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Commun. Theor. Phys. 56 (2011) 10091015 Vol. 56, No. 6, December 15, 2011

Application of the Generalized Differential Quadrature Method in Solving Burgers

Equations

R. Mokhtari,1, A. Samadi Toodar,2 and N.G. Chegini2

1Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

2Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran

(Received May 26, 2011; revised manuscript received August 22, 2011)

Abstract The aim of this paper is to obtain numerical solutions of the one-dimensional, two-dimensional and coupledBurgers equations through the generalized differential quadrature method (GDQM). The polynomial-based differentialquadrature (PDQ) method is employed and the obtained system of ordinary differential equations is solved via the totalvariation diminishing RungeKutta (TVD-RK) method. The numerical solutions are satisfactorily coincident with theexact solutions. The method can compete against the methods applied in the literature.

PACS numbers: 02.60.Cb, 02.30.JrKey words: generalized differential quadrature method (GDQM), total variation diminishing RungeKutta

(TVD-RK) method, Burgers equations

1 Introduction

Exact and numerical solutions of the nonlinear par-

tial differential equations and systems of nonlinear partial

differential equations play an important role in the phys-

ical sciences as well as in the engineering fields, see e.g.

Refs. [15] and their bibliographies.

Burgers equation is one of the very few nonlinear par-

tial differential equations that can be solved exactly using

the HopfCole transformation. Nevertheless, numerical

methods for solving such equations have practical signifi-

cance and drawn the attention of many scientists.

Burgers equation was formulated by Bateman in

1915,[6]

and later treated by Burgers.[7]

This equationis also called the nonlinear advection-diffusion equation,

and can be regarded as a qualitative approximation of

the NavierStocks equations. It retains the nonlinear as-

pects of the governing equation in many practical trans-

port problems such as aggregation interface growth, the

formation of large-scale structures in the adhesion model

for cosmology, turbulence transport, shock wave theory,

wave processes in thermo-elastic medium, transport and

dispersion of pollutants in rivers and sediment transport.

One-dimensional Burgers equation is as follows[8]

u

t + uu

x 2u

x2 = 0 , 0 < x < 1 , t > 0 , (1)

with the initial condition

u(x, 0) = f(x) , 0 x 1 ,

where = 1/Re in which Re is the Reynolds number.

Some exact/numerical solutions of the one-dimensional

Burgers equation are obtained by an explicit Backlund

transformation,[9] tanh-coth method,[10] differential trans-

formation method,[11] variational iteration method,[12]

homotopy analysis method,[13] modified local Crank

Nicolson method,[14] element-free characteristic Galerkin

method,[15] least-squares quadratic B-spline finite ele-

ment method,[16] a sixth-order compact finite differencescheme,[17] cubic B-spline quasi-interpolation method,[18]

finite element method,[19] reproducing kernel method[20]

and so on.Two-dimensional Burgers equation is as follows[14]

u

t+ u

u

x+ u

u

y=

2u

x2+

2u

y2,

0 < x , y < 1 , t > 0 , (2)with the initial condition

u(x,y, 0) = f(x, y) , 0 x , y 1 .

Some exact/numerical solutions of two-dimensional Burg-

ers equation reported in the literature are: mod-ified local CrankNicolson method,[14] Adomian de-

composition method,[21] generalized (G/G)-expansion

method,[22] lattice Boltzmann method,[23] Eulerian

Lagrangian method,[24] artificial boundary method[25] and

etc.

The coupled Burgers equation which we consider is as

follows[2627]

u

t

2u

x2+ u

u

x+

(uv)

x= 0 ,

0 < x < 1 , t > 0 ,

v

t

2v

x2+ v

v

x+

(uv)

x= 0 ,

0 < x < 1 , t > 0 , (3)

Corresponding author, E-mail: [email protected]

c 2011 Chinese Physical Society and IOP Publishing Ltd

http://www.iop.org/EJ/journal/ctp http://ctp.itp.ac.cn

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1010 Communications in Theoretical Physics Vol. 56

with the initial conditions

u(x, 0) = f(x) , v(x, 0) = g(x) , 0 x 1 ,

where is a real constant, and are arbitrary constants

depending on the system parameters such as Peclet num-ber, Stokes velocity of particles due to gravity and Brown-

ian diffusivity. Some exact/numerical solutions of coupled

Burgers equations are obtained via differential transfor-

mation method,[11] variational iteration method,[12] anddirect method.[28]

The differential quadrature method was first proposedby Bellman and coworkers[29] and then generalized by Shuand Richards[30] and since then it called GDQM. ABCs

of the GDQM and its development have been explainedin the Shus book.[31] Some recent applications of the

method can be found in Refs. [5,3235] and references

cited therein.In this paper, after applying PDQ method to discretize

the space variable, we use TVD RungeKutta method fordiscretizing the time variable to obtain a fully discretized

scheme to solve one-dimensional, two-dimensional, andcoupled Burgers equations. The paper is followed by il-lustrating some numerical examples and concluded by abrief summary.

2 Analysis of the Method

In this section, we deal with implementing the method

of PDQ and proceed to employ the TVD RungeKutta

method.

2.1 PDQ Method

For one dimensional case, if a function U(x) is suf-ficiently smooth over the interval [xL, xR], its first and

second order derivatives at the grid points xi for i =1, 2, . . . , N are approximated as follows

Ux(xi) =

Nj=1

aijU(xj) , (4)

Uxx(xi) =Nj=1

bijU(xj) , (5)

where aij and bij represent weighting coefficients of the

first and second order derivative approximations, respec-

tively. Following the key idea of PDQ,[31] the weighting

coefficients aij and bij for i, j = 1, 2, . . . , N are determined

as follows

aij =M(1)(xi)

(xi xj)M(1)(xj), i = j , (6)

aii = N

j=1, j=i

aij , (7)

bij = 2aij

aii 1

xi xj

, i = j , (8)

bii = N

j=1, j=i

bij , (9)

where

M(1)(xi) =

Nk=1,k=i

(xi xk) .

Using derivative approximations (4) and (5), Eq. (1)

leads to

u

t(xi, t) = u(xi, t)

Nj=1

aiju(xj , t)+Nj=1

biju(xj , t), (10)

in which i = 1, 2, . . . , N and weighting coefficients aij and

bij are obtained from Eqs. (6)(9). The system of ordi-

nary differential equations (10) is solved by using third

order TVD RungeKutta method.

Furthermore, using derivative approximations (4) and

(5), Eq. (3) leads to

U

t(xi, t) =

Nj=1

bijU(xj , t) + 2U(xi, t)

Nj=1

aijU(xj , t)

V(xi, t)

Nj=1

aijU(xj , t) + U(xi, t)

Nj=1

aijV(xj , t)

,

V

t(xi, t) =

Nj=1

bijV(xj , t) + 2V(xi, t)

Nj=1

aijV(xj , t)

V(xi, t)

Nj=1

aijU(xj , t) + U(xi, t)

Nj=1

aijV(xj , t)

, (11)

in which i = 1, 2, . . . , N and weighting coefficients aij and

bij are obtained from Eqs. (6)(9). The system of ordi-

nary differential equations (11) is solved by using third

order TVD RungeKutta method.

For two-dimensional case, if U(x, y) is a two-

dimensional function defined on a rectangular domain, its

first and second order partial derivatives at the grid points

(xi, yj) for i = 1, 2, . . . , N and j = 1, 2, . . . , M are approx-

imated as follows

Ux(xi, yj) =Nk=1

axkj(xk , yj) , (12)

Uy(xi, yj) =Mk=1

ayik(xi, yk) , (13)

Uxx(xi, yj) =Nk=1

bxkj(xk, yj) , (14)

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No. 6 Communications in Theoretical Physics 1011

Uyy(xi, yj) =Mk=1

byik(xi, yk) , (15)

where axij and ayij represent weighting coefficients of the

first order derivative approximations with respect to x and

y, respectively and bxij and byij represent weighting coeffi-

cients of the second order derivative approximations with

respect to x and y, respectively. Following the key idea

of PDQ,[31] the weighting coefficients axij, ayij, b

xij , and b

yij

are determined as follows

axij =M(1)(xi)

(xi xj)M(1)(xj), i = j ,

axii = N

j=1,j=i

axij ,

i, j = 1, 2, . . . , N , (16)

ayij =P(1)(yi)

(yi yj)P(1)(yj), i = j ,

ayii = M

j=1,j=i

ayij ,

i, j = 1, 2, . . . , M , (17)

bxij = 2axij

axii

1

xi xj

, i = j ,

bxii = N

j=1,j=i

bxij ,

i, j = 1, 2, . . . , N , (18)

byij = 2ayij

ayii

1

yi yj

, i = j ,

byii = M

j=1,j=i

byij,

i, j = 1, 2, . . . , M , (19)

where

M(1)(xi) =N

k=1,k=i

(xixk) , P(1)(yi) =

Mk=1,k=i

(yiyk) .

Using derivative approximations (12)(15), Eq. (2)

leads to

u

t(xi, yj, t) = u(xi, yj , t)

Nk=1

axiju(xk, yj , t)

u(xi, yj , t)Mk=1

ayiju(xi, yk, t)

+

Nk=1

bxiju(xk, yj , t)

+

Mk=1

byiju(xi, yk, t) , (20)

in which i = 1, 2, . . . , N , j = 1, 2, . . . , M , and the weight-ing coefficients axij, a

yij , b

xij , and b

yij are obtained from

Eqs. (16)(19). The system of ordinary differential equa-

tions (20) is solved by using third order TVD Runge

Kutta method.

2.2 TVD RungeKutta Method

The TVD RungeKutta method is used to solve a sys-

tem of ordinary differential equations such as

ut = L(u) , (21)

resulting from a method of lines approximation of the hy-perbolic conservation law ut = f(u)x, where the spatialderivative f(u)x is approximated by a TVD finite differ-

ence or finite element approximation denoted by L(u),which has the property that the total variation of the nu-merical solution T V(u) = j |uj+1 uj|, does not in-crease, i.e. T V(un+1) T V(un).Proposition 1 (Ref. [36]) The optimal third order TVD

RungeKutta method for solving Eq. (21) is given by

u(1) = un + tL(un) ,

u(2) =

3

4un +

1

4u(1) +

1

4tL(u(1)) ,

un+1 =

1

3un +

2

3u(2) +

2

3tL(u(2)) , (22)

with a CourantFriedrichsLevy (CFL) coefficient c = 1.

3 Numerical Examples

In this section we apply the polynomial based differ-ential quadrature method (PDQ) to different examples.

Accuracy of the method is measured by using the maxi-

mum and relative error norms which are defined by

E = max0jN

{|(u(xj , tn) U(xj , tn))|} ,

E =

N

j=1(u(xj , tn) U(xj , tn))2

Nj=1(U(xj , tn))

2,

where u(xj , tn) and U(xj , tn) are the exact and numerical

solutions at space xj and time tn, respectively.

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3.1 One-Dimensional Burgers Equation

Example 1 Consider the one-dimensional Burgers equa-

tion (1), with the solitary wave solution[3738]

u(x, t) = c/ + (2/) tanh(x ct) ,

where and c are arbitrary constants. Initial and bound-

ary conditions are established from the exact solution. In

Table 1, numerical results are compared with the methodof Chebyshev spectral collocation (CSC).[38]

Table 1 Comparison of numerical solutions of Exam-ple 1 by maximum and relative error norms for variousvalues of and at time t = 0.25 and for x [0, 1].

Method ||E|| ||E||

0.01 7.64105 5.49104

PDQ 1 0.001 7.68107 5.93106

0.0001 7.69109 5.98108

0.01 7.62105 5.04104

CSC[38] 0.001 7.82107 5.85106

0.0001 8.94108 4.31107

0.01 7.64104 5.49104

PDQ 0.1 0.001 7.68106 5.93106

0.0001 7.69108 5.98108

0.01 7.62104 5.04104

CSC[38] 0.001 7.99106 5.88106

0.0001 1.67106 4.31107

It must be pointed out that in Ref. [39], GDQM has

been applied to discretize a similar problem. Unfortu-

nately, established numerical results do not possess high

accuracy because of using the explicit Euler method in

solving the resulting system of differential equations. We

solve this system through TVD-RK to obtain more accu-

rate numerical results.

Example 2 Consider the one-dimensional Burgers equa-

tion (1), with the initial condition

u(x, 0) = sin(x) , 0 x 1 ,

and the boundary conditions u(0, t) = u(1, t) = 0. The

exact solution was found in terms of the infinite series by

Cole[40] as

u(x, t) = 2j=1 jaj sin(jx)exp(j

22t)

a0 + 2j=1 aj cos(jx)exp(j

22t) ,

where

a0 =

10

exp[(2)1(1 cos(x))]dx ,

aj = 2

10

exp[(2)1(1 cos(x))] cos(jx)dx .

In Table 2, numerical results at time t = 0.1 with

= 1 and parameters x = 0.1 and t = 0.001

are compared with the methods of cubic B-spline quasi-

interpolation (CBQI),[41] cubic B-spline (CB),[42] implicit

finite difference (IFdM),[43] boundary element (BEM),[44]

modified Adomian decomposition (ADM),[45] finite dif-

ference (DFDM),[46] least-squares quadratic B-spline fi-

nite element (BFEM),[16] and local discontinuous Galerkin

(LDG).[47] In Table 3, numerical results of PDQ method

at time t = 1 and = 0.1 with parameters x = 0.05

and t = 0.001 are compared with the methods of cubic

B-spline quasi-interpolation (CBQI),[41] implicit finite dif-

ference (IFDM),[43] boundary element (BEM),[44] Hon and

Maos scheme (HM),[48] multiquadric quasi-interpolation

(MQQI)[49] and local discontinuous Galerkin (LDG).[47]

In Fig. 1, we display the numerical results with t = 0.1

for = 1, 0.1. And also the numerical results of the PDQmethod, for t = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 together with the

initial data, are illustrated in Fig. 2, which correspond to

= 0.1, 0.01, and 0.0001.

Fig. 1 (Color online) Numerical results at t = 0.1 for = 1 (a) and = 0.1 (b).

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Fig. 2 (Color online) Numerical results at t = 1, h = 0.05 for = 0.1 (a), = 0.01 (b), and h = 0.03 and= 0.0001 (c).

Table 2 Comparison of numerical results of Example 2 at time t = 0.1 for = 1.

Method x = 0.1 x = 0.3 x = 0.5 x = 0.7 x = 0.9

PDQ (present) 0.109 53 0.291 89 0.371 57 0.309 90 0.120 68

CBQI[41] 0.109 51 0.291 82 0.371 47 0.309 81 0.120 65CB[42] 0.109 52 0.291 84 0.371 49 0.309 83 0.120 65

IFDM[43] 0.110 09 0.293 35 0.373 42 0.311 44 0.121 28

BEM[44] 0.109 31 0.291 24 0.370 70 0.309 11 0.120 31

ADM[45] 0.109 81 0.292 62 0.372 49 0.310 66 0.120 98

BFEM[16] 0.109 78 0.292 38 0.372 12 0.310 44 0.120 97

DFDM[46] 0.109 47 0.291 70 0.371 33 0.309 70 0.120 61

LDG (k = 1)[47] 0.109 54 0.291 89 0.371 57 0.309 90 0.120 68

LDG (k = 2)[47] 0.109 54 0.291 89 0.371 57 0.309 90 0.120 69

Exact 0.109 54 0.291 90 0.371 58 0.309 91 0.120 69

Table 3 Comparison of numerical results of Example 2 at time t = 1 for = 0.1.

Method x = 0.1 x = 0.3 x = 0.5 x = 0.7 x = 0.9

PDQ (present) 0.066 31 0.192 78 0.291 91 0.308 09 0.146 06

CBQI[41] 0.066 28 0.192 69 0.291 75 0.307 91 0.145 83

HM[48] 0.0664 0.1928 0.2919 0.3079 0.1459

IFDM[43] 0.066 89 0.194 45 0.294 48 0.311 07 0.147 69

BEM[44] 0.066 44 0.192 63 0.291 39 0.307 11 0.145 07

MQQI (c = 7.2 103)[49] 0.071 24 0.193 39 0.285 17 0.292 88 0.1354 2

LDG (k = 1)[47] 0.066 31 0.192 78 0.291 91 0.308 08 0.146 06

LDG (k = 2)[47] 0.066 32 0.192 78 0.291 91 0.308 09 0.146 06

Exact 0.066 32 0.192 79 0.291 91 0.308 10 0.146 06

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3.2 Two-Dimensional Burgers Equation

Example 3 Consider the two-dimensional Burgers equa-

tion (2), with the following exact solution[50]

u(x, y, t) =1

1 + exp((x + y t)/2).

The initial and boundary conditions are taken from the

exact solution. In Table 4, numerical results at times

t = 0.05 and t = 0.25 with = 0.1, 0.01 and parame-ters N = M = 10 and t = 0.001 are compared with the

methods of Chebyshev spectral collocation (CSC)[38] and

lattice Boltzmann (LBM).[23]

Table 4 Comparison of numerical results of Example 3at different times.

Method N M t t ||E||10 10 0.05 0.005 0.1 2.70105

PDQ10 10 0.05 0.0005 0.01 2.73107

10 10 0.25 0.005 0.1 1.20104

10 10 0.25 0.0005 0.01 1.21106

10 10 0.05 0.005 0.1 1.28106

CSC[38]

30 30 0.05 0.0005 0.01 4.14105

30 30 0.25 0.005 0.1 4.32103

10 10 0.25 0.1 1.06102

LBM[23] 20 20 0.25 0.1 3.07103

80 80 0.25 0.01 5.89102

3.3 Coupled Burgers Equations

Example 4 Consider the coupled Burgers equations (3)

for = 2 and different values of and at times t = 0.5

and t = 1.0. The exact solution of the equation is given

by Ref. [37] as

u(x, t) = a0(1 tanh(A(x 2At))) ,

v(x, t) = a0

2 12 1

tanh(A(x 2At))

, (23)

where a0 = 0.05 and A = a0(4 1)/(4 2). The

initial and boundary conditions are taken from the ex-

act solutions. In Tables 5 and 6, numerical results are

compared with the methods of Chebyshev spectral collo-

cation (CSC),[38] Fourier pseudo-spectral (FPM)[51] and

cubic B-spline collocation (CBC).[52] In Fig. 3, we display

the numerical and the exact solutions for u and v values

when N = 10, = 0.1, t = 1, = 1, = 2, and a = 0.1.

Table 5 Comparisons of errors at different times foru(x, t) of Example 4.

Method t ||E|| ||E||0.5 0.1 0.3 1.00104 2.02103

PDQ0.3 0.03 2.52104 5.07103

1 0.1 0.3 2.01104 4.03103

0.3 0.03 5.04104 1.00102

FPM[51] 0.5 0.1 0.3 9.619104 3.245105

0.3 0.03 4.310104 2.733105

1 0.1 0.3 1.153103 2.405105

0.3 0.03 1.268103 2.832105

0.5 0.1 0.3 4.38105 1.44103

CSC[38] 0.3 0.03 4.58105 6.68104

1 0.1 0.3 8.66105 1.27103

0.3 0.03 9.16105 1.30103

0.5 0.1 0.3 4.16105 6.73104

CBC[52]0.3 0.03 4.59105 7.32104

1 0.1 0.3 8.25105

1.32103

0.3 0.03 9.18105 1.45103

Table 6 Comparisons of errors at different times forv(x, t) of Example 4.

Method t ||E|| ||E||

0.5 0.1 0.3 3.80105 1.56103

PDQ0.3 0.03 1.85104 1.59103

1 0.1 0.3 7.58105 3.10103

0.3 0.03 3.67104 3.15103

0.5 0.1 0.3 3.33104 2.74105

FPM[51] 0.3 0.03 1.14103 2.45104

1 0.1 0.3 1.16103 3.74105

0.3 0.03 1.63103 4.52104

0.5 0.1 0.3 4.9910

5 5.4210

4

CSC[38]0.3 0.03 1.81104 1.20103

1 0.1 0.3 9.92105 1.29103

0.3 0.03 3.62104 2.35103

0.5 0.1 0.3 1.48104 9.04104

CBC[52]0.3 0.03 5.72104 1.59103

1 0.1 0.3 4.77105 1.25103

0.3 0.03 3.61104 2.25103

Fig. 3 (Color online) Numerical and exact solutions of Example 4 for N = 10, = 0.1, t = 1, = 1, = 2, and a = 0.1.

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4 Summary

In this paper, the polynomial-based generalized dif-

ferential quadrature method is applied to the Burgers

and coupled Burgers equations and the obtained system

of ordinary differential equations is solved via the TVD

RungeKutta method. In comparison with methods ap-

plied in the literature such as spectral methods, this ap-

proach is conservative and produce reasonable numerical

results. As mentioned in Ref. [53], not only the applied

method achieves the high accuracy and spectacular con-

vergence rates of spectral methods but also its particular

advantage lies in its ease of implementation and its abil-

ity to use more general approximating polynomials, i.e.

the standard orthogonal polynomials of spectral methods

need not be used.

References

[1] C.Q. Dai, Q. Yang, and Y.Y. Wang, Commun. Theor.Phys. 55 (2011) 622.

[2] R. Mokhtari, Commun. Theor. Phys. 55 (2011) 204.

[3] R. Mokhtari and M. Mohammadi, Int. J. Nonlinear Sci.Numer. Simul. 10 (2009) 779.

[4] R. Mokhtari and M. Mohammadi, Comput. Phys. Com-mum. 181 (2010) 1266.

[5] R. Mokhtari, A. Samadi Toodar, and N.G. Chegini, Chin.

Phys. Lett. 28 (2011) 020202.[6] H. Bateman, Mon. Weather Rev. 43 (1915) 163.

[7] J.M. Burgers, Adv. Appl. Mech. 1 (1948) 171.

[8] J.M. Burgers, The Nonlinear Diffusion Equation, Reiedl,Dordtrecht (1974).

[9] Z.S. LU, Commun. Theor. Phys. 44 (2005) 987.

[10] A.M. Wazwaz, Appl. Math. Comput. 190 (2007)11981206.

[11] R. Abazari and A. Borhanifar, Appl. Math. Comput. 59(2010) 2711.

[12] M.A. Abdou and A.A. Soliman, Appl. Math. Comput.181 (2005) 245251.

[13] M.M. Rashidi, G. Domairry, and S. Dinarvand, Appl.Math. Comput. 14 (2009) 708717.

[14] P. Huang and A. Abduwali, Appl. Math. Comput. 59(2010) 2452.

[15] X.H. Zhang, J. Ouyang, and L. Zhang, Eng. Anal. Bound-ary Elem. 33 (2009) 356.

[16] S. Kutluay, A. Esen, and I. Dag, Appl. Math. Comput.167 (2004) 21.

[17] M. Sari and G. Gurarslan, Appl. Math. Comput. 208(2009) 475.

[18] C.G. Zhu and R.H. Wang, Appl. Math. Comput. 208(2009) 260.

[19] E.N. Aksan, Appl. Math. Comput. 170 (2005) 895.

[20] S.S. Xie, S. Heo, S. Kim, G. Woo, and S. Yi, J. Comput.Appl. Math. 214 (2008) 417.

[21] H. Zhu, H. Shu, and M. Ding, Appl. Math. Comput.60

(2010) 840.

[22] L.L. Zhang and X.Q. Liu, Commun. Theor. Phys. 52(2009) 784.

[23] Y. Duan and R. Liu, Appl. Math. Comput. 206 (2007)432.

[24] D.L. Young, C.M. Fan, S.P. Hu, and S.N. Atluri, Eng.Anal. Boundary Elem. 32 (2008) 395.

[25] X. Wu and J. Zhang, Appl. Math. Comput. 56 (2008)242.

[26] S.E. Esipov, Phys. Rev. E 52 (1995) 3711.

[27] C.J. Bing, G.X. Guo, and Q.Z. Jun, Chin. Phys. 19 (2010)090403.

[28] L. Huang, Commun. Theor. Phys. 45 (2006) 781.

[29] R. Bellman, B.G. Kashef, and J. Casti, J. Comput. Phys.10 (1972) 40.

[30] C. Shu and B. Richards, Int. J. Numer. Methods Fluids15 (1992) 791.

[31] C. Shu, Differential Quadrature and Its Application inEngineering, Springer-Verlag London Limited (2000).

[32] G. Meral and M. Tezer-Sezgin, Int. J. Numer. Meth.Biomedical Eng. 27 (2011) 485.

[33] G. Gurarslan and M. Sari, Int. J. Numer. Meth. Biomed-ical Eng. 27 (2011) 69.

[34] Y. Zhao, Z. Zong, and Z.R. Li, Appl. Math Mech.-ENGLISH EDITION 32 (2011) 349.

[35] Q.A. Shen, Int. J. Comput. Math. 88) 285.

[36] S. Gottlieb and C.W. Shu, Math. Comput. 67 (1998) 73.

[37] A.A. Soliman, Phys. Lett. A 361 (2006) 394.

[38] A.H. Khater, R.S. Temsah, and M.M. Hassan, Appl.Math. Comput. 222 (2008) 333.

[39] M. Mestrovic, Generalized Differential QuadratureMethod for Burgers Equation, Paper No. PVP2003-1905pp. 211-214, doi:10.1115/PVP2003-1905 ASME 2003 Pre-ssure Vessels and Piping Conference (PVP2003) July 20-24, 2003, Cleveland, Ohio, USA.

[40] J.D. Cole, Quart. Appl. Math. 9 (1951) 225.[41] C.G. Zhu and R.H. Wang, Appl. Math. Comput. 208

(2009) 260.

[42] I. Dag, D. Irk, and B. Saka, Appl. Math. Comput. 163(2005) 199.

[43] A.R. Bahadir, Int. J. Appl. Math. 8 (1999) 897.

[44] A.R. Bahadir and M. Saglam, Appl. Math. Comput. 160(2005) 663.

[45] S. Abbasbandy and M.T. Darvishi, Appl. Math. Comput.163 (2005) 1265.

[46] K. Pandey, L. Verma, and A.K. Verma, Appl. Math. Com-put. 215 (2009) 2206.

[47] Z.G. Zhong, Y.X. Jun, and W. Di, Appl. Math. Comput.216 (2010) 3671.

[48] Y.C. Hon and X.Z. Mao, Appl. Math. Comput. 95 (1998)37.

[49] R. Chen and Z. Wu, Appl. Math. Comput. 172 (2006)472.

[50] O. Goyon, Int. J. Numer. Methods Fluids 22 (1996) 937.

[51] A. Rashid and A.I.B.M. Ismail, Comput. Methods Appl.Math. 9 (2009) 412420.

[52] R.C. Mittal and G. Arora, Commun Nonlinear Sci. Nu-mer. Simul. 16 (2011) 1304.

[53] G. Mansell, W. Merryfield, B. Shizgal, and U. Weinert,

Comput. Meth. Appl. Mech. Eng. 104 (1993) 295.

[quadratura] 11216

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