2013lectures 8a odes
TRANSCRIPT
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Ordinary Differential Equations
=
+ + =
= + + =
2
2
2
Example falling brick
1/ 2
Order since highest derivative is
Spring mass damper
0
Order
Set of 1st order ODEs
, 0
dv Kg vdt m
d x dx m c Kx
dt dt
dx dv v m cv Kx
dt dt
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Sets of ODEs
So ODE problems can be reduced to a set of N first-order
ODEs
Not completely specified needs boundary conditions initial value problems
boundary value problems
Niyyyxfdx
xdyNi
i ,...,1),...,,()(
21 == equationsfor
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General Procedure
Re-write the dyand dxterms as yand x and multiply
by x
Literally doing this is Eulers method
Niyyyxfdx
xdyNi
i ,...,1),...,,()(
21 == equationsfor
xyxfyy
yxfx
xy
yxf
dx
xdy
iiii +=
=
=
+ ),(
),()(
),()(
1
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Tank mixing problem
( )
( ) tccV
Vcc
tccV
V
dt
tdc
iinin
ii
inin
+=
=
+
tank
tank
&
&
1
)()(
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Mixing tank
0.0185
0.05515
0.391000.1950
0.61150
1.4300
0.0113
0.03610
0.1130
Error Et
at t=600
t
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Quantify the error
( ) ( )
order?iserrorSo
oncentredforseriesTaylorWrite
...6
)(
2
)()()()(
321
1
+
+
++=+
+
ttc
ttc
ttctctc
cc
iiiii
ii
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Not quite accumulation of errors
( )
( ) Ntc
t
tc
EEEEE
N
i
Ntotal
2
2
321
2
2
)(
...
=
++++=
errordaccumulatetotalsoerror,thishasstepeachBut
order1stwaserrorthethatbeforesawWe
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Euler Error
Local error
lk=yk-y*k-1(tk)
Global error
ek=yk-y(tk)
y
t
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Euler Stability (1)
0
1
0
( )
Euler
(1 )
So that
(1 )
x
i i i
i
k
k
dyy y x y e
dx
y y y x
y x
y y x
+
= =
= +
= +
= +
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Euler Stability (2)
0 0Real solution ( ) Euler (1 )
For negative
If small
If big
x k
k
dyy y x y e y y x
dx
x
x
= = = +
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In terms of integration
dxyxfxyxxy
dxyxfxyxxy
dxyxfdxdx
dy
xxx
yxf
dx
dy
xx
x
xx
x
xx
x
xx
x
+
+
++
+=+
=+
=
+
=
),()()(
),()()(
),(
),(
fromsidesbothIntegrate
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Runge-Kutta
slope)a(akafunctionincrementis
formGeneral
integraltheevaluatetoquadratureGaussianuseOr
betweenlocationsatevaluatedfofvaluesonbased
integralforfitl)(polynomiaorderhigheruseCould
hyy
hxx
dxyxfxyhxyyxfdx
dy
ii
hx
x
+=
+
+=+=
+
+
1
),()()(),(
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R-K General form
)...,(
),(
),(
),(
,,,
11,122,111,11
22212123
11112
1
21
2211
1
hkqhkqhkqyhpxfk
hkqhkqyhpxfk
hkqyhpxfk
yxfk
aaa
where
kakaka
hyy
nnnnninin
ii
ii
ii
n
nn
ii
+
+++++=
+++=
++=
=
+++=
+=
M
K
K
constants
:asWrite
formGeneral
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R-K 1st Order Form
( ) hyxfyy
yxfka
ka
hyy
iiii
ii
ii
),(1
),(constant1
where
formGeneral
1
1
1
11
1
+=
=
=
=
+=
+
+
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RK 4th Order( )
),(
)2
1,2
1(
)2
1,
2
1(
),(
226
1
34
23
12
1
43211
hkyhxfk
hkyhxfk
hkyhxfk
yxfk
hkkkkyy
ii
ii
ii
ii
ii
++=
++=
++=
=
++++=+y(x)
xi xi+1 x
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RK4 if f(x,y)=f(x)
( )
( )hhxfhxfxf
hkkkkdxxf
dxxfyyxfdx
dy
xfyf
iii
hx
x
hx
xii
)()2/(4)(6
1
226
1)(
)()(
)(),(
4321
1
++++=
+++=
+==
=
+
+
+
hasRK4
If
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Example y=x+y, y(0)=0
4.3896.3935.7155.6545.0503.2501.8
3.2505.0524.4984.4483.9532.3531.6
2.3533.9553.5013.4613.0551.6551.4
1.6553.0572.6852.6522.3201.1201.2
1.1202.3222.0171.9901.7180.7181
0.7181.7201.4701.4481.2260.4260.8
0.42551.2271.0231.0040.8220.2220.6
0.22210.8230.6560.6410.4920.0920.4
0.09180.4930.3560.3440.2210.02140.2
0.021400.2220.110.1000
yn=y+1/6(k
1+2k
2+2k
3+k
4)hk
4=f(x+h,y+hk
3)k
3=f(x+h/2,y+h/2k
2)k
2=f(x+h/2,y+h/2k
1)k
1=f(x,y)yx
0
0.5
1
1.5
2
2.5
3
3.5
0 0.5 1 1.5 2
x
y
Euler
analytical
RK4