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Ordinary Differential Equations

00 )0(,)0(with 0),,( yyyyxyyxfy ′=′=>′=′′

Initial Value Problem (IVP)

Boundary Value Problem (BVP)

ba ybyyaybxayyxfy ==<<′=′′ )(,)(with ),,(

Initial Value Problem,1st order

0)0(with 0),( yytytfy =>=′

htyhtyy )()( ofion approximat difference forward −+

=′

),()( )( ythftyhty +=+⇒

within solution for Seek 0 nttt ≤≤

Mtthhtt

Miihtt

nii

i

01

0

and

,...,1,0−

==−

=+=

+

ionapproximatEuler ),( 1 iiii ythfyy +=⇒ +

2

Example

22),(

1 with 30for )( Find

1)0( 2

1iiii

iiiiiytytyythfyy

htty

yyty

+=

−+=+=

=≤≤

=−

=′

+

(0.75+2)/2=1.37533(1+0.5)/2=0.7522(0+1)/2=0.511100yitii

As h gets smaller we approach the exact solution.

Geometric interpretation of Euler’s Method

Φ

Step size, h

t

y

t0,y0

True value

y1, Predictedvalue

3

2),(

13)(

1)0(2

1ii

iiiiiytyythfyy

htty

yyty

−+=+=

=≤≤

=−

=′

+

with 0for Find

4

5

Error in Euler

Euler approximation

→ Local discretization error

→ Global discretization error

Final global error:

6

How to improve accuracy of Euler’s Method?

( ) ...!32

)()(32

+′′′+′′+′+=+ ηyhyhyhxyhxy

Consider Taylor series

and compute the derivatives as

.....

( ) ( )ηyhfffhhfxyhxy yx ′′′++++=+!32

)()(32

For Taylor’s formula of order N

Local discretization error = O (hN+1)

Global discretization error = O (hN)

7

)4(4

)3(3

)2(2

!4!32)()( yhyhyhyhxyhxy +++′+=+

Final global error:

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Taylor’s method is cumbersome from numerical point of view since higher derivatives need to be calculated.Alternative way to improve accuracy is to use several function evaluations:

slope at the beginning of

step

slope at the end of step

↓↓

Local DE Global DE

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Modified Euler Method(use two slopes sequentially)

Runge-Kutta Method : accuracy of Taylor N=4, no high derivatives,

several function evaluations

Find ai, bi by matching the Runge-Kutta method to N=4 Taylor method.This results in 11 equations for 13 unknowns.

2 of ai, bi are selected and the rest are solved in terms of the selected ones.

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11

error in Simpson ~ O (h5); accumulated error in Runge-Kutta after M steps ~ O (h4 )

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For k2,k3,k4=0 we recover Euler’s method

Remark:

Find A, B, P, Q by matching the Runge-Kutta method to N=2 Taylor method:

let

→

}

→

13

We need to select one of A,B,P or Q

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Sytem of ODEs

within solution for Seek 0 nttt ≤≤

Mtthhtt

Mkkhttn

kk

i

01

0 ,...,1,0−

==−

=+=

+ and

Euler’s approximation

Runge-Kutta method of order=4 (RK4)

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Higher order ODEs

Reduce the ODE to a system of lower order ODEs

→

16

Boundary Value Problem (BVP)

Linear BVP

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Solution of

is given as

where u and v are the solutions of the following IVPs:

proof:

→

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