonation. Theprotonated form of hydrazones physically adsorbs onthe positively charged mild steel surface throughchloride irons (Cl�). On the other hand, directadsorption takes place by donor–acceptor interac-tions between lone pair of electrons on nitrogen,oxygen and sulphur atoms and pi-electrons of ben-zene and heterocyclic rings and vacant d-orbitals ofFe atom. Among CBTH and CBFH, sulphur bearingCBTH shows better inhibition efficiency compared tooxygen bearing CBFH because oxygen is more elec-tronegative compared to sulphur.

4. Conclusion

Synthesized hydrazone derivatives showed goodinhibition property against MS corrosion in 0.5 M HClmedium. Hydrazones proved efficient at lower

Figure 6. SEM images of MS surface (a) polished (b) immersed in 0.5 M HCl (c) immersed in 0.5 M HCl in the presence ofCBTH (d) immersed in 0.5 M HCl in the presence of CBFH.

Table 5. Percentage weights of elements obtained from EDX spectra.Weight percentage of elements detected

Mild steel surface under investigation Fe C O N S Al Cl

Immersed in 0.5 N HCl 57.0 – 41.6 – – 1.46 –Immersed in 2mM of CBTH 22.95 7.48 50.87 16.35 0.06 – 2.28Immersed in 2mM of CBFH 8.50 23.51 44.56 22.02 0.10 – 1.41

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 53

temperatures than higher temperature. The adsorp-tion behavior of both hydrazones followed Langmuirisotherm. Polarization study revealed that the hydra-zones acted as cathodic and anodic inhibitors.Nyquist plots, Bode plots and morphological studies(SEM and EDX) confirmed the presence of protectiveinhibitory film. Quantum chemical studies provedthat electronic interaction of CBTH is better than thatof CBFH.

Disclosure statement

No potential conflict of interest was reported bythe authors.

Funding

One of the authors (T. K. C) received meritorious fellowship(RFSMS) from University Grants Commission, New Delhi(Order No. MGC/003/2015-16 dated 04/08/2015) and this isgratefully acknowledged.

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ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 55

ORIGINAL ARTICLE

Optimization of methyl esters production from non-edible oils usingactivated carbon supported potassium hydroxide as a solid base catalyst

Abdelrahman B. Fadhil, Akram M. Aziz and Marwa H. Altamer

Industrial Chemistry Research Laboratory, Department of Chemistry, College of Science, Mosul University, Mosul, Iraq

ABSTRACTThe present investigation reports transesterification of non-edible oils, waste cooking oil(WCO) and waste fish oil (WFO) with methanol using activated carbon supported potassiumhydroxide (KOH) as a solid base catalyst. Activated carbon was prepared from polyethyleneterephthalate waste and loaded with KOH by wet impregnation method to prepare KOH/ACsolid base catalyst. X-ray diffraction and Scanning Electron Microscopy (SEM) were utilized tocharacterize the resulting solid base catalyst. Retention method was utilized to measure thespecific surface area of the activated carbon and its derived solid base catalyst, whileHammett indicator method was followed to determine the basic strength of the preparedsolid base catalyst. The prepared catalyst has granular and porous structures with a high bas-icity and a superior catalytic performance for the transesterification reaction.Transesterification reaction variables were arranged to obtain the best biodiesel yield. Thehighest methyl ester yield from WCO (88.12% w/w with an ester content of 96.68% w/w)was obtained by employing 3.5wt.% KOH/AC catalyst, 9:1 methanol to oil molar ratio, 65 �Creaction temperature, the reaction time of 180min and the stirring rate of 600 rpm, whilemaximum methyl ester yield from WFO (92.66% w/w with 96.98% w/w ester content) wasproduced with 3.0wt.% KOH/AC catalyst, 9:1 methanol to oil molar ratio at 65 �C for150minutes of the reaction and 600 rpm. The prepared solid base catalyst was recoverableand thermally stable giving a yield of 70wt.% after the 5th cycle. The fuel properties of theraw oils were significantly enhanced after transesterification with methanol in the presenceof KOH/AC, and were in conformity with the ASTM D 6751 limits as well. Therefore, the pre-pared KOH/AC composite may be considered as a promising solid base catalyst for transes-terification of non-edible oils with methanol.

ARTICLE HISTORYReceived 20 December 2016Accepted 13 August 2017

KEYWORDSBiodiesel; non-edible oils;heterogeneous solid basecatalyst; KOH/AC solid basecatalyst; analysis of catalyst;evaluation of fuel properties

1. Introduction

Biodiesel (BD) is a renewable, biodegradable and sus-tainable liquid bio-fuel. It can typically be producedfrom triglycerides (vegetable oils or animal fats)through base-catalyzed transesterification reactionwith methanol or ethanol (Takase et al., 2014). BD isan effective alternative to petro diesel fuel. It can beblended with diesel fuel or used in a pure form. Theuse of BD as a fuel reduces pollutant levels and prob-able carcinogens produced upon combustion of fossilbased fuels (Encinar, Gonz�alez, Pardal, & Mart�ınez,2010; Li et al., 2013; Takase et al., 2014). However, thehigh price of BD could be ascribed to the highlyexpensive raw materials used in its production. BDproduction through homogeneous catalysis alsoincreases its production cost due to soap formation,non-reusability of the catalyst, and handling and sep-aration problems. Consequently, heterogeneouscatalysis was applied at a commercial level becauseof its advantages, such as ease of separation fromthe reaction mixture, selectivity, and appreciable

catalytic activity in reuse. Heterogeneous catalysisbrings up some problems, such as the mass transferdiffusional resistance which decreases the reactionrate to a large extent in comparison to homogeneouscatalysis. To overcome this problem, catalyst supportwhich can provide higher surface area and high num-ber of the active sites was utilized in transesterifica-tion reaction of oils or fats to yield theircorresponding fatty acid alkyl ester (Dhawane et al.,2016; Kaur & Ali, 2014; Zabeti, Daud, & Aroua, 2009).

Metal oxides and modified metal oxides wereused as catalysts in various applications, such asphotocatalytic degradation reactions of industrialdyes (Puna et al., 2013; Puna et al., 2014; Saravanan,Gracia, et al., 2015; Saravanan, Gupta, et al., 2015;Saravanan, Khan, et al., 2015; Saravanan et al., 2016)as well as preparation of nanoparticles modified elec-trode for the determination of drugs (Devaraj et al.,2016). They were also used as solid base catalysts forsynthesis of biodiesel.

CONTACT Abdelrahman B. Fadhil abdelrahmanbasil@yahoo.com Industrial Chemistry Research Laboratory, Department of Chemistry, Collegeof Science, Mosul University, Mosul, Iraq� 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

University of BahrainARAB JOURNAL OF BASIC AND APPLIED SCIENCES2018, VOL. 25, NO. 2, 56–65https://doi.org/10.1080/25765299.2018.1449414

Alumina, alumina/silica, CaO, MgO and NaY weretested as supports for different alkali salts, like KOH,NaOH, KF and K2CO3. However, using activated car-bon as a catalyst support is highly effective in bothliquid and vapor phase reactions. Activated carbonhas higher micro-porous surface and larger activesites than the other adsorbents making it a suitablecatalyst support for the impregnation of base cata-lysts. Different catalysts, such as potassium carbonate,ferrous sulfate, potassium hydroxide, calcium oxide,and potassium fluoride were easily dispersed on thesupport surface to enhance the transesterificationreaction (Baroutian, Aroua, Abdul Raman, & Sulaiman,2010; Dias et al., 2013; Hindryawati, Maniam, Md, &Chong, 2014; Li et al., 2013; Noiroj et al., 2009; Wan &Hameed, 2010). The use of industrial wastes for acti-vated carbon synthesis reduces the cost of the cata-lyst support, which in turns reduces the productioncost of BD. Among industrial wastes, preparation ofthe activated carbon from polyethylene terephthalatehas been widely recommended (Almaz�an et al., 2010;Brateka et al., 2013; Lian, Xing, & Zhu, 2011). High sur-face area activated carbons were prepared from poly-ethylene terephthalate waste, which is a suitablematerial for supporting catalysts, like potassiumhydroxide. Waste cooking oils, non-edible oils andanimal fats were utilized as potential feedstocks forreducing the production cost of BD (Ali & Fadhil,2013; Encinar et al., 2010; Fadhil & Ahmed, 2016;Fadhil, Aziz, & Al-Tamer, 2016a, 2016b; Hameed, Goh,& Chin, 2009). Nevertheless, supporting of potassiumhydroxide on the activated carbon prepared frompolyethylene terephthalate waste to prepare KOH/ACsolid base catalyst has not yet reported in the litera-ture. Furthermore, transesterification reaction of non-edible oils, such as waste cooking oil and waste fishoil with methanol using potassium hydroxide on theactivated carbon prepared from polyethylene tereph-thalate waste has not published yet in literatureas well.

Solid base catalyst produced by loading potassiumhydroxide on the activated carbon prepared frompolyethylene terephthalate waste polyethylene ter-ephthalate waste tested for transesterification reac-tion of non-edible oils, such as waste cooking oil andwaste fish oil with methanol. The prepared solidbase catalyst was characterized by different techni-ques to assess its suitability as a solid base catalystfor transesterification reaction. Percentage of KOHloaded onto the activated carbon, amount of solidbase catalyst (%w/w of oil), methanol to oil molarratio, reaction temperature and period were theexperimental conditions optimized during the pre-sent investigation. Recycling and evaluation of thesolid catalyst was also studied. Properties of the pro-duced biodiesels referring to ASTM standard testmethods were determined as well.

2. Materials and methods

2.1. Materials

Waste cooking oil (WCO) and fish waste were obtainedfrom restaurants and fish slaughterhouse, respectivelyrun in the city of Mosul, Nineveh Governorate, northof Iraq in August 2013. Polyethylene terephthalatewaste (consumed transparent plastic water bottles)were taken from the university restaurant and used inthe preparation of the activated carbon (AC). Reagentgrade methanol (99%) and potassium hydroxide werepurchased from BDH (UK). Other chemicals wereobtained from Merck (Kenilworth, NJ). All chemicalswere of analytical reagent grade and used as receivedwithout any further purification.

2.2. Preparation of feedstocks

Waste cooking oil was mixed with anhydrous sodiumsulfate and left overnight for moisture removal. Theoil was then filtered to remove particles of sodiumsulfate and kept in a dark container for further use.The waste fish oil (WFO) was extracted from fishwaste as explained elsewhere (Fadhil & Ahmed,2016). The average molecular weight of the oils wascalculated based on their fatty acid compositions.The acid values of the raw oils were determined bythe titration method following ASTM D664 testmethod, whereas their iodine values were measuredas per Hanus method.

2.3. Preparation of the activated carbon andKOH/AC solid catalyst

The AC was prepared from the polyethylene tereph-thalate waste (soft drink bottles) following the pro-cedure given previously by Fadhil and Ahmed (2016).Briefly, PET waste was cut, carbonized at 500 �C andsteam activated at 750 �C (Fadhil & Ahmed, 2016).The KOH/AC solid base catalyst was prepared by wetimpregnation method. The AC was immersed in KOHsolutions of different concentrations (25–100%). Theimpregnated samples were stirred for 6 h at 600 rpm.After impregnation, the mixture was filtered using afilter paper (Whatman No.1) and oven-dried at 105 �Cuntil a constant weight was obtained. The dried cata-lysts were calcined at 450 �C for 2 h. The amount ofKOH loaded on the AC was determined gravimetri-cally (Fadhil, Al-Tikrity, & Albadree, 2015; Wan &Hameed, 2010).

2.4. Characterization of the catalyst

Scanning Electron Microscopy was utilized using aFEI Quanta 200 FEI Co. Ltd. at the accelerating volt-age of 20-kV to investigate the morphologies of theAC and the solid base catalyst derived from it. X-ray

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 57

diffractograms obtained using a reflection scan withnickel-filtered CuKa radiation at 40 kV and 70mA(PW 3040/60 X’PERT PRO ANALYTICAL 2009, TheNetherland) were used to analyze the crystal struc-ture of the newly prepared solid catalyst. The meas-urements were performed at (2h) between 20� and80�. The specific surface areas of the activated car-bon and the catalyst were determined by the reten-tion method (Fadhil & Saeed, 2016). The basicproperties of the catalysts were evaluated by aminetitration using Hammett indicators (Tabak et al.,2007; Takase et al., 2014).

2.5. Transesterification of non-edible oil usingKOH/AC solid base catalyst

Transesterification experiments of the non-edible oilswith methanol using the prepared solid base catalystwere performed in a (250mL) three-neck round-bot-tom flask with a condenser and a thermometer. Theround was filled with (50g) of the oil. The catalystwas mixed with the appropriated amount of metha-nol and then added to the oil. The mixture wasrefluxed with stirring at 600 rpm for a given reactiontime. As the reaction was completed, the catalystwas separated from the reaction mixture throughcentrifugation. The filtrate was transferred to a sepa-rating funnel and allowed to settle, to ensure theseparation of the methyl esters and glycerol com-pletely. After removal of the glycerol, excess metha-nol was recovered from the methyl ester layer bydistillation under vacuum using a rotary evaporator.The methyl content on the purified BD, was deter-mined by column chromatography following methodproposed by (Bindhu et al., 2012). A glass column(18� 46 cm) packed with silica gel (60–120 mesh)was loaded with (1.0 g of BD) dissolved in hexane.The methyl ester fraction was eluted with (300mL) ofa mixture of (hexane: diethyl ether 99.5:0.5% v/v).After evaporation of the solvent system, amount ofmethyl esters separated was calculated and the estercontent was determined on a weight bases. The BDyield was calculated by the following equation:

BD yield ð%Þ ¼ Weight of the product� Ester content ð%ÞTotal weight of oil used ðgÞ � 100

2.6. Analysis and characterization of biodiesel

The fuel properties of the methyl esters including theflash point (D93), cloud point (ASTM D 2500), pourpoint (ASTM D 97), kinematic viscosity at 40 �C (ASTMD445), density at 15.6 �C (ASTM D 5002), acid value(ASTM D664), refractive index at 20 �C (D1747 – 09)and carbon residue (ASTM D 4530) were determined.AOCS Cc 17-95 was followed to determine the soapcontent of the prepared biodiesel, whereas (Pisarello,

Dall-Csta, Veizaaga, & Querini, 2010) was used fordetermining the total and free glycerin in the pro-duced methyl esters. Each property was measured intriplicate and the result was presented as the mean-±standard deviation (SD).

2.7. Regeneration and reusability of the catalyst

Regeneration of the resulting solid base catalyst wasperformed in a Soxhlet apparatus using a mixture ofhexane and acetone so as to remove BD, glyceroland the oil molecules attached to the catalyst surface.After the regeneration, the regenerated catalyst wastested for transesterification of the non-edible oilswith methanol using the optimal conditions obtainedduring the study (Fadhil et al., 2016a, 2016b).

3. Results and discussion

3.1. Feedstocks properties

As we stated earlier, different non-edible oils were uti-lized for BD production using the solid base catalyst.Considering WCO, in Iraq, sunflower oil is the mainvegetable oil used for cooking and frying. The acid val-ues of WCO and WFO were found to be 2.0 and1.23mg KOH/g, respectively. In consequence, one-steptransesterification reaction of both oils with methanolusing the prepared solid base catalyst is possible.

3.2. Selection of the optimal catalyst andits analysis

The amount of KOH loaded onto the AC increasedwith increasing KOH concentration in the impregnationsolution as shown in Table 1. In addition, the surfaceof the AC decreased with increasing the amount ofKOH loaded on the AC. However, the solution whichcontains 75wt.% of KOH exhibited the highest loadingof KOH on the AC. Furthermore, this sample showedthe lowest surface area among all the samples, due tothe blocking of the AC pores with KOH molecules(Fadhil et al., 2016b). It is also clearly seen from Table1 that the increase in the amount of KOH in theimpregnation solution led to an increase in the basicstrength of the catalyst. The highest basicity(9.3<H_< 15) of KOH/AC catalyst was obtained withthe sample which contains 75wt.% KOH in the impreg-nation solution (a loading of 28.75% w/w). Beyond thisconcentration, no significant increase in the basicstrength of the catalyst was observed.Transesterification of WCO and WFO was carried outusing 3.0% w/w of the catalyst, 6:1 methanol to oilmolar ratio, 60� C reaction temperature, 120minutes ofreaction and 600 rpm rate of stirring under optimumconditions selected for KOH/AC catalyst as shown inTable 1. The methyl ester yield increased in parallel

58 A. B. FADHIL ET AL.

with the amount of KOH loaded on the AC. The cata-lyst loaded with the highest amount of KOH (28.75%w/w) resulted in the highest conversion for bothoils. Consequently, the catalyst loaded with 28.75%w/w KOH was chosen as the best catalyst, and thusit was utilized in the next transesterificationexperiments.

In order to characterize the phase and the struc-ture of this KOH/AC solid base catalyst, the XRD ana-lysis of the catalyst was conducted as shown inFigure 1. The X-ray diffractograms of the AC showedbroad diffraction peaks at 2h¼ 24.24� and 43.43�

which are ascribed to the typical amorphous carbonwith polycyclic aromatic sheets. They also indicatethe destruction of the PET and generation of the ran-domly arranged amorphous carbon structures duringcarbonization and activation. After the KOH loading,many peaks appeared at different positions(2h¼ 23.97�, 30.26�, 34.33�, 35.31�, 38.15�, 39.84�,41.49�, 44.36�, 48.31�, 53.64� and 54.08�). The peaksobserved at 2h¼ 30.26�, 41.49� and 53.64� indicatethe presence of K2O phase (Li et al., 2013).

Figure 2 shows the SEM images of the parent ACand the solid base catalyst derived from it. The SEMimage indicates many micropores present on the sur-face of AC which are produced as a result of thesteam activation. The micropores on the AC surfaceprovide high surface area of the adsorbent. It is alsoclear from the SEM image of the AC that the steamactivation was able to build the pores on the carbonsurface which is necessary for loading the catalystmolecules. The SEM image of the derived catalystshows that the micropores on the carbon surfacewere blocked by the KOH molecules, which confirmsthat KOH molecules were significantly adsorbed ontothe AC surface, suggesting that the obtained solidenables the heterogeneous-catalyzed transesterifica-tion process to proceed for BD synthesis.

3.3. Optimization of transesterification parameters

The low acid values of the tested non-edible (WCOand WFO) enabled direct development of BD

through KOH/AC-catalyzed transesterification reaction.Therefore, the process parameters, such as theamount of the solid base catalyst, methanol tooil molar ratio, the temperature and the durationof the transesterification process were optimized.Transesterification of WCO and WFO using KOH/ACcatalyst was investigated by testing different amountsof the catalyst ranged from 1.0 to 5.0% wt.% whereincrements by 0.50% were applied as depicted inFigure 3(a). The molar ratio of methanol to oil, tem-perature, time and the stirring rate of the reactionwere arranged as 6:1, 60 �C, 120minutes and600 rpm, respectively. It was observed that the methylester yield increased progressively with increasing theamount of the catalyst due to the higher the amountof the catalyst provides more number of the activecenters, which in turn resulted in higher conversion.However, transesterification of WCO by the said solidcatalyst required (3.5% w/w) of the catalyst to pro-duce maximum yield of methyl ester comparing toWFO which exhibited the highest yield of methylester at (3.0% w/w) of the catalyst. This difference inthe optimum amount of the catalyst necessary forachieving maximum conversion of various feedstockscould be ascribed to the acid values of the raw oilsas well as the type of the oil. The WFO is a virgin oiland was not subjected to any thermal treatment incomparison to WCO which was subjected to the fry-ing process which affects the chemical compositionof the oil significantly. However, the amount of thesolid catalyst more than the optimum amount causedhigher mass transfer resistance due to the highly vis-cous mixture, resulting in lower methyl ester yield(Agarwal, Chauhan, Chaurasia, & Singh, 2012; Takaseet al., 2014). Findings obtained in the present studycorroborate with those reported by other researchers(Agarwal et al., 2012; Fadhil et al., 2016b; Takaseet al., 2014). Based on these data, the subsequentexperiments were conducted using (3.5% w/w) ofKOH/AC catalyst for WCO and (3.0% w/w) for WFO.

Different methanol to oil molar ratios (3:1, 6:1,9:1,12:1, 15:1 and 18:1) were tested on transesterification

Table 1. Properties of KOH/AC solid base catalyst.BD yield (wt.%)�

Surface area (m2/g) Loading (wt.%) Basic strengthKOH/AC

KOH (wt.%) KOH/AC KOH/AC KOH/AC WCO WFO

25 542 14.3 7.2 <H_<9.8 38.55 ± 1.22 42.33 ± 1.2250 438 21.52 7.2 <H_<9.8 48.55 ± 1.0 53.56 ± 1.575 431 28.75 9.8 <H_<15 63.23 ± 1.11 73.12 ± 1.11100 431 28.75 9.8 <H_<15 61.14 ± 1.24 70.12 ± 1.24

Properties of the authentic activated carbon in comparison to its derived catalyst.

Property Authentic AC Catalyst (28.75 wt.% KOH/AC)

Surface area (m2/g) 1105 431.0Density (g/mL) 0.4584 0.5566pH 6.80 12.50

�3.0 wt.% KOH/ACþ 6:1 (methanol:oil molar ratio) 60 �C reaction temperatureþ120minutes reaction time.

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 59

of WCO and WFO by the KOH/AC solid catalystFigure 3(b). The other parameters were chosen as(3.5wt. % KOH/AC) for WCO and (3.0wt. % KOH/AC)for WFO, 60 �C reaction temperature, 120minutes ofreaction period and 600 rpm rate of stirring. The con-version yield was found to be very low at the lowermethanol to oil molar ratio (3:1) due to incompletionof the reaction as shown in Figure 3(b) (Agarwalet al., 2012; Takase et al., 2014). Since the transesteri-fication is a reversible reaction, and the higher

methanol to oil molar ratio pushes the reaction for-ward the products, the methyl ester yield increasesprogressively. In addition, methanol facilitates thesolubility of phases, thus favouring a smooth reac-tion with limited mass transfer resistance (Girish,Niju, Begum, & Anantharaman, 2013). It was noticedthat both oils have given the highest methyl estersyield at 9:1 methanol to oil molar ratio, while molarratios higher than the optimal decreased the methylesters yield as shown in Figure 3(b), due to the

Figure 1. XRD of the AC and KOH/AC solid base catalyst.

60 A. B. FADHIL ET AL.

difficulty of glycerol separation as a result of thedilution of the oil with the excess methanol, andthus reduces the methyl ester yield (Dias et al., 2013;Zabeti et al., 2009). Accordingly, the value of 9:1 wasestablished as the optimum methanol to oil molarratio for transesterification of both oils in the pres-ence of the KOH/AC catalyst.

Transesterification of non-edible oils on KOH/ACwas carried out at different temperatures (30, 40, 50,60, 65 and 70 �C) as shown in Figure 3(c). The cata-lyst amount, the methanol to oil molar ratio, thereaction period and the stirring rate were selected as3.5wt. % KOH/AC for WCO and 3.0wt. % KOH/AC forWFO, 9:1, 120minutes and 600 rpm, respectively.As shown in Figure 3(c), the lower temperatures(30–50 �C) were accompanied by lower conversion,which could be ascribed to the higher viscosity ofthe oil at lower temperatures, causing lower masstransfer resistance among oil-methanol-catalystphases (Dias et al., 2013; Fadhil et al., 2016b;Hindryawati et al., 2014; Takase et al., 2014; Zabetiet al., 2009). The conversion yield increased withincreasing the reaction temperature and reached themaximum value at 65 �C. On the other hand, thereaction temperatures higher than 65 �C reduced the

methyl ester yield due the evaporation and formingbubbles of methanol, which might inhibit the inter-face interaction (Al-Jammal, Al-Hamamre, & Alnaief,2016; Encinar et al., 2010; Takase et al., 2014). Thus,value of 65 �C was chosen as the optimal reactiontemperature based on this observation.

Because of the importance of the reaction periodon the transesterification reaction from the econom-ical point of view, transesterification of WCO andWFO was tested at different time intervals (30–210minutes) by 30minutes increments Figure 3(d).The catalyst amount, the methanol to oil molar ratio,the reaction temperature and stirring rate wereselected as 3.5wt. % KOH/AC for WCO and 3.0wt. %KOH/AC for WFO, 9:1, 65 �C and 600 rpm, respect-ively. It was observed that higher reaction time wasaccompanied by a higher methyl ester yield.However, WCO gave maximum methyl ester yieldafter 180minutes of the reaction, whereas WFOexhibited the highest yield of methyl ester in150minutes time. The difference seen in the opti-mum reaction time necessary for achieving the max-imum conversion of various feedstocks could beexplained by the different acid values of the raw oils.Furthermore, WFO is a virgin oil and was not sub-jected to any thermal treatment in opposition toWCO which was subjected to the frying process at ahigher temperature leading to significant changes inthe chemical composition. It was observed that con-duction of the reaction at durations longer than theoptimum period reduced the methyl ester whichmay be due to the hydrolysis of some of the estersto their corresponding free fatty acids (Fadhil et al.,2016a, 2016b; Hindryawati et al., 2014).

3.4. Reusability of the solid catalyst

The most important characteristics of the solid basecatalysts are their recoverability and reusability whichin turn affect the economics of using heterogeneouscatalysis for biodiesel production (Encinar et al.,2010; Fadhil et al., 2015). Based on this fact, the uti-lized catalyst was regenerated and examined fortransesterification of WCO and WFO with methanolfor further cycles (five cycles) by applying the opti-mum conditions obtained during the present studyas shown in Table 2.

The results pointed out that the methyl ester yieldwas reduced by increasing the cycle number whenthe regenerated catalyst used. However, good con-version (>70%) was obtained up to the 5th cycle.The reduction in the methyl ester yield with increas-ing the regeneration runs could be attributed to theleaching of active metals from the support surface.As a result, the number of the active sites availablefor the reaction will be less in number. Theseachievements agree with the previously reported

Figure 2. SEM images obtained for the AC and KOH/ACsolid base catalyst.

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 61

data by other researchers (Al-Jammal et al., 2016;Girish et al., 2013; Takase et al., 2014 ). Moreover,deactivation of the catalyst could also be due to theadsorption of oil, methyl ester, glycerol and free fattyacids on the surface of catalyst (Girish et al., 2013).Table 2 also presents the influence of the catalystreusability on the acid value of the methyl ester pro-duced using the regenerated solid base catalyst. Itwas also noticed that the acid value of the producedmethyl ester increases with the increase of the num-ber of recycles, which could also be ascribed to thedecrease of the number of active sites available forthe reaction (Fadhil et al., 2016b).

3.5. Comparison of the prepared catalyst withother solid catalysts

Table 3 compares the results obtained for the trans-esterification of WCO and WFO with methanol usingKOH/AC solid base catalyst with other solid base cat-alysts prepared by the impregnation of variouspotassium salts onto different supports, such as theAC, zirconia, CaO, alumina, etc. The comparisonshows that the amount of the potassium salt (KOH)

loaded on the AC is almost higher than thatobserved for other solid catalysts. This could beattributed to the higher surface area of the AC thanthe other supports. Although the optimum reactiontemperature and time obtained using the newlydeveloped solid catalyst were comparable to manysolid catalysts reported in literature; the solid basecatalyst developed in the present study gave the

Table 2. Effect of reusability of the catalysts on the yieldand acid vale of biodiesels.

Effect of the catalyst reusability on methyl ester yield

Yield (wt.%)

Cycle WCO WFO

1 88.12 92.662 85.13 88.043 80.44 82.224 76.2 76.35 70.21 72.22

Effect of the catalyst reusability on the acid value

Acid value (mg KOH/g)

Cycle WCO WFO

1 0.15 0.112 0.24 0.153 0.29 0.214 0.36 0.265 0.41 0.31

40

45

50

55

60

65

70

75

1 2 3 4 5 6

ME

yie

ld (

% w

/w)

Catalyst concentration (% w/w)

WCO

WFO

(a) (b)

(d)(c)

40

45

50

55

60

65

70

75

80

85

3 8 13 18 23

ME

yie

ld (

% w

/w)

Methanol/oil molar ratio

WCO

WFO

50

55

60

65

70

75

80

85

90

30 40 50 60 70 80

ME

yie

ld (

% w

/w)

Reaction temperature (°C)

WCO

WFO

60

65

70

75

80

85

90

95

30 80 130 180 230 280

ME

yie

ld (

% w

/w)

Time (minutes)

WCO

WFO

Figure 3. Influence of (a) catalyst amount, (b) methanol to oil molar ratio, (c) reaction temperature and (d) reaction time onthe methyl ester yield.

62 A. B. FADHIL ET AL.

maximum yield of methyl ester with a lower amountof the catalyst and a lower methanol to oil molarratio than the other solid catalysts. Hence, it can beconsidered as an attractive catalyst for making BDfuels. However, the differences in the optimal condi-tions required for maximum yield of BD from variousoil feedstocks by using different solid base catalystscould be attributed to a number of factors, such asthe acid values of raw oils utilized in the productionof BD, the reactivity of the utilized solid base cata-lyst, the type of the ion loaded onto the support andthe utilized support (carbon based catalyst, claybased catalyst, etc.) and mode of heating (conven-tional, microwave, or sonication) employed duringthe transesterification process.

3.6. Properties of biodiesels produced usingKOH/AC solid base catalyst

The fuel properties of biodiesels produced fromWCO and WFO by transesterification with methanolusing KOH/AC solid base catalyst were measured inaccordance with ASTM standard methods and theresults were compared with those of standard ASTMD6751 BD (Table 4).

The density, kinematic viscosity, flash point andrefractive index values of the raw oils were signifi-cantly reduced after transesterification with methanolin the presence of the solid base catalyst, whichreflects the reactivity of the prepared solid base cata-lyst and proves that the values of the assessed prop-erties satisfy the limits prescribed by ASTM standard

Table 3. Comparison of biodiesels produced using the prepared solid catalyst in comparison with other heteroge-neous catalysts.

Oil Catalyst

Catalystloading(wt.%)

Catalystamount(wt.%)

CH3OH/oilmolar ratio

Temperature(�C)

Time(minutes)

BD yield(wt.%) Reference

WCO KOH/AC 28.75 3.5 9:1 65 180 88.12 This workWFO KOH/AC 28.75 3.0 9:1 65 150 92.66 This workWCO KOH/zarconia 32.0 6.0 15:1 60 120 90.80 (Takase et al., 2014)WCO KBr/CaO 25.0 4.0 12:1 65 180 78.90 (Mahesh,

Ramanathan,Begum, &Narayanan, 2015)

WCO Modified coalfly ash

– – 10.7:1 (Xiang, Wang, &Jiao, 2016)

WCO KOH/Alumina 15 5.0 9:1 70 120 96.80 (Noiroj et al., 2009)WPCO KF/AC – 3.0 8.8:1 175 60 83 (Hindryawati

et al., 2014)WCO Sea sand – 7.5 12:1 60 360 95.40 (Mucino et al., 2014)WCO Li, Na, K/Rice

husk silica– 3.0 9:1 65 180 96.50 (Li et al., 2011)

WCO CaO from whitebivalveclam shells

– 8.0 18:1 65 180 95.84 (Girish et al., 2013)

WCO Ionicliquid catalyst

– 1.0 9:1 70 6 98.1 (Lin, Yang, Chen, &Lin, 2013)

WCO KOH/Zeolite – 11.5 50 240 96.7 (Wan Omar &Amin, 2011)

Rapeseed oil Li-CaO – 5.0 12:1 – 240 >93.0 (Puna et al., 2014)Waste cooking

palm oilCsM-SiO2 25 3.0 20:1 65 180 90 (Amania, Asif, &

Hameed, 2016)Jatropha cru-

cas oilZr-CaO 15 5.0 15:1 65 120 >99 (Kaur & Ali, 2014)

Soybean oil Na2ZrO3 – 3.0 >3:1 65 180 98.30 (Santiago-Torres,Romero-Ibarra, &Pfeiffer, 2014)

Table 4. Physico-chemical properties of the obtained biodiesels.Biodiesel

Property Test method ASTM D6751 WCO WFO

Ester content % (w/w) (Bindhu et al., 2012) 96.50 96.68 ± 1.0 96.98 ± 1.60Density at 16 �C (g/ml) ASTM D4052-91 0.9000 0.8887 ± 0.0012 0.8822 ± 0002Kinematic viscosity at 40 �C (mm2/s) ASTM D445 5.0 4.78 ± 0.45 3.11 ± 0.12Flash point (�C) ASTM D93 93min. 170 ± 1.0 130 ± 1.0Cloud point (�C) ASTM D2500 – �3.0 ± 0.5 4.0 ± 0.5Pour point (�C) ASTM D2500 – �7.0 ± 0.5 �2.0 ± 0.5Acid value mg KOH/ g ASTM D664 0.50 max. 0.15 ± 0.03 0.11 ± 0.02Refractive index at 20 �C D1747 – 09 – 1.4559 ± 0.0002 1.4545 ± 0.0002Total glycerin (wt.%) (Pisarello et al., 2010) 0.25 max. 0.13 ± 0.02 0.10 ± 0.01Free glycerin (wt.%) (Pisarello et al., 2010) 0.02 max. 0.0051 ± 0.0010 0.0043 ± 0.0010Carbon residue (wt.%) ASTM D4530 0.05 max. 0.035 ± 0.001 0.024 ± 0.001Soaps (ppm) AOCS Cc 17-95 5.0 0.88 ± 0.11 0.55 ± 0.10Water content (%) ASTM D 95 – <0.01 <0.01Methanol content (%) ASTM D4530 – 0.014 ± 0.002 0.012 ± 0.001

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 63

D 6751 BD. Another evidence on the reactivity of theprepared solid base catalyst is the acid values ofobtained biodiesels which are much lower thanthose of the corresponding raw oils. The pour pointsof the prepared biodiesels indicate that the pro-duced fuels will be convenient for use in cold wea-ther conditions. Other properties of the producedbiodiesels, such as the content of the total glycerin,free glycerin, water, menthol and soaps were muchlower than the standard values, which also assuresthe high conversion level of the utilized oils to theirmethyl esters in the presence of the newly preparedsolid base catalyst.

Conclusions

The solid base catalyst prepared by the impregnationof activated carbon originating from polyethylene ter-ephthalate waste with KOH was utilized for transester-ification of different non-edible oils, namely WCO andWFO with methanol. KOH/AC-catalyzed transesterifi-cation of WCO gave the highest yield of methyl ester(88.12% w/w) using 3.5wt.% of KOH/AC catalyst, 9:1methanol to oil molar ratio, 65 �C reaction tempera-ture,180minutes of the reaction time and 600 rpmrate of stirring, whilst maximum methyl ester yieldsfrom WFO (92.66% w/w) was produced by utilizing3.0wt.% of KOH/AC catalyst, 9:1 methanol to oil molarratio, 65 �C reaction temperature, 150minutes of thereaction time and 600 rpm rate of stirring. The proper-ties of the produced biodiesels were very local tothose of no. 2 diesel fuel which enables us to use it inthe injection engines without any modification.Moreover, the catalysts were effective for at least fivecycles with a yield greater than 70wt.%.

Acknowledgements

Facilities presented by Department of Chemistry, Collegeof Science, Mosul University, Mosul, Iraq for conductingthis research work are highly appreciated.

Disclosure statement

No potential conflict of interest was reported bythe authors.

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ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 65

ORIGINAL ARTICLE

New similarity solutions for the generalized variable-coefficients KdVequation by using symmetry group method

Rehab M. El-Shiekha,b

aDepartment of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt; bDepartment of Mathematics, College ofScience and Humanities at Howtat Sudair, Majmaah University, Kingdom of Saudi Arabia

ABSTRACTIn this paper, a generalized variable-coefficients KdV equation (gvcKdV) arising in fluidmechanics, plasma physics and ocean dynamics is investigated by using symmetry groupanalysis. Two basic generators are determined, and for every generator, the admissible formsof the variable coefficients and the corresponding reduced ordinary differential equations areobtained. Finally, by searching for solutions to those reduced ordinary differential equations,many new exact solutions for the gvcKdV equation have been found.

ARTICLE HISTORYReceived 6 February 2017Accepted 9 October 2017

KEYWORDSSymmetry group method;generalized variable-coefficients KdV equation;tanh-function method

1. Introduction

This paper is devoted to studying the generalizedvariable-coefficients KdV equation (gvcKdV), which isgiven by (Wang, 2006)

ut þ g1uxxx þ g2u3þ g3u

2þ g4uþ g5� �

ux þ g6uþ g7 ¼ 0;

(1)

where, giðtÞ with i¼ 1;2; . . .7 are arbitrary functionsof t. When g2 ¼ 0; Equation (1) is derived by consid-ering the time-dependent basic flow and boundaryconditions from the well-known Euler equation withan earth rotation see (Tang, Huang, & Lou, 2006) andanalytical solitonic solution is obtained for it by con-sidering g3 ¼ g5 ¼ g7 ¼ 0, which means that Equation(1) not really solved but only the known famous vari-able coefficients KdV equation. Furthermore, manyphysical and mechanical situations governed byEquation (1) like pressure pulses in fluid-filled tubesof special value in arterial dynamics, trapped quasi-one-dimensional Bose–Einstein condensates, ion-acoustic solitary waves in plasmas and the effect of abump on wave propagation in a fluid-filled elastictube; moreover, a model for strongly nonlinearinternal waves in the ocean (Alam & Ali Akbar, 2015;Bibi & Mohyud-Din, 2014; El-Shiekh, 2012; Hu, Tan, &Hu, 2016) and references therein.

Recently, Wang used the semi-inverse method toobtain the variational principle for Equation (1) butno solutions obtained. Therefore, the most importanttarget for this paper is obtaining new exact solutionsfor Equation (1) under some constraints among the

variable coefficients by using the symmetrygroup analysis.

2. Symmetry method

Recently, many methods have been investigated todeal with nonlinear partial differential equations likebilinear representation, B€acklund transformationmethods (El-Shiekh, 2015; L€u & Peng, 2013; L€u, Lin, &Qi, 2015a; L€u & Lin, 2016), tanh function and sine–cosine methods (Bibi & Mohyud-Din, 2014; El-Shiekh,2015; El-Wakil, Abulwafa, El-hanbaly, El-Shewy, &Abd-El-Hamid, 2016; Hu et al., 2016; Moussa &El-Shiekh, 2011), direct reduction method (El-Shiekh,2012, 2015, 2017) and the symmetry group analysis(El-Sayed, Moatimid, Moussa, El-Shiekh, & El-Satar,2014; El-Sayed et al., 2015; Moatimid, El-Shiekh, &Al-Nowehy, 2013; Moussa & El-Shiekh, 2010, 2012).

Symmetry method is one of the new modificationof Lie group analysis; it is more easy and simple incalculations than Lie method (Moatimid et al., 2013;Wang, Liu, & Zhang, 2013; Wang, Kara, & Fakhar,2015; Wang, 2016) and can be briefly described inthe following steps:

Suppose that the differential operator L can bewritten in the form

LðuÞ ¼ opuotp

� HðuÞ; (2)

where, u ¼ uðt; xÞ and H may depend on t, x, u andany derivative of u as long the derivative of u does

CONTACT Rehab M. El-Shiekh rehab_el_shiekh@yahoo.com, r.abdelhaim@mu.edu.sa Department of Mathematics, Faculty of Education,Ain Shams University, Cairo, Egypt� 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

University of BahrainARAB JOURNAL OF BASIC AND APPLIED SCIENCES2018, VOL. 25, NO. 2, 66–70https://doi.org/10.1080/25765299.2018.1449343

not contain more than ðp� 1Þ; t derivatives. We willconsider the symmetry operator (called infinitesimalsymmetry) in the form

SðuÞ ¼ Aðt; x; uÞ ouot

þXni¼1

Biðt; x; uÞ ouoxi þ Cðt; x; uÞ;

(3)

and the Fr�echet derivative of L(u) is given by

FðL; u; vÞ ¼ dde

Lðuþ evÞje¼ 0 (4)

With these definitions, we will computethe following:

i. FðL; u; vÞ;ii. FðL; u; SðuÞÞ;iii. Substitute H(u) for (o

puotpÞ in FðL; u; SðuÞÞ;

iv. Set this expression to zero and perform a poly-nomial expansion;

v. Solve the resulting partial differential equations.Once this system of partial differential equationsis solved for the coefficients of S(u), Equation (2)can be used to obtain the functional form ofthe solutions.

3. Determination of symmetries

In order to find the symmetries of Equation (1), weset the following symmetry operator

S uð Þ ¼ A x; t; uð Þut þ B x; t; uð Þux þ C x; t; uð Þ: (5)

Calculating the Fr�echet derivative F L; u; vð Þ of L uð Þin the direction of v, given by Equation 4ð Þ, andreplacing v by S uð Þ in F, we get

F L; u; S uð Þð Þ ¼ St þ g1Sxxx þ 3g2u2uxSþ g2u3Sxþ g3u2Sx þ 2g3u uxSþ g4uxS

þ g4uSx þ g5Sx þ g6Sþ g7:

(6)

Substituting the values of different derivatives ofS(u) in F with the aid of Maple program, we get apolynomial expansion in ux; ut; uy; uxut ,… ,etc. Onmaking use of Equation (1) in the polynomial expres-sion for F, rearranging terms of various powers ofderivatives of u and equating them to zero, weobtain

Ax ¼ Au ¼ Bu ¼ Cu u ¼ Cxu ¼ 0;

3g1Bx � Ag1ð Þt ¼ 0;

Ct � Ag7ð Þt � Ag6ð Þtuþ g1Cxxx þ g2Cxu3 þ g3Cxu2

þg4Cxuþ g5Cx � g6Cuu� g7Cu þ g6C ¼ 0;

g2Bxu3 þ g5Bx � Ag5ð Þt � Ag2ð Þtu3 � Ag3ð Þtu2

þ2g3uC þ Bt þ g4C þ 3g2u2C � Ag4ð Þtuþg4Bxuþ g3Bxu2 ¼ 0:

(7)

On solving system 7ð Þ; the infinitesimal A, B and Cin the above equations are:

A ¼ 1C0ðtÞ ½3c1CðtÞþ c2�; dCðtÞ

dt¼ g1ðtÞ;

B ¼ c1xþ c3;

C ¼ c4uþ c5;

(8)

where, ci; i ¼ 1; 2; . . .; 5; are arbitrary constants. Thefunctions gi ¼ gi tð Þ; i ¼ 1; 2; . . .; 7; are governed bythe following equations:

Ag2ð Þt � c1 þ 3c4ð Þg2 ¼ 0;

Ag3ð Þt � c1 þ 2c4ð Þg3 � 3c5g2 ¼ 0;

Ag4ð Þt � c1 þ c4ð Þg4 � 2c5g3 ¼ 0;

Ag5ð Þt � c1g5 þ c5g4 ¼ 0;

Ag6ð Þt ¼ 0;

Ag7ð Þt þ c4g7 þ c5g6 ¼ 0:

(9)

The symmetry Lie algebra of Equation (1) is gener-ated by the operators

V1 ¼ 3C tð ÞC0 tð Þ

oot

þ xoox

;

V2 ¼ 1C0 tð Þ

oot

;

V3 ¼ oox

;

V4 ¼ uoou

;

V5 ¼ oou

:

(10)

and the commutator table of it is given by

Now, we are going to search for a one-dimen-sional optimal system of the Lie algebra generatedby the operators (10) as follows:

Consider a general element of V ¼X5

i¼1aiVi, and

checking whether V can be mapped to a new elem-ent V�under the general adjoint transformationAd exp eVið Þð ÞVj ¼ Vj � e Vi; Vj½ � þ e2

2 Vi; Vi; Vj½½ ��; to sim-plify it as much as possible.

The adjoint table

V1 V2 V3 V4 V5V1 0 �3V2 �V3 0 0V2 3V2 0 0 0 0V3 V3 0 0 0 0V4 0 0 0 0 �V5V5 0 0 0 V5 0

V1 V2 V3 V4 V5V1 V1 exp 3eð ÞV2 exp eð ÞV3 V4 V5V2 V1 � 3eV2 V2 V3 V4 V5V3 V1 � eV3 V2 V3 V4 V5V4 V1 V2 V3 V4 exp eð ÞV5V5 V1 V2 V3 V4 � eV5 V5

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 67

Following Olver (1986), we can deduce the follow-ing basic fields which form an optimal system forthe gvcKdV,

i. V1 þ k1V4;ii. V2 þ k2V3 þ k3V4;iii. V3 þ k4V5;iv. V4;v. V5;

where, ki; i ¼ 1; . . .; 4 are arbitrary constants. Thecases (iii), (iv) and (v) give trivial reductions.Therefore, we will discuss the first and second caseby only using the following characteristic equation:

dtA x; t; uð Þ ¼

dxB x; t; uð Þ ¼

�duC x; t; uð Þ : (11)

By solving Equation (11) for both generators (I)and (II)

where, n1; . . .; n6 and m1; . . .;m6 are arbi-trary constants.

4. Reductions and exact solutions

In this section, the primary focus is on the reductionsassociated with the two generators (i) and (ii), andtheir solutions.

Generator (I)

Corresponding to this generator, the gvcKdV isreduced to the following ordinary differentialequation.

3F000 þ n1F3F0 þ n2F2F0 þ n3FF0 þ ðn4 � fÞF0þ ðn5 � k1ÞF þ n6 ¼ 0:

(12)

To solve Equation (12), we seek a special solutionin the form

F ¼ A0 þ A1f�2

3; (13)

where, A0 and A1 are arbitrary constants to be deter-mined. Substituting Equation (13) into Equation (12)and equating the coefficients of different powers off to zero, we get a system of algebraic equations,solutions of which give rise to the relations on theconstants as

A0 ¼ 3n62

; A1 ¼ 60n6n2

� �13

; n1 ¼ � 2n29n6

;

n3 ¼ � 32n2n6; n4 ¼ 3

4n2n

26; k1 ¼ 2

3þ n5:

(14)

Finally, we get the following exact solution for thegvcKdV

u1 x; tð Þ ¼

C tð Þ� 2

3þn5ð Þ3

3n62

þ 60n6n2

� �13

x þ 23þ n5

�� �C tð Þ�1

3

��23

!

(15)

Generator (II)

The reduced nonlinear ordinary differential equationcorresponding to this case is

F000 þm1F3F0 þm2F

2F0 þm3FF0 þ ðm4 � k3ÞF0

þ ðk4 �m5ÞF þm6 ¼ 0:(16)

Herein, we apply the modified extended tanhfunction method (El-Shiekh, 2015) to obtain rationalexact solitary wave solutions to Equation (16). Letus assume that Equation (16) has a solution inthe form

F fð Þ ¼ A0 þXNi¼1

Ai/i þ Bi/

�i; (17)

where, / fð Þ is a solution of the following Riccatiequation

/0 ¼ r þ /2; (18)

which has the following solutions

/ðfÞ ¼ � ffiffiffiffiffiffi�rp

tanhð ffiffiffiffiffiffi�rp

fÞ; r< 0;

/ðfÞ ¼ � ffiffiffiffiffiffi�rp

cothð ffiffiffiffiffiffi�rp

fÞ; r< 0;

/ðfÞ ¼ ffiffir

ptanð ffiffi

rp

fÞ; r > 0;

/ðfÞ ¼ � ffiffir

pcotð ffiffi

rp

fÞ; r > 0;

/ðfÞ ¼ � 1f; r ¼ 0:

(19)

Substituting Equation (17) into Equation (16) andby balancing the linear term with the greatest non-linear term, we get

N ¼ 23

(20)

Therefore,

F fð Þ ¼ A0 þ A1/23 þ B1/

�23 : (21)

The similarityvariable f

Similaritysolution u x; tð Þ integrability conditions

The first generator

ðx þ k1ÞC tð Þ�13 ; F fð ÞC tð Þ

�k13 ; g2 tð Þ ¼ n1

3 C0tð ÞC tð Þ�2

3 þk1 ;

g3 tð Þ ¼ n23 C

0tð ÞC tð Þ23ðk1�1Þ

;

g4 tð Þ ¼ n33 C

0tð ÞC tð Þ13ðk1�2Þ

;

g5 tð Þ ¼ n43 C

0tð ÞC tð Þ�2

3;

g6 tð Þ ¼ n53 C

0tð ÞC tð Þ�1

;

g7 tð Þ ¼ n63 C

0tð ÞC tð Þ�ð1þk1

3 Þ;The second generator

k2C tð Þ � x; F fð Þe�k3C tð Þ g2 tð Þ ¼ m1C0tð Þe3k3C tð Þ;

g3 tð Þ ¼ m2C0tð Þe2k3C tð Þ;

g4 tð Þ ¼ m3C0tð Þek3C tð Þ;

g5 tð Þ ¼ m4C0tð Þ;

g6 tð Þ ¼ m5C0tð Þ;

g7 tð Þ ¼ m6C0tð Þe�k3C tð Þ;

68 R. M. EL-SHIEKH

Substituting Equation (21) into Equation (16) andequating the powers of /j; j ¼ 0;� 11

3 ;�3;� 73 ; . . . to

zero, we obtain a system of algebraic equations. Bysolving that system with maple program yields thefollowing solution

A1 ¼ �25

9m1

� �13

; B1 ¼ 15m1

� �23 3m3m1�m2

2

63m1

� �;

r ¼ 6142

ffiffiffiffiffiffiffiffiffiffiffi5

14m1

rm2

2�3m1m3

m1

� �32

;

A0 ¼ � m2

3m1;m6 ¼ 0; k3 ¼ m5;

k2 ¼ m4 þ 11323m2

1

� 98m32 � 441m1m2m362

ffiffiffiffiffi70

pm2

2 � 3m1m3

� �32

(22)

Substituting Equation (22) into Equation (21),we get

F fð Þ ¼ � m2

3m1� 2

59m1

� �13

/23 þ 15

m1

� �23 3m3m1 �m2

2

63m1

� �/

�23 ;

(23)

where, / is given by Equation (18). SubstitutingEquation (23) into the similarity solution correspond-ing to this case, we obtain

u2 x; tð Þ ¼ e�k3C tð Þ"�m2

3m1þ 2

59m1

� �13

r13 tanh

23ð ffiffiffiffiffiffi�rp

k2C tð Þ � xð Þ

�15m1

23 3m3m1�m2

263m1

r13 tanh

23ð ffiffiffiffiffiffi�rp

k2C tð Þ � xð ÞÞ

#

(24)

u3 x; tð Þ ¼ e�k3C tð Þ"�m2

3m1þ 2

59m1

� �13

r13 coth

23ð ffiffiffiffiffiffi�rp

k2C tð Þ � xð ÞÞ

þ15m1

23 3m3m1�m2

263m1

r13 coth

23ð ffiffiffiffiffiffi�rp

k2C tð Þ � xð ÞÞ

#

(25)where

r ¼ � 142

ffiffiffiffiffiffiffiffiffiffiffi5

14m1

rm2

2 � 3m1m3

m1

� �32

and

k2 ¼ m4 þ 11323m2

198m3

2 � 441m1m2m3 � 2ffiffiffiffiffi70

pm2

2 � 3m1m3

� �32

;

u4 x; tð Þ ¼ �e�k3C tð Þ"m2

3m1þ 2

59m1

� �13

r13 tan

23ð ffiffi

rp

k2C tð Þ � xð ÞÞ

�15m1

23 3m3m1�m2

263m1

r13 tan

23ð ffiffi

rp

k2C tð Þ � xð ÞÞ

#

(26)

u5 x; tð Þ ¼ �e�k3C tð Þ"m2

3m1þ 2

59m1

� �13

r13 cot

23ð ffiffi

rp

k2C tð Þ � xð ÞÞ

�15m1

23 3m3m1�m2

263m1

r13 cot

23ð ffiffi

rp

k2C tð Þ � xð ÞÞ

#;

(27)

where

r ¼ 142

ffiffiffiffiffiffiffiffiffiffiffi5

14m1

rm2

2 � 3m1m3

m1

� �32

and

k2 ¼ m4 þ 11323m2

198m3

2 � 441m1m2m3 þ 2ffiffiffiffiffi70

pm2

2 � 3m1m3

� �32

;

u6 x; tð Þ ¼ �e�k3C tð Þ m2

3m1þ 2

59m1

� �13

k2C tð Þ � xð Þ�23

!;

(28)

where

r ¼ 0; k2 ¼ �m32

27m21þm4 and m3 ¼ m2

2

3m1:

5. Conclusion

In this paper, we have applied the symmetry groupanalysis to the gvcKdV. This application leads to twononequivalent generators; for every generator in theoptimal system, the admissible forms of the coeffi-cients and the corresponding reduced ordinary dif-ferential equation are obtained. The search forsolutions to those reduced ordinary differential equa-tions using tanh function method has yielded manyexact new solutions that were not obtained before.

Disclosure statement

No potential conflict of interest was reported bythe author.

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70 R. M. EL-SHIEKH

ORIGINAL ARTICLE

New interaction solutions and nonlocal symmetries for the(2þ 1)-dimensional coupled Burgers equation

Hengchun Hu and Yueyue Li

College of Science, University of Shanghai for Science and Technology, Shanghai, China

ABSTRACTThe nonlocal symmetries for the coupled (2þ 1)-dimensional Burgers system are obtainedwith the truncated Painlev�e expansion method. The nonlocal symmetries can be localized tothe Lie point symmetries by introducing auxiliary dependent variables. The finite symmetrytransformations related with the nonlocal symmetries are computed. The multi-solitary wavesolution of the (2þ 1)-dimensional coupled Burgers system are presented. By using the con-sistent tanh expansion method, many interaction solutions among solitons and other typesof nonlinear excitations of a (2þ 1)-dimensional coupled Burgers system can be obtained,which include soliton-cnoidal waves, multiple resonant solutions, soliton-error functionwaves, soliton-rational waves, and soliton-periodic waves.

ARTICLE HISTORYReceived 4 June 2017Accepted 18 September 2017

KEYWORDSNonlocal symmetries;consistent tanh expansion;soliton-cnoidalwave solutions

1. Introduction

Recently, symmetry reduction approach with nonlo-cal symmetries related to Darboux transformations,B€acklund transformation and residual symmetries hasbeen successfully used to find some types of inter-action solutions among different types of nonlinearexcitations including the solitons, cnoidal waves, Airywaves and Bessel waves for a number of integrablesystems such as the Kadomtsev-Petviashvili (KP)equation, the Burgers equation, the AKNS system,the modified Kadomtsev-Petviashvili (mKP) equationand the coupled integrable dispersionless equation.From the results of nonlocal symmetry reduction,Lou found that the symmetry related to the Painlev�etruncated expansion is just the residue with respectto the singular manifold in the Painlev�e analysis pro-cedure and called residual symmetry (Gao, Lou,and Tang, 2013; Hu, Lou, and Chen, 2012; Lou, 2013;Lou, Hu, and Chen, 2012).

On the other hand, the consistent tanh expansionmethod (CTE) is proposed to identify CTE solvablesystems, which is a special simplified form of theconsistent Riccati expansion (CRE) method defined inRef. (Lou, 2015). Some interaction solutions betweensolitons and other nonlinear excitations can be foundwith the help of the CTE method for many integrablesystems including the Broer-Kaup (BK) system, theBoussinesq-Burgers equations and the (2þ 1)-dimen-sional Boiti-Leon-Pempinelli (BLP) equation (Chenand Lou, 2013; Chen, Hu, and Zhu, 2015; Cheng, Lou,

Chen, and Tang, 2014; Hu, Tan, and Hu, 2016; Lou,Cheng, and Tang, 2014). There are many methods toconstruct the exact solutions for different integrablesystems, such as the Boussinesq equation, theunsteady KdV equation, the Benney-Luke equation,etc. (Akbar and Aliz, 2011; Akbar, Ali, and Mohyud-Din, 2013; Akter and Akbar, 2015; Alam and Akbar,2013, 2015; Alam, Hafez, Akbar, and Roshid, 2015;Alam, Hafez, Belgacem, and Akbar, 2015; Islam, Khan,Akbar, and Mastroberardino, 2014; Khan and Akbar,2013, 2016).

In this paper, we focus on the following (2þ 1)-dimensional coupled Burgers system

ut � 2uux � vxx ¼ 0; vyt � uxxy � 2uvxy � 2uxvy ¼ 0;

(1)

which deserves the name as a coupled Burgerssystem because it is reduced to the standardBurgers equation by setting u¼ v and x¼ y (El-Wakil,Abulwafa, El-hanbaly, El-Shewy, Abd-El-Hamid, 2016;Kumar and Kumar, 2014; Su, 2017; Vaneeva, Posta,Sophocleous, 2017). The Painlev�e analysis and infinitemany symmetries are studied by Wang (Wang, Liang,Tang, 2014) and finite symmetry group and localizedstructures are given in (Lei and Yang, 2013).

The paper is organized as follows. In Section 2, thenonlocal symmetries for the coupled (2þ 1)-dimen-sional Burgers system are obtained with the truncatedPainlev�e expansion, then we localize it by introducingsome dependent variables. The multi-solitary wavesolution of the coupled (2þ 1)-dimensional Burgers

CONTACT Hengchun Hu hhengchun@163.com, hhengchun@aliyun.com College of Science, University of Shanghai for Science andTechnology, Shanghai, China� 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

University of BahrainARAB JOURNAL OF BASIC AND APPLIED SCIENCES2018, VOL. 25, NO. 2, 71–76https://doi.org/10.1080/25765299.2018.1449417

system is obtained using the finite symmetry transfor-mations. In Section 3, the coupled (2þ 1)-dimensionalBurgers system is proved to be CTE solvable andmany interaction solutions between solitons andother nonlinear excitations are given out directly.Section 4 is devoted to a short summary anddiscussions.

2. Nonlocal symmetries of the (2þ1)-dimensional coupled Burgers system

In order to find the nonlocal symmetries of the(2þ 1)-dimensional coupled Burgers system, we trun-cate the Laurent series as

u ¼ u0/

þ u1; v ¼ v0/þ v1; (2)

where the function / is an arbitrary singular mani-fold and the functions u0, u1, v0, v1 are determinedby substituting (2) into the Equation (1) and vanish-ing all the coefficients of each power of /, we have

u0 ¼ d/x; v0 ¼ /x; u1 ¼ /t � d/xx

2/x;

v1 ¼ d/t � /xx

2/x; d2 ¼ 1:

(3)

Substituting (2) and (3) into the coupled Burgerssystem (1), the manifold / satisfies the Schwarzianform

Ct ¼ CCx þ 2dCxx � Sx; (4)

where

C ¼ /t

/x; S ¼ 2/x/xxx � 3/2

xx

2/2x

are Schwarzian variables and keep form invariantunder the M€obious transformation

/ ! a/þ bc/þ d

; ad 6¼ bc: (5)

According to the definition of residual symmetries(Lou, 2013), the nonlocal symmetries of the (2þ 1)-dimensional coupled Burgers Equation (1) can beread out from the truncated Painlev�e analysis

ru ¼ d/x; rv ¼ /x: (6)

To find the group of the nonlocal symmetry (6),

fu; vg ! f�u; �vg ¼ fu; vg þ efru;rvg; (7)

we need to solve the initial value problem

d�ude

¼ d�/x; �uje¼0 ¼ u; (8a)

d�vde

¼ �/x; �vje¼0 ¼ v; (8b)

with e being an infinitesimal parameter. It is difficultto solve the Equation (8) for the new functions �u and�v due to the intrusion of the function �/ and its deriv-atives. To solve this initial value problem, we prolong

the coupled Burgers system (1) such that the nonlocalsymmetries become the local symmetries for the pro-longed system by introducing another three newdependent variables as the following

/x ¼ f ; /t ¼ g; fx ¼ h: (9)

Then the nonlocal symmetry (6) for the coupledBurgers system (1) becomes a Lie point symmetry ofthe prolonged system (1), (3) and (9) and it is verifiedthat the Lie point symmetries of the prolonged sys-tem have the form

ru ¼ df ; rv ¼ f ; r/ ¼ �/2; rf ¼ �2/f ;

rg ¼ �2/g; rh ¼ �2f 2 � 2/h:(10)

Correspondingly, the initial value problem of (10)becomes

d�ude

¼ d�f ;d�vde

¼ �f ;d�/de

¼��/2;d�fde

¼�2�/�f ;

d�gde

¼�2�/�g;d�hde

¼�2�f2 � 2�/�h;

(11)

�uje¼0 ¼ u; �vje¼0 ¼ v; �/je¼0 ¼ /; �f je¼0 ¼ f ;

�gje¼0 ¼ g; �hje¼0 ¼ h:(12)

The solution of the initial value problem (11) and(12) for the enlarged system (1), (3) and (9) can bewritten as

�u ¼ uþ def

1þ e/; �v ¼ v þ ef

1þ e/;

�/ ¼ /1þ e/

; �f ¼ f

ð1þ e/Þ2 ;(13)

�g ¼ g

ð1þ e/Þ2 ;�h ¼ h

ð1þ e/Þ2 �2ef 2

ð1þ e/Þ3 : (14)

Using the finite symmetry transformation (13) and(14), we can obtain a new solution from the initialsolution. For example, we take the trivial solutionu¼ v¼ 0 of the coupled Burgers system (1) and themulti-solitary wave solution for (4) is supposed as

/ ¼ 1þXNn¼1

expðknx þ wntÞ; (15)

where kn, wn are arbitrary constants. The multi-soli-tary wave solution (15) is the solution of (4) onlywith the relation

wn ¼ dk2n: (16)

A solution of the coupled Burgers system (1)presents in the following form by using (9), (13)and (14)

u ¼ d

XN

n¼1knexpðknx þ dk2ntÞ

1þXN

n¼1expðknx þ dk2ntÞ

; (17a)

u ¼XN

n¼1knexpðknx þ dk2ntÞ

1þXN

n¼1expðknx þ dk2ntÞ

: (17b)

72 H. HU AND Y. LI

3. Consistent tanh expansion solvability andinteraction solutions for the Equation (1)

The author introduced the consistent tanh expansionmethod in (Lou, 2015), that is to say, for a Painlev�eintegrable system, we want to find the solution inthe form

u ¼XJ

j¼0

uj tanhjðxÞ;

where J can be determined by the leading orderanalysis of the integrable system. We can take thefollowing truncated tanh function expansion for theEquation (1)

u ¼ u0 þ u1tanhðxÞ; v ¼ v0 þ v1tanhðxÞ: (18)

Substituting the expression (18) into the coupledBurgers system (1), we obtain two complicatedexpression for the function tanhx.

ð2u21xx�2v1x2xÞtanh3ðxÞþð2u0u1xxþ2v1xxx�2u1u1x

þv1xxx�u1xtÞtanh2ðxÞþð2v1x2

x�v1xx�2u21xx�2u1u0xþu1t�2u0u1xÞtanhðxÞþu0t�2v1xxxþu1xt�2u0u0x�v1xxx�2u0u1xx�v0xx ¼ 0;

(19)

ð6u1xyx2x�6u1v1xxxyÞtanh4ðxÞþð�4u1xxxxy

�2u1xyx2xþ2v1xyxtþ2u1v1xxy

�2u1yx2x�4u1xxxxyþ2u1xv1xyþ2u1v1xxy

�4u0v1xxxyþ4u1v1yxxÞtanh3ðxÞþð�2u1xv1y�v1yxtþu1xxxy�v1xytþ2u1xyxx

�8u1xyx2x�2u1v1xyþ2u1xxxy

�v1txyþu1yxxxþ2u0v1yxxþ2u0v1xxyþu1xxxy

þ8u1v1xxxyþ2u0xv1xyþ2u0v1xxy

þ2u1v0yxxÞtanh2ðxÞþð�2u0v1xy�2u1v1xxy

þ4u1xxxxy�2u1xv0y�2v1xyxt

þ2u1xyxxxþ4u0v1xxxy�2u1v0xyþv1ytþ4u1xxxxy

�2u1xv1xy�u1xxy�2u0xv1y�2u1v1xxy�4u1v1yxxþ2u1yx2

xÞtanhðxÞ�2u1v0yxx

þv0yt�u0xxyþv1yxt�2u0v1xxy

�2u0v0xy�2u0xv0yþ2u1xyx2x�u1xxxy�u1yxxx

�2u0v1xxy�2u1xyxx�2u0v1yxx

�u1xxxyþv1txy�2u1xxxy�2u1v1xxxyþv1xyt

�2u0xv1xy ¼ 0:

(20)

Setting the coefficients of different powers to zeroin the above complicated systems (19)–(20), we havenine over-determined equations for only fiveundetermined functions u0, u1, v0, v1 and x. It is for-tunate that the over-determined system is consistentand possesses the following solution

u1 ¼ dxx; v1 ¼ xx; u0 ¼ xt � dxxx

2xx; v0 ¼ dxt � xxx

2xx;

(21)

and the function x only needs to satisfy the condi-tion

Ct ¼ CCx þ 2dCxx � Sx þ 4xxxxx; (22)

where

C ¼ xt

xx; S ¼ 2xxxxxx � 3x2

xx

2x2x

:

In summary, we have the following theorem:

Theorem 3.1. If x is a solution of the consistentcondition (22), then

u ¼ dxx tanhðxÞ þ xt � dxxx

2xx;

v ¼ xx tanhðxÞ þ dxt � xxx

2xx;

(23)

constitute a CTE solution of the coupled (2þ 1)-dimen-sional Burgers system (1).

According to Theorem 3.1, we can find variousinteraction solutions among different types of nonlin-ear excitations of the Equation (1) by solving the xEquation (22). It is interesting to see that Equation(22) can be written as

xt ¼ dxxx 6 2x2x þ fxx; (24a)

ft ¼ dfxx þ ffx: (24b)

It is not difficult to find that the four equations in(24) have similarity solutions, therefore, it is sufficientto investigate the solutions of one case of (24).Namely, d¼�1 and the Equation (24) becomes

xt ¼ �xxx þ 2x2x þ fxx; (25a)

ft ¼ �fxx þ ffx: (25b)

It is clear that x can be obtained by solving thevariable coefficient potential Burgers equation (25a)via a fixed solution of the Burgers equation (25b).Therefore, the corresponding solution of the coupledBurgers system (1) can be obtained from the CTEexpression (23).

3.1. Single soliton solutions

In (25a), we take a trivial solution

x ¼ k0x þ l0y þ w0t þ d0; f ¼ w0 � 2k20k0

; (26)

with k0, l0, w0, d0 being arbitrary constants. Then sub-stituting (26) into (23) with d¼�1 yields the singlesoliton solution of the coupled Burgers system (1)

u ¼ �k0 tanhðk0x þ l0y þ w0t þ d0Þ þ w0

2k0; (27a)

v ¼ k0 tanhðk0x þ l0y þ w0t þ d0Þ � w0

2k0: (27b)

3.2. Soliton-cnoidal waves

In order to obtain the interaction solutions betweensoliton and cnoidal wave of the coupled Burgers sys-tem (1), letting

x ¼ k0x þ l0y þ w0t þWðk1x þ l1y þ w1tÞ; (28)

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 73

where

Wðk1x þ l1y þ w1tÞ ¼ WðXÞ ¼ W

satisfies

W21X ¼ C0 þ C1W1 þ C2W

21 þ C3W

31 þ C4W

41 ; W1 ¼ WX ;

(29)

with

C0 ¼ k20ð�2k1C3k0 þ k21C2 þ 12k20Þk41

; (30a)

C1 ¼ k0ð�3k1C3k0 þ 2k21C2 þ 16k20Þk31

; (30b)

C4 ¼ 4; (30c)

w0 ¼ w1k0k1

; (30d)

and C2, C3 are arbitrary constants and substituting(30) and (28) into the Equation (23) with d¼�1, weobtain

u ¼ �ðk0 þ k1W1Þ tanh�k0x þ l0y þ w1k0

k1t þW

�

þ w1

2k1þ k21W1X

2ðk0 þ k1W1Þ ;(31a)

v ¼ ðk0 þ k1W1Þ tanh�k0x þ l0y þ w1k0

k1t þW

�

� w1

2k1� k21W1X

2ðk0 þ k1W1Þ :(31b)

It is obvious that the Equation (29) is an equationfor the definition of the elliptic functions, which canbe expressed in terms of Jacobi elliptic functions.Here, we write down two types of soliton-cnoidalwave solutions of (22) with d¼�1. The first one is asimple solution as

W1 ¼ l0 þ l1snðmX; nÞ; (32)

where sn(mX, n) is the usual Jacobi elliptic sine func-tion. Substituting (32) and (30) into (29) yields

C2 ¼ 8ð2k21l20 � 2k1l0k0 � k20Þk21

; (33a)

C3 ¼ �16l0; (33b)

n ¼ 1; (33c)

l1 ¼k0 þ k1l0

k1; (33d)

m ¼ �2ðk0 þ k1l0Þk1

: (33e)

Hence, one kind of soliton-cnoidal wave solutionsis obtained by taking (32) and

W ¼ l0X þ l1

ðXX0

snðmY; nÞdY; (34)

with the parameter requirement (33) into the generalsolution (31). Then we can discuss the soliton-cnoidalwaves for the (2þ 1)-dimensional coupled Burgers

Equation (1) by selecting the proper arbi-trary constants.

The second type of the soliton-cnoidal wave inter-action solutions is to select

x ¼ k0x þ l0y þ w0t þ AEf ðsnðk1x þ l1y þ w1t;mÞ; nÞ;(35)

where Ef is the first incomplete elliptic integral andsn(z, m) is the usual Jacobi elliptic sine function.Substituting (35) into (22) with d¼�1 and settingthe coefficients of different powers of Jacobi ellipticfunctions into zero, we can find eight arbitrary con-stant solutions except m¼ n. Then substituing (35)with m¼ n into (23) with d¼�1, we can obtain thesoliton-cnoidal wave interaction solutions of thecoupled Burgers Equation (1) with the proper arbi-trary constants in the same way.

3.3. Solitons and potential Burgerswave solutions

To find out the interaction solutions of the coupledBurgers system (1), we consider x in the form

x ¼ k0x þ l0y þ w0t þ g; (36)

where g is a function of x, y and t.Applying the transformation (36) to (25a), we

have

gt ¼ �gxx þ 2g2x þ ðf þ 4k0Þgx þ 2k20 � w0 þ fk0:

(37)

For further simplicity, we take f as the simplestconstant solution

f ¼ w0 � 2k20k0

; (38)

on account of which, (37) becomes a constant coeffi-cient potential Burgers equation

gt ¼ �gxx þ 2g2x þw0

k0þ 2k0

� �gx: (39)

After substituting (36) and (39) into (23) withd¼�1, we get the interaction solution between asoliton and a potential Burgers wave of the coupledBurgers system (1)

u ¼ �ðk0 þ gxÞtanhðk0x þ l0y þ w0t þ gÞ þ gx þ w0

2k0;

(40a)

v ¼ ðk0 þ gxÞtanhðk0x þ l0y þ w0t þ gÞ � gx � w0

2k0:

(40b)

It is well known that the potential Burgers equa-tion has many types of known exact solutions, suchas resonant soliton solutions, and error function solu-tion. In the following, we use the known solutions of(39) to construct the interaction solutions between asoliton and potential Burgers waves.

74 H. HU AND Y. LI

Example 1: Multiple resonant soliton solutions

Equation (25a) possesses the following multiplewave solution

g ¼ � 12ln½1þ

Xni¼1

expðkix þ liy þ witÞ�; (41)

where ki, li are arbitrary constants while wi are deter-mined by the dispersion relations

wi ¼ kik0

ð2k20 þ w0 � k0kiÞ:

Substituing (41) into (40), the (nþ 1) resonant soli-ton solutions of the coupled Burgers system (1) canbe directly obtained.

Example 2: Interaction solution between a solitonand an error function wave

It can be proved that the potential BurgersEquation (39) possesses an error function solution

g¼�12ln erf

iðxþ yþw1tÞ2

ffiffit

p� �� �

; i2 ¼ 1; w1 ¼ 2k0þw0

k0;

(42)

where the error function erf(x) is defined by

erfðxÞ� 2ffiffiffip

pðx0expð�n2Þdn; (43)

which means that solution (40) and (42) present thesoliton-error function interaction solution of thecoupled Burgers system (1).

Example 3: Soliton interactions with rational waves

It is known that the potential Burgers system (39)has a general solution

g ¼ � 12ln

Ð1�1 Fðx þ y þ 2k0t þ w0

k0t þ i2

ffiffit

pnÞexpð�n2Þdn

(44)

where

F � FðzÞ � Fðx þ y þ 2k0t þ w0

k0t þ i2

ffiffit

pnÞ<expðn2Þ; n ! 1

(45)

is an arbitrary function of the indicated variable.If we take F as a polynomial solution of z,

F ¼XNn¼1

cnzn (46)

with arbitrary constants cn, the g wave (44) becomes

g¼�12ln

XNn¼1

hnXnj¼0

½1þð�1Þj�ðitÞj=2ðxþ yþw1tÞn�j

Cð1þ j=2ÞCðn� jþ1Þ

24

35;

hn� 12

ffiffiffip

pcnCðnþ1Þ;

(47)

where C(x) is the usual Gamma function. The CTEsolution (40) along with (47) becomes an interactionsolution between a soliton and a rational wave.

Example 4: Soliton interactions with periodic waves

We consider the arbitrary function F in (44) as thefollowing special simple form

F ¼XNj¼1

cjffiffiffip

p cos½ajðz þ djÞ�expðbjzÞ (48)

where aj, bj, cj, dj are arbitrary constants. In this case,we have

g ¼ � 12ln

XNj¼1

cjcos½ajðx þ y þ dj þ ðw1 � 2bjÞtÞ�8<:

exp½bjx þ bjy þ ðw1bj � b2j þ a2j Þt�);

(49)

which means solution (40) with (49) is an interactionsolution among solitons and periodic waves.

4. Conclusion and discussion

In summary, the nonlocal symmetries for the coupled(2þ 1)-dimensional Burgers system are obtained byusing the truncated Painlev�e expansion. To solve theinitial value problem related to the nonlocal symme-tries, we prolong the coupled (2þ 1)-dimensionalBurgers equation so that nonlocal symmetriesbecome the local Lie point symmetries for the pro-longed system. The finite symmetry transformationsof the prolonged coupled (2þ 1)-dimensionalBurgers equation is derived by solving the Lie’s firstprinciple. In the meanwhile, the coupled (2þ 1)-dimensional Burgers equation is studied by means ofthe CTE method and proved to be CTE solvable.Many exact interaction excitations such as the soli-ton-cnoidal waves, the multiple resonant soliton sol-utions, soliton-error function waves, soliton-rationalwaves and soliton-periodic waves of the coupled(2þ 1)-dimensional Burgers equation are explicitlyconstructed with the help of the CTE method byselecting different solutions to the x equation. Thesenew interaction wave solutions are presented analyt-ically with the proper constant selections. The moreinteraction excitations from CTE method about othercoupled integrable systems will be worth of fur-ther study.

Disclosure statement

No potential conflict of interest was reported bythe authors

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 75

Funding

The work is supported by National Natural ScienceFoundation of China [Nos. 11071164 and 11201302],Shanghai Natural Science Foundation [No. 10ZR1420800],the Hujiang Foundation of China [B14005].

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76 H. HU AND Y. LI

ORIGINAL ARTICLE

Application of two different algorithms to the approximate long waterwave equation with conformable fractional derivative

Melike Kaplana and Arzu Akbulutb

aDepartment of Mathematics, Kastamonu University, Art-Science Faculty, Kastamonu, Turkey; bDepartment of Mathematics-Computer, Eskisehir Osmangazi University, Art-Science Faculty, Eskisehir, Turkey

ABSTRACTThe current paper devoted on two different methods to find the exact solutions with variousforms including hyperbolic, trigonometric, rational and exponential functions of fractional dif-ferential equations systems with conformable farctional derivative. We have employed themodified simple equation and exp(�UðnÞÞ method here for the approximate long waterwave equation. We have adopted here the fractional complex transform accompanied byproperties of conformable fractional calculus for reduction of fractional partial differentialequation systems to ordinary differential equation systems.

ARTICLE HISTORYReceived 24 June 2017Accepted 2 December 2017

KEYWORDSExact solutions; symboliccomputation; fractionaldifferential equations

1. Introduction

Fractional differential equations, containing the frac-tional differentiation are generalizations of classicaldifferential equations of integer order. Recently,these equations have a great deal of attention andapplied in many research fields, such as nonlinearcontrol theory, electromagnetic theory, fluid mechan-ics, signal processing, electrochemistry and mathem-atical biology (Kilbas, Srivastava, & Trujillo, 2006;Miller & Ross, 1993). Moreover, various applicationsof the fractional calculus can be found in plasmaphysics turbulence, stochastic dynamical system,image processing, fluid dynamics and astrophysics.Fractional partial differential equations are becomingincreasingly popular due to their practical applica-tions in various fields of science and engineering.There are two major approaches in the theoreticalformulation of initial value problems for fractionaldifferential equations (Dai, Wang, & Liu, 2016; Imran,Khan, Ahmad, Shah, & Nazar, 2017; Saqib, Ali, Khan,Sheikh, & Jan, in press; Sheikh, Ali, Khan, & Saqib,2016; Shah & Khan, 2016). One of them is based onthe interpretation of the initial condition of fractionalsystems as a distributed initial condition (Agarwal,O’Regan, Hristova, & Cicek, 2017).

It has great importance to obtain exact solutionsof the nonlinear fractional differential equations toseek the wave phenomena that they describe. Inorder to solve the nonlinear fractional partial differ-ential equations, a general method cannot bedefined so far. Also, in recent years, several effective

methods have been applied to obtain exact solutionsof these equations such as sub-equation method(Alzaidy, 2013; Mohyud-Din, Nawaz, Azhar, & Akbar,2017; Ray & Sahoo, 2015; Sahoo & Ray, 2015), tanhmethod (Ray & Sahoo, 2015), simplest equationmethod (Taghizadeh, Mirzazadeh, Rahimian, &Akbari, 2013), Jacobi elliptic function expansionmethod (Tasbozan, Cenesiz, & Kurt, 2016),Kudryashov method (Demiray, Pandir, & Bulut, 2014;Eslami, 2016; Hosseini, Mayeli, & Ansari, 2017;Sonmezoglu, Ekici, Moradi, & Zhou, 2017), trial equa-tion method (Ekici et al., 2016; Odabasi & Misirli,2015; Pandir, Gurefe, & Misirli, 2013), exp-functionmethod (Bekir, Guner, Aksoy, & Pandir, 2015; Zhanget al., 2010), first integral method (Eslami, Fathi,Mirzazadeh, & Biswas, 2014; Eslami & Rezazadeh,2016; Mirzazadeh, Eslami, & Biswas, 2014; Younis,2013), (G0/G)-expansion method (Ray & Sahoo, 2017),modification of the truncated expansion method(Mirzazadeh & Eslami, 2013), functional variablemethod (Bekir, Guner, Bhrawy, & Biswas, 2015), vari-able separation method (Wang & Dai, 2015; Wang,Zhang, & Dai, 2016), modified tanh-function method(Wang & Dai, 2016; Wang et al., 2016), Laplace trans-form method (Ali et al., 2016; Ali, Saqib, Khan, &Sheikh, 2016), homotopy analysis method (Kurt,Tasbozan, & Cenesiz, 2016), ansatz method (Zayed &Al-Nowehy, 2017) and so on (Biswas, Bhrawy,Abdelkawy, Alshaery, & Hilal, 2014; Guner & Eser,2014; Korkmaz, 2017; Mirzazadeh, 2016; Mirzazadeh& Eslami, 2013; Saqib et al., in press; Sheikhet al., 2016).

CONTACT Melike Kaplan mkaplan@kastamonu.edu.tr Department of Mathematics, Kastamonu University, Art-Science Faculty,Kastamonu, Turkey� 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

University of BahrainARAB JOURNAL OF BASIC AND APPLIED SCIENCES2018, VOL. 25, NO. 2, 77–84https://doi.org/10.1080/25765299.2018.1449348

In this paper, by use of the properties of fractionalcalculus, we propose two different methods to seekexact solutions of fractional partial differential equa-tions with conformable derivative. Based on a travel-ing wave transformation, certain fractional partialdifferential equation systems can be turned intoanother fractional ordinary differential equation sys-tems with respect to one new variable.

2. Brief of conformable fractional derivative

Recently, the authors Khalil et al. introduced a newsimple and intriguing definition of the fractionalderivative called conformable fractional derivative(Khalil, Al Horani, Yousef, & Sababheh, 2014). Thisderivative is well-behaved and obeys the Leibniz ruleand chain rule. Let us review the conformable frac-tional derivative (Cenesiz, Baleanu, Kurt, & Tasbozan,2017; Chung 2015).

Definition 1. Suppose f : 0;1½ Þ ! R be a function.Then, the conformable fractional derivative of f oforder a; 0<a � 1; is defined as

ðTafÞðtÞ ¼ lime!0

fðt þ et1�aÞ � fðtÞe

for all t > 0: Some useful properties can be listedas follows:

1. Linearity: Taðaf þ bgÞ ¼ aðTafÞ þ bðTagÞ, forall a; b 2 R

2. Leibniz rule: TaðfgÞ ¼ fTaðgÞ þ gTaðfÞ3. Let f be a differentiable and a�conformable dif-

ferentiable function and g be a differentiable func-tion defined in the range of f. Then

Taðf �gÞðtÞ ¼ t1�ag0ðtÞf 0ðgðtÞÞ:Moreover, the following rules are hold.TaðtpÞ ¼ ptp�a; for all p 2 RTaðkÞ ¼ 0, for all constant functions fðtÞ ¼ k

Taðf=gÞ ¼ gðTafÞ � fðTagÞg2

:

Additively, if f is differentiable, then TaðfÞðtÞ¼ t1�a df

dt ðtÞ:

3. The fractional complex transformation andthe modified simple equation method

Consider the fractional differential equation withconformable derivative:

Fðu;Dat u;D

ax u;D

at D

at u;D

at D

ax u;D

axD

ax u; . . .Þ ¼ 0; 0< a< 1;

(1)

To find the exact solution of Equation (1), the fol-lowing fractional complex transformation can beintroduced:

uðx; tÞ ¼ UðnÞ; n ¼ kxa

a� c

ta

a; (2)

where k and c are constants to be determined.Under the transformation Equation (2), we canrewrite Equation (1) in the following nonlinear ordin-ary differential equation

QðU;U0;U00;U000; . . .Þ ¼ 0: (3)

Then we integrate Equation (3) as many times aspossible with respect to n and set the integrationconstant as zero.

Firstly, according to the modified simple equationmethod (Kaplan, Bekir, Akbulut, & Aksoy, 2015), theexact solution of Equation (3) can be represented bya polynomial in w0 nð Þ

w nð Þ as follows

U nð Þ ¼Xmn¼0

anw0 nð Þw nð Þ

" #n(4)

where an; n ¼ 0; 1; 2; . . .;mð Þ are unknown constantssuch that am 6¼ 0; and w is an unknown function ofn to be calculated. Here the positive integer m,called as the balancing number is determined bythinking the homogeneous balance principlebetween the highest order nonlinear term with thehighest order derivative term which appears inEquation (3).

We obtain a polynomial of w�j nð Þ with the deriva-tives of w nð Þ.by substituting Equation (4) intoEquation (3). Then, by equating the coefficients ofw�j nð Þ to zero, where j � 0; we get a system whichcan be solved to find an n ¼ 0; 1; 2; . . .;mð Þ; c andu nð Þ: Finally we substitute the values of an; k, c andw nð Þ into Equation (4) to find the exact solution ofEquation (1).

4. The fractional complex transformation andthe exp ð�UðnÞÞ method

Secondly, we introduce the expð�UðnÞÞ method forfinding different types of exact solutions to nonlinearfractional differential equations with conformablefractional derivative (Kaplan & Bekir, 2017). To reducethe considering Equation (1) to a nonlinear ordinarydifferential equation, we follow the same procedure.

According to the expð�UðnÞÞ method, we seekthe exact solution of Equation (3) in the followingform:

U nð Þ ¼Xmn¼0

anðexpð�UðnÞÞÞn; (5)

where an am 6¼ 0ð Þ are constants to be determinedlater, and UðnÞ satisfies the following auxiliary ordin-ary differential equation:

U0ðnÞ ¼ expð�UðnÞÞ þ lexpðUðnÞÞ þ k (6)

One can know that the auxiliary equationEquation (6) has different solutions as follows:

Case 1 (Hyperbolic function solutions): When k2 �4l > 0 and l 6¼ 0;

78 M. KAPLAN & A. AKBULUT

U1ðnÞ ¼ ln�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

ptanh

� ffiffiffiffiffiffiffiffiffiffik2�4l

p2 ðnþ CÞ

�� k

2l

0@

1A(7)

Case 2 (Trigonometric function solutions): Whenk2 � 4l< 0 and l 6¼ 0,

U2ðnÞ ¼ ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4l� k2

ptan� ffiffiffiffiffiffiffiffiffiffi

4l�k2p

2 ðnþ CÞ�� k

2l

0@

1A

(8)

Case 3 (Hyperbolic function solutions): When k2 �4l > 0; m¼ 0 and k 6¼ 0,

U3ðnÞ ¼ �lnk

coshðkðnþ CÞÞ þ sinhðkðnþ CÞÞ � 1

� �

(9)

Case 4 (Rational function solutions): When k2 �4l ¼ 0; l 6¼ 0 and k 6¼ 0,

U4ðnÞ ¼ ln � 2ðkðnþ CÞ þ 2Þk2ðnþ CÞ

!(10)

Case 5: When k2 � 4l ¼ 0; l ¼ 0 and k ¼ 0,

U5ðnÞ ¼ lnðnþ CÞ (11)

Here C is the integration constant. Also we bal-ance the highest order linear term with the highestorder nonlinear term in Equation (5) to find the bal-ancing number N.

Substituting Equation (5) into Equation (3) andcollecting all terms with the same order ofexpð�UðnÞÞnðn ¼ 0; 1; 2; . . .Þ together, we get a poly-nomial in expð�UðnÞÞ. Equating each coefficient ofthis polynomial to zero yields a set of algebraicequations for an; k; k; l and c. Solving the equationsystem, we can construct a variety of exact solutionsfor Equation (1).

5. Applications

5.1. Modified simple equation method

The fractional approximate long water wave (ALW)equation is known as

Dat u� uDa

x u� Dax v þ aD2a

x u ¼ 0;Dat v � Da

x uvð Þ � aD2ax v ¼ 0:

(12)

Yan has found three types of travelling wave solu-tions of this equation via the fractional sub-equationmethod (Yan, 2015). Also Guner et al. used ðG0=GÞ�expansion method to establish the exact solutions ofEquation (12). By using the transformation:

uðx; tÞ ¼ UðnÞ; vðx; tÞ ¼ VðnÞ;

n ¼ kxa1

a1� c

ta2

a2;

(13)

Equation (12) reduces a nonlinear ordinary differen-tial equation system, which reads

�cU0 � kUU0 � kV 0 þ ak2U00 ¼ 0;�cV 0 � k UVð Þ0 � ak2V 00 ¼ 0:

Here prime denotes the derivative with respect ton: Then, we can integrate this system once withrespect to n

�cU� k2U2 � kV þ ak2U0 ¼ 0;

�cV � kUV � ak2V 0 ¼ 0:(14)

If we balance the highest-order derivative termU0000 and the non-linear term ðU0Þ2 of Equation (14),we obtain the balancing number as m ¼ 1:

So, we assume that our solution is in the follow-ing form:

U nð Þ ¼ a0 þ a1w0

w

!;

V nð Þ ¼ b0 þ b1w0

w

!þ b2

w0

w

!2 (15)

By substituting Equation (15) into Equation (14)and collecting all the terms with the same power ofe�UðnÞ; ðn ¼ �4;�3; . . .; 0Þ. Then by equating eachcoefficient of the above system to zero yields a setof the following algebraic equations as follows:

From the first equation, we find:

w0 : � ka202

� ca0 � kb0 ¼ 0;

w�1 : �kb1 � ca1 � ka0a1ð Þw0 þ ak2a1w00 ¼ 0;

w�2 : � 12ka21 � ak2a1 � kb2

� �w0� �2 ¼ 0;

(16)

and from the second equation, we get

w0 : �ka0b0 � cb0 ¼ 0;w�1 : �cb1 � ka0b1 � ka1b0ð Þw0 � ak2b1w

00 ¼ 0;

w�2 : �ka0b2 � cb2 � ka1b1 þ ak2b1ð Þ w0� �2 � 2ak2b2w0w00 ¼ 0;

w�3 : �ka1b2 þ 2ak2b2ð Þ w0� �3 ¼ 0:

(17)

Then it is easy to obtain from the first equationsof the systems above:

a0 ¼ 0; b0 ¼ 0;

a0 ¼ � 2ck; b0 ¼ 0;

a0 ¼ � ck; b0 ¼ c2

2k2;

and

a1 ¼ 0; b2 ¼ 0;a1 ¼ �2ak; b2 ¼ 0;a1 ¼ 2ak; b2 ¼ �4a2k3

a0 ¼ � 2ck; b0 ¼ 0;

from the last equations of the systems. Then, weonly consider the result: a1 ¼ 2ak; b2 ¼ �4a2k3

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 79

satisfied. Since a1 6¼ 0; b2 6¼ 0 the first and secondcases are omitted.

Case 1: a0 ¼ 0; b0 ¼ 0:

By substituting these solutions into the remainingequations we get:

b1 ¼ �4ac:

Then we find

w nð Þ ¼ C1 þ C2exp�cnak2

� �:

Therefore, we find the exact solutions of theapproximate long water wave equation with con-formable fractional derivative as

U nð Þ ¼ �2cC2exp

�cnak2

� �

k C1 þ C2exp�cnak2

� �� � ;

V nð Þ ¼4c2C2exp

�cnak2

� �

k2 C1 þ C2exp�cnak2

� �� ��4c2C2

2exp�cnak2

� �

k2 C1 þ C2exp�cnak2

� �� �2 :

where n ¼ k xa1a1

� c ta2a2

(Figures 1 and 2).

Case 2: a0 ¼ � 2ck ; b0 ¼ 0;

By substituting these solutions into the remainingequations we get:

b1 ¼ 4ac:

Then we find

w nð Þ ¼ C1 þ C2expcnak2

� �:

Therefore, we find the exact solutions of theapproximate long water wave equation with

conformable fractional derivative as

U nð Þ ¼ � 2ckþ

2cC2expcnak2

� �

k C1 þ C2expcnak2

� �� � ;

V nð Þ ¼4c2C2exp

cnak2

� �

k2 C1 þ C2expcnak2

� �� ��4c2C2

2expcnak2

� �

k2 C1 þ C2expcnak2

� �� �2 :

where n ¼ k xa1a1

� c ta2a2

(Figures 3 and 4).

Case 3: When a0 ¼ � ck ; b0 ¼ c2

2k2 ; w nð Þ yields anabsurd solution. Hence, the case is discarded.

5.2 Exp (�U(n)) method

According to the expð�UðnÞÞ method, we assumethat the exact solutions of the approximate longwater wave equation with conformable fractionalderivative as

U nð Þ ¼ a0 þ a1expð�UðnÞÞ;V nð Þ ¼ b0 þ b1expð�UðnÞÞ þ b2 expð�UðnÞÞð Þ2:

(18)

Equation (18) is substituted into Equation (12) andthen all the terms with the same power of e�UðnÞ;ðn ¼ �3; . . .; 0Þ are collected. By equating each coef-ficient to zero yields a set of the following algebraicequations respectively:

e�2n : � ka212

� kb2 � ak2a1 ¼ 0;

e�n : �ca1 � kb1 � ak2a1k� ka0a1 ¼ 0;

e0n : �kb0 � ca0 � ak2a1l� ka202

¼ 0:

Figure 1. Graph of the uðx; tÞ corresponding to the values a ¼ 0:5; 1 from left to right when C2 ¼ �1;C1 ¼ 1; c ¼ 1; k ¼ 2; a ¼ 4:

80 M. KAPLAN & A. AKBULUT

and

e�3n : �ka1b2 þ 2ak2b2 ¼ 0;e�2n : 2k2ab2k� cb2 � ka1b1 � ka0b2 þ ak2b1 ¼ 0;e�n : �ka1b0 þ ak2b1kþ 2ak2b2l� cb1 � ka0b1 ¼ 0;e0n : �ka0b0 � cb0 þ ak2b1l ¼ 0:

Then by solving the set of algebraic equations withthe aid of Maple, we get

a0 ¼ 2k26

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

p2

!ak; a1 ¼ 2ak; b0 ¼ �4a2k2l

b1 ¼ �4a2k2k; b2 ¼ �4a2k2; k ¼ k;

(19)

c ¼ ak2 kl� k2 k26

ffiffiffiffiffiffiffiffiffiffik2�4l

p2

� �þ 2l k

26ffiffiffiffiffiffiffiffiffiffik2�4l

p2

� �h i

�lþ k k26

ffiffiffiffiffiffiffiffiffiffik2�4l

p2

� � (20)

Lets us discuss the following different cases:Case 1 (Hyperbolic function solutions): When k2 �

4l > 0 and l 6¼ 0;

u1ðx; tÞ ¼ 2k26

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

p2

!ak

þ 2ak

ln

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

ptanh

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

p2

ðnþ CÞ�� k

2l

!!;

Figure 2. Graph of the vðx; tÞ corresponding to the values a ¼ 0:5; 1 from left to right when C2 ¼ �1; C1 ¼ 1;c ¼ 1; k ¼ 2; a ¼ 4:

Figure 3. Graph of the uðx; tÞ corresponding to the values a ¼ 0:5; 1 from left to right when C2 ¼ �1; C1 ¼ 1;c ¼ 1; k ¼ 2; a ¼ 4.

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 81

v1ðx; tÞ ¼ �4a2k2l

� 4a2k2k

ln

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

ptanh

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

p2

ðnþ CÞ�� k

2l

!!

� 4a2k2 ln

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

ptanh

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

p2

ðnþ CÞ�� k

2l

!!2

;

where

n ¼ kxa1

a1� c

ta2

a2; c ¼ ak2 kl� k2 k

26ffiffiffiffiffiffiffiffiffiffik2�4l

p2

� �þ 2l k

26ffiffiffiffiffiffiffiffiffiffik2�4l

p2

� �h i

�lþ k k26

ffiffiffiffiffiffiffiffiffiffik2�4l

p2

� � ;

C is an integration constant.Case 2 (Trigonometric function solutions): When

k2 � 4l< 0 and l 6¼ 0,

u2ðx; tÞ ¼ 2k26

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

p2

!ak

þ 2ak

ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4l� k2

ptan� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4l� k2p

2ðnþ CÞ

�� k

2l

!!;

v2ðx; tÞ ¼ �4a2k2l

� 4a2k2k

ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4l� k2

ptan� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4l� k2p

2ðnþ CÞ

�� k

2l

!!

� 4a2k2 ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4l� k2

ptan� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4l� k2p

2ðnþ CÞ

�� k

2l

!!2

;

where

n ¼ kxa1

a1� c

ta2

a2; c ¼ ak2 kl� k2 k

26ffiffiffiffiffiffiffiffiffiffik2�4l

p2

� �þ 2l k

26ffiffiffiffiffiffiffiffiffiffik2�4l

p2

� �h i

�lþ k k26

ffiffiffiffiffiffiffiffiffiffik2�4l

p2

� � ;

C is an integration constant.

Case 3 (Hyperbolic function solutions): When k2 �4l > 0; m¼ 0 and k 6¼ 0,

u3ðx; tÞ ¼ 2k26

ffiffiffiffiffik2

p

2

!ak

þ 2ak

� ln

k

coshðkðnþ CÞÞ þ sinhðkðnþ CÞÞ � 1

!!;

v3ðx; tÞ ¼ �4a2k2k � ln

k

coshðkðnþ CÞÞ þ sinhðkðnþ CÞÞ � 1

!!

� 4a2k2

� ln

k

coshðkðnþ CÞÞ þ sinhðkðnþ CÞÞ � 1

!!2

;

where

n ¼ kxa1

a1� c

ta2

a2; c ¼ ak2 �k2 k

26ffiffiffiffik2

p2

� �þ 2l k

26ffiffiffiffik2

p2

� �h i

k k26

ffiffiffiffik2

p2

� � ;

C is an integration constant.Case 4 (Rational function solutions): When k2 �

4l ¼ 0; l 6¼ 0 and k 6¼ 0,

u4ðx; tÞ ¼ kak þ 2ak

ln

� 2ðkðnþ CÞ þ 2Þ

k2ðnþ CÞ

!!;

v4ðx; tÞ ¼ �4a2k2l� 4a2k2k

ln

� 2ðkðnþ CÞ þ 2Þ

k2ðnþ CÞ

!!

�4a2k2 ln

� 2ðkðnþ CÞ þ 2Þ

k2ðnþ CÞ

!!2

;

where

n ¼ kxa1

a1� c

ta2

a2; c ¼

ak2 kl� k2 k2

� �þ 2l k

2

� �h i

�lþ k k2

� � ;

C is an integration constant.

Figure 4. Graph of the vðx; tÞ corresponding to the values a ¼ 0:5; 1 from left to right when C2 ¼ �1;C1 ¼ 1; c ¼ 1; k ¼ 2; a ¼ 4.

82 M. KAPLAN & A. AKBULUT

Case 5: When k2 � 4l ¼ 0; m¼ 0 and k ¼ 0; cyields an absurd solution. Hence, the caseis discarded.

Note that, our solutions are different from thegiven ones in (Guner, Atik, & AytuganKayyrzhanovich, 2017). Also, we can say that the sol-utions obtained in this paper via two different meth-ods are different from each others. If we comparethem, the solution process in the modified simpleequation method is difficult from the expð�UðnÞÞmethod and it gives more fresh solutions. On theother hand, expð�UðnÞÞ method gives the exact sol-utions with different forms.

6. Conclusions

In this work, by use of the fractional calculus for con-formable fractional derivative, and the modified sim-ple equation and a new expð�UðnÞÞ method isproposed to seek exact solutions of the fractional dif-ferential equation systems. The modified simpleequation method has proved its validity in that morefresh solutions for fractional partial differential equa-tion systems can be obtained. Since, the auxiliaryequation of this method is not a solution of any pre-defined functions. Moreover, by using theexpð�UðnÞÞ method more general exact solutionswith different forms including solitary wave solutions,periodic wave solutions and rational solutions. As anapplication the approximate long water wave equa-tion has been considered and abundant exact solu-tions are verified.

Disclosure statement

No potential conflict of interest was reported bythe authors.

ORCID

Melike Kaplan http://orcid.org/0000-0001-5700-9127

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84 M. KAPLAN & A. AKBULUT

ORIGINAL ARTICLE

Exact solutions of the classical Boussinesq system

Hong-Qian Sun and Ai-Hua Chen

College of Science, University of Shanghai for Science and Technology, Shanghai, China

ABSTRACTIn this paper, we study exact solutions of the classical Boussinesq (CB) system, whichdescribes propagations of shallow water waves. By using the bilinear form, with exponentialexpansions, we obtain solitary wave solutions of the CB system. Based on asymptotic analysismethod, we study the elastic and elastic–inelastic–coupled interactions of the obtained soli-tary wave solutions. With extended three–wave method, we obtain the periodic solitary solu-tion of the CB system. And with polynomial expansions, we get the rational solutions of theCB system. These interesting exact solutions may be useful in the study of some phenomenaappeared in shallow water waves.

ARTICLE HISTORYReceived 7 April 2017Accepted 20 August 2017

KEYWORDSClassical Boussinesq system;bilinear form; solitary wavesolution; periodic solitarysolution; rational solution

1. Introduction

The study of the exact solutions has been an inter-esting issue in both experimental and theoreticalresearch, which can explain the nonlinear phenom-ena of fluid dynamics, elastica dynamics, plasma, etc.In recent years, different types of exact solutions ofnonlinear partial differential equations have beenobtained, such as soliton (Cai, Bai, & Luo, 2017; Ekici,Mirzazadeh, & Eslami, 2016; Korkmaz, 2017a, 2017b;Ling, Feng, & Zhu, 2016; Vijayajayanthi, Kanna,Lakshmanan, & Murali, 2016; Wazwaz, 2017), periodicand quasiperiodic solutions (Dai, Lin, Fu, & Zeng,2010; Fan & Hon, 2008; He & Abdou, 2007; Lax, 1976)and rational solutions (L€u, Ma, Chen, & Khalique,2016a; L€u, Ma, Zhou, & Khalique, 2016b; Shi, Zhao, &Ma, 2015; Zhang & Ma, 2015).

To seek the exact solutions of the nonlinear par-tial differential equations, many methods have beenproposed, for instance the inverse scattering trans-formation (IST) (Ablowitz & Segur, 2000), B€acklundtransformation (Rogers & Schief, 2002), Painlev�e ana-lysis method (Chowdhury, 1999), Darboux transform-ation (DT) (Gu, Hu, & Zhou, 2005), Hirota directmethod (Hirota, 2004), the tanh-function method(Parkes & Duffy, 1996; Zhang, Xu, & Li, 2002), theimproved F-expansion method (Islam, Khan, Akbar, &Mastroberardino, 2014; Wang & Zhang, 2005), themodified simple equation method (Akter & Akbar,2015; Khan & Akbar, 2013; Khan, Akbar, & Alam,2013), the (G0=G)-expansion and extend (G0=G)-expan-sion method (Akbar & Ali, 2011; Akbar, Ali, &Mohyud-Din, 2013; Alam & Akbar, 2013, 2015; Alam,

Hafez, Belgacem, & Akbar, 2015b; Zayed & Shorog,2010), the Exp-function method (He & Abdou, 2007;Khan & Akbar, 2014), the generalized Kudryashovmethod (Khan & Akbar, 2016), the expð�UðgÞÞ–expansion and expðUðgÞÞ method (Alam, Hafez,Akbar, & Roshid, 2015a; Roshid & Rahman, 2014), theextended three–wave method (Dai et al., 2010; Li,Dai, & Liu, 2011; Singh & Gupta, 2016; Wang, Dai, &Liang, 2010).

In this paper, based on the bilinear form, we con-sider exact solutions including solitary wave solution,periodic solitary solution and rational solution of theclassical Boussinesq (CB) system (see Wu & Zhang,1996)

ut þ ð1þ uÞv½ �x þ14vxxx ¼ 0;

vt þ vvx þ ux ¼ 0;

8<: (1)

where, u is the elevation of the water wave and vis the surface velocity of water along x-direction.This system was introduced in Wu and Zhang(1996), which is derived from Euler equation. Thissystem can be used to study the run-up of oceanwaves such as tsunami waves on dykes and dams.A good understanding of exact solutions is helpfulin harbour and coastal design. Therefore, findingmore type of solutions is very important in fluiddynamics.

In Li, Ma, and Zhang (2000); Li and Zhang (2001,2003); Zhang, Chang, and Li (2009); Zhang andLi (2003); Zhang, Zhao, and Chen (2015), byusing Darboux transformation, the authors obtainedthe bidirectional solitons on water and the

CONTACT Ai-Hua Chen ahchen79@sina.com College of Science, University of Shanghai for Science and Technology, Shanghai, China� 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

University of BahrainARAB JOURNAL OF BASIC AND APPLIED SCIENCES2018, VOL. 25, NO. 2, 85–91https://doi.org/10.1080/25765299.2018.1449416

elastic–fusion–coupled interaction of the CB system(1). In Roshid and Rahman (2014); Zayed and Shorog(2010); Zhang et al. (2002), by using of Tanh func-tion, extended ðG0=GÞ–expansion and expð�UðgÞÞ–expansion, the authors obtained the travelingwave solutions including hyperbolic solution, trig-onometric solution, periodic solution of the CBsystem (1).

Through the following proper transformation

u ¼ 12lnðfgÞ½ �xx � 1; v ¼ � lnðf=gÞ½ �x; (2)

the CB system (1) can be transformed into the bilin-ear form

Dt � 12D2x

� �f � g ¼ 0; ðDxDt � 1

2D3xÞf � g ¼ 0: (3)

In Section 2, based on the bilinear form in (3) andassuming f and g have exponential expansions, wecan obtain solitary wave solutions of the CB system(1), including the bidirectional soliton and the elas-tic-inelastic-coupled solitary wave solution. By usingasymptotic analysis method (Chakravarty & Kodama,2008; Lambert, Musette, & Kesteloot, 1987; Wang,Tian, Li, Wang, & Jiang, 2014; Zhang & Chen, 2016),we analyze the interactions of the obtained solu-tions. In Section 3, by using extended three–wavemethod, we obtain single soliton solution, periodicsolitary wave solution, and solitary wave solutionwith fission of the CB system (1). In Section 4, takingf and g as polynomial expansions (L€u et al., 2016a,2016b; Shi et al., 2015; Zhang & Ma, 2015), we getrational solutions of the CB system (1). In Section 5,we give our conclusions.

2. Solitary wave solutions of the CB system

In this section, based on parameter expansions andtaking f and g as exponential forms, solitary wavesolutions of the CB system (1) are obtained accord-ing to the bilinear form (3). Then by using asymp-totic analysis method, we study propagations andinteractions of the solitary wave solutions of the CBsystem (1).

In order to get the bilinear derivative Equation (3),we assume that f and g have the following asymmet-ric form expansions of the bookkeeping parameter e

f ¼ 1þ ef1ðx; tÞ þ e2f2ðx; tÞ þ e3f3ðx; tÞ þ � � � ; (4)

g ¼ eg1ðx; tÞ þ e2g2ðx; tÞ þ e3g3ðx; tÞ þ � � � : (5)

Substituting (4) and (5) into the bilinear form (3),we have

f ¼ 1þXNi¼1

hieni ; g ¼XNi¼1

eni þX

1�i<j�N

lijeniþnj ; (6)

where,

ni ¼ kix � k2i2t þ ai; hi ¼ k21

k2i; lij ¼ k21ðki � kjÞ2

k2i k2j

;

(7)

ki and ai ði ¼ 1; 2; � � � ;NÞ are arbitrary constants.Substituting (6) into the transformation (2), we canobtain the multi-solitary wave solutions of the CBsystem (1). In the following discussions, we givedetailed analysis for the cases of N¼ 1, 2, 3.

For N¼ 1, based on (6), we obtain

f ¼ 1þ en1 ; g ¼ en1 ; n1 ¼ k1x � k212t þ a1: (8)

Substituting f and g into (2), we get the singlesoliton solution of (1)

u ¼ k218sech2

n12� 1; v ¼ k1

2� k1

2tanh

n12; (9)

where, u is a bell-soliton solution and v is a kink-soli-ton solution with velocity k1=2 (see Figure 1).

For N¼ 2, we have

f ¼ 1þ en1 þ k21k22

en2 ;

g ¼ en1 þ en2 þ ðk1 � k2Þ2k22

en1þn2 ;

ni ¼ kix � k2i2t þ ai; i ¼ 1; 2:

(10)

Substituting f and g into (2), we get the solution(u, v) of the CB system (1). We consider the collisions

Figure 1. Plots of the single soliton solution (9) of the CB system (1) with k1 ¼ 6; a1 ¼ 3.

86 H.-Q. SUN AND A.-H. CHEN

of the solution u through the asymptotic analysismethod. The analysis for v is similar and is omittedfor brevity. Without loss of generality, we let0< k1 < k2 and we have the following asymptoticexpressions.

1. Before the interactions (t ! �1):

að Þ n1 � 0; n2 � þ1; u�01 ¼k218sech2

n12þ D�

01

� �� 1;

(11)

bð Þ n2 � 0; n1 � �1; u�02 ¼k228sech2

n22þ D�

02

� �� 1;

(12)

cð Þ n2 � n1 � 0; n1 � �1; n2 � �1;

u�12 ¼ðk2 � k1Þ2

8sech2

n2 � n12

� 1;(13)

2. After the interactions (t ! þ1):

að Þ n1 � 0; n2 � �1; uþ01 ¼k218sech2

n12� 1;

(14)

bð Þ n2 � 0; n1 � þ1; uþ02 ¼k228sech2

n22þ Dþ

02

� �� 1;

(15)cð Þ n2 � n1 � 0; n1 � þ1; n2 � þ1;

uþ12 ¼ðk2 � k1Þ2

8sech2

n2 � n12

þ Dþ12

� �� 1;

(16)

where, D�01 ¼ lnððk2 � k1Þ=k2Þ, D�

02 ¼ lnðk1=k2Þ,Dþ02 ¼ lnððk2 � k1Þ=k2Þ, Dþ

12 ¼ lnðk1=k2Þ.Thus, the solution u in (2) has the following

asymptotic expression

u ! u�01 þ u�02 þ u�12; t ! �1;uþ01 þ uþ02 þ uþ12; t ! þ1:

�(17)

From (11)–(17), we find that after the collision, thespeeds and the amplitudes of the three-bell-type sol-itons of u remain unchanged except the phase shiftsand then the collision is elastic (see Figure 2).Therefore, the solution u is a solitary wave solutionwith elastic interaction.

For N¼ 3, based on (6), we obtain

f ¼ 1þ en1 þ k21k22

en2 þ k21k23

en3 ;

g ¼ en1 þ en2 þ en3 þ ðk1 � k2Þ2k22

en1þn2

þ ðk1 � k3Þ2k23

en1þn3 þ ðk2 � k3Þ2k21k22k

23

en2þn3 ;

8>>>>>>>><>>>>>>>>:

(18)

where, ni ¼ kix � tk2i =2þ ai (i¼ 1, 2, 3). Substituting fand g into (2), we get the solution (u, v) of the CBsystem (1). Without loss of generality, we let0< k1 < k2 < k3 and then we have the followingasymptotic expression

u ! u�02 þ ~u�

03 þ u�13; t ! �1;

~uþ01 þ uþ

02 þ ~uþ12 þ uþ

13 þ ~uþ23; t ! þ1;

�

(19)

where,

u�02 ¼

k228sech2

n22þ D

�02

� �� 1;

~u�03 ¼

k238sech2

n32þ ~D

�03

� �� 1;

u�13 ¼

ðk3 � k1Þ28

sech2n3 � n1

2� 1;

(20)

~uþ01 ¼

k218sech2

n12� 1;

uþ02 ¼

k228sech2

n22þ D

þ02

� �� 1;

~uþ12 ¼

ðk2 � k1Þ28

sech2n2 � n1

2þ ~D

þ12

� �� 1;

(21)

uþ13 ¼

ðk3 � k1Þ28

sech2n3 � n1

2þ D

þ13

� �� 1;

~uþ23 ¼

ðk3 � k2Þ28

sech2n3 � n2

2þ ~D

þ23

� �� 1;

(22)

and D�02 ¼ lnðk1ðk3 � k2ÞÞ=k2k3Þ, ~D

�03 ¼ lnðk1=k3Þ,

Dþ02 ¼ lnððk2 � k1Þ=k2Þ, ~D

þ12 ¼ lnðk1=k2Þ, D

þ13 ¼ ln

ðk1ðk3 � k2Þ=k3ðk2 � k1ÞÞ, ~Dþ23 ¼ lnðk2=k3Þ.

From the asymptotic expression in (19), wefind that there are three waves before collision andthere are five waves after collision. Takingk1 ¼ 1; k2 ¼ 5=2; k3 ¼ 7=2, a1 ¼ a3 ¼ 0, for different

Figure 2. Plots for the three-soliton solution (17) of the CB system (1) with k1 ¼ 3; k2 ¼ 5; a1 ¼ �1, a2 ¼ 2.

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 87

values of a2, we will find different interaction phe-nomena. If we let a2 ¼ �20, we find that u�

13 splitsinto ~uþ

12 and ~uþ23; ~u

�03 splits into ~uþ

01 and uþ13; u

�02 is

transformed into uþ02 which is invariant except the

phase shift. If a2 ¼ 0, before and after collision,u�02; u

þ02 and u�

13, uþ13 are the same in velocities and

amplitudes except the phase shifts, and then theinteraction is an elastic collision. After interaction,~u�03 splits into three solitary waves ~uþ

01; ~uþ12 and ~uþ

23,and the interaction is an inelastic collision. Ifa2 ¼ 20, u�

02 splits into ~uþ01 and ~uþ

12; ~u�03 splits into

uþ02 and ~uþ

23; u�13 is transformed into uþ

13 which isinvariant except phase shift. Although there are dif-ferent interaction behaviours depending on the val-ues of a2, there are always three solitary wavesbefore interaction and five after interactions.Therefore, the solution u is a solitary wavesolution with elastic–inelastic–coupled interaction(see Figure 3).

3. Periodic solitary wave solutions of the CBSystem

In this section, based on the bilinear form (3), byusing the extended three–wave method, we find thesingle soliton solution, periodic solitary solution andsolitary wave solution of the CB system (1).

We formulate solutions of the bilinear form (3) as

f ¼ a1eg1 þ a2cosg2 þ a3coshg3 þ e�g1 ;

g ¼ b1eg1 þ b2cosg2 þ b3coshg3 þ b4e�g1 ;(23)

where, gi ¼ kix þ wit, i¼ 1, 2, 3. Substituting (23)into the bilinear form (3) and equating the coeffi-cients of e�g1cosg2; e

�g1coshg3, � � � to zeros, weobtain a system of algebraic equations with theunknowns of ai, ki, wi (i¼ 1, 2, 3) and bj(j ¼ 1; 2; 3; 4) which can be solved by usingMathematica. And we have the following differentsets of solutions.

Case 1. a2 ¼ b1 ¼ b2 ¼ b3 ¼ 0; w1 ¼ k21, k3¼ k1,w3 ¼ k21.

In this case, from (23), we have

f ¼ a1ek1xþk21 t þ a3coshðk1x þ k21tÞ þ e�ðk1xþk21 tÞ;

g ¼ b4e�ðk1xþk21 tÞ:

(24)

Through the transformation (2), we get the solu-tion of the CB system (1)

u ¼ �1þ ð2þ a3Þð2a1 þ a3Þk212 ða1 � 1Þsinhðk1x þ k21tÞ þ ð1þ a1 þ a3Þcoshðk1x þ k21tÞ� �2 ;

v ¼ � ð2a1 þ a3Þk1a1 þ e�2ðk1xþk21 tÞ þ a3e�ðk1xþk21 tÞcoshðk1x þ k21tÞ

:

8>>>><>>>>:

(25)

If we let a1 ¼ 15; a3 ¼ 1; b4 ¼ 2 and k1 ¼ �2, u isa bell-soliton and v is a kink-soliton, whose velocityis different from that of the solution (9).

Case 2. a1 ¼ a2 ¼ a3 ¼ b1 ¼ b3 ¼ 0; w2 ¼ �k1k2,w1 ¼ ðk22 � k21Þ=2.

In this case, from (23), we have

f ¼ e�g1 ; g ¼ b2cosg2 þ b4e�g1 : (26)

And from the transformation (2), we get the solu-tion of the CB system (1)

u ¼ �1þ b2eg1 b2k22eg1 þ 2b4k1k2sing2 þ b4ðk22 � k21Þcosg2

� �2 b2eg1cosg2 þ b4ð Þ2 ;

v ¼ b2eg1ðk1cosg2 � k2sing2Þb2eg1cosg2 þ b4

;

8>>><>>>:

(27)

where, g1 ¼ k1x þ ðk22 � k21Þt=2; g2 ¼ k2x � k1k2t.This is a periodic solitary wave solution includingexponential and trigonometric functions. The contourplots of the solution (27) are shown in Figure 4.

Case 3. a1 ¼ a2 ¼ a3 ¼ b1 ¼ b2 ¼ 0; w1 ¼ �ðk21 þ k23Þ=2,w3 ¼ �k1k3.

Substituting these parameters into (23), we have

f ¼ e�g1 ; g ¼ b3coshg3 þ b4e�g1 : (28)

Figure 3. Contour plots of the solution u in (19) of the CB system (1) with k1 ¼ 1; k2 ¼ 5=2; k3 ¼ 7=2, a1 ¼ a3 ¼ 0.

88 H.-Q. SUN AND A.-H. CHEN

Then by the transformation (2), we get the solu-tion of the CB system (1)

u ¼ �1þ b3eg1 b3k23eg1 þ 2b4k1k3sinhg3 þ b4ðk21 þ k23Þcoshg3

� �2ðb3eg1coshg3 þ b4Þ2

;

v ¼ b3eg1ðk1coshg3 þ k3sinhg3Þb3eg1 coshg3 þ b4

;

8>>><>>>:

(29)

where, g1 ¼ k1x � ðk21 þ k23Þt=2 and g3 ¼ k3x � k1k3t.When we take k1 ¼ 1; k3 ¼ 4; b3 ¼ 1; b4 ¼ 5,this is a solitary wave solution with fission (seeFigure 5).

4. Rational solutions of the CB system

In this section, based on polynomial solutions of thebilinear form (3), we consider rational solutions ofthe CB system (1).

For simplicity, we formulate a polynomial solutionof the standard bilinear form (3) as

f ¼Xni¼0

fiðtÞxi; g ¼ 1: (30)

Substituting (30) into the bilinear form (3), we get

Xni¼0

fi0ðtÞxi � 1

2

Xn�2

i¼0

ði þ 2Þði þ 1Þfiþ2ðtÞxi ¼ 0;

Xn�1

i¼0

ði þ 1Þfiþ10ðtÞxi � 1

2

Xn�3

i¼0

ði þ 3Þði þ 2Þði þ 1Þfiþ3ðtÞxi ¼ 0;

8>>>><>>>>:

(31)

from which we can deduce the general formula of f as

f ¼Xnk¼0

xn�kXai¼0

abiþ1bciþ1ðn� 2i � cÞ!

ða� iÞ!2a�iðn� kÞ! ta�i

!; (32)

where, a ¼ ½k=2�; b ¼ modðk þ 1; 2Þ; c ¼ modðk; 2Þ, aiand bi (i � 0) are arbitrary constants. From the trans-formation (2), we find the rational solutions of theCB system (1).

For n¼ 1, (32) reads

f ¼ a1x þ b1: (33)

Then the rational solution of the CB system (1) is

u ¼ �1� a212ða1x þ b1Þ2

; v ¼ � a1a1x þ b1

: (34)

For n¼ 2, (32) reads

f ¼ a1x2 þ b1x þ a1t þ a2: (35)

The rational solution of the CB system is

u ¼ �1� ð2a1x þ b1Þ22ða1x2 þ b1x þ a1t þ a2Þ2

þ a1a1x2 þ b1x þ a1t þ a2

;

v ¼ � 2a1x þ b1a1x2 þ b1x þ a1t þ a2

:

8>>><>>>:

(36)

For n¼ 3, (32) is

f ¼ a1x3 þ b1x

2 þ ð3a1t þ a2Þx þ b1t þ b2: (37)

And the rational solution of the CB system is

u ¼ �1� ð3a1x2 þ 2b1x þ 3a1t þ a2Þ22ða1x3 þ b1x2 þ ð3a1t þ a2Þx þ b1t þ b2Þ2

þ 3a1x þ b1a1x3 þ b1x2 þ ð3a1t þ a2Þx þ b1t þ b2

;

v ¼ � 3a1x2 þ 2b1x þ 3a1t þ a2a1x3 þ b1x2 þ ð3a1t þ a2Þx þ b1t þ b2

:

8>>>>>>>><>>>>>>>>:

(38)

The contour plots of u in (34), (36) and (38) areshown in Figure 6.

5. Conclusions

In this paper, based on the bilinear form, via theexponential expansion, the expanded three-wavemethod and polynomial expansion, explicit exact sol-utions of the classical Boussinesq system are derived,which include solitary wave solution, periodic solitarysolution and rational solution. By using of theasymptotic analysis method, the interactions of theobtained solitary wave solutions are discussed indetail. In addition, plots of the obtained solutions arerevealed with the help of Mathematica. These

Figure 5. Plots of the solitary wave solution (29) of the CB system (1) with k1 ¼ 1; k3 ¼ 4; b3 ¼ 1; b4 ¼ 5.

Figure 4. Contour plots of the periodic solitary wave solu-tion (27) of the CB system (1) with k1 ¼ �1; k2 ¼ 3,b2 ¼ �2; b4 ¼ 9.

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 89

interesting exact solutions may be useful in thestudy of some phenomena appeared in shallowwater waves.

Acknowledgements

We are most grateful to the anonymous referees for point-ing out many useful references and the help in improvingthe original manuscript.

Disclosure statement

No potential conflict of interest was reported by theauthors.

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ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 91

Volume 25 Issue 2 August 2018

Contents

Arab Journal of Basic and Applied Sciencesالمجلة العربية للعلوم ا� ساسية والتطبيقية

Original articles Evaluation of newly synthesized hydrazones as mild steel corrosion inhibitors by adsorption, electrochemical, quantum chemical and morphological studies Turuvekere K. Chaitra , Kikkeri N. Mohana & Harmesh C. Tandon . . . . . . . . . . . . . . . . . . . . . . . 45 Optimization of methyl esters production from non-edible oils using activated carbon supported potassium hydroxide as a solid base catalyst Abdelrahman B. Fadhil , Akram M. Aziz & Marwa H. Altamer . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 New similarity solutions for the generalized variable-coeffi cients KdV equation by using symmetry group method Rehab M. El-Shiekh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 New interaction solutions and nonlocal symmetries for the (2 + 1)-dimensional coupled Burgers equation Hengchun Hu & Yueyue Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Application of two diff erent algorithms to the approximate long water wave equation with conformable fractional derivative Melike Kaplan & Arzu Akbulut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Exact solutions of the classical Boussinesq system Hong-Qian Sun & Ai-Hua Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85