010.141 NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS MODULE 6

ENGINEERING MATHEMATICS II

010.141

NUMERICAL METHODS FOR

DIFFERENTIAL EQUATIONS

MODULE 6

INTRODUCTION

Ø The methods which we will cover are:

• Methods for First-Order Differential Equation o Euler Method

o Improved Euler Method o Runge-Kutta

o Adams-Bashforth o Adams-Moulton

• Methods for Higher Order Equations and for Systems

METHODS FOR FIRST-ORDER DIFFERENTIAL EQUATION

In this section, we will assume that the problem to be solved is expressed as:

y' = f ( x, y ) y ( x₀ ) = y₀ Initial Value Problem and f is such that the problem has a unique solution.

We shall start with step-by-step methods, i.e., x₀ ® x₁ = x₀ + h ® x₂ = x₀ + 2h, ×××

where "h" is the step size.

METHODS FOR FIRST-ORDER DIFFERENTIAL EQUATION (cont)

Step-by-step methods' formulation is derived from Taylor series

(

)

( )

( )

( )

y x h y h

x y' x

y h

x y

2

EULER'S METHOD

For instance, the Euler method also called Euler-Cauchy is derived directly from the first two terms of the Taylor expansion:

( )

( ) ( ) ( )

( ) ^{x h f} ( ^{x, y} )

y

x y' h x

y h

x y

y x, f y

+

»

+

» +

¢ =

EULER'S METHOD

The iteration process follows:

( )

( )

( ) ( ) ( )

( )

(

)

1 2

0 0

0 1

0 0

0 0

1

0 0

y , x f h y

y

y , x f h y

y

y , x f h x

y h

x y y

y 0

y i.e.

known y

+

=

+

=

+

= +

=

=

®

(

)

n 1

n

y h f x , y

y

= +

EULER'S METHOD

EULER'S METHOD

Example

Consider the D.E.

wyth i.c. y(0) = 0

This D.E. has solution:

y(x) = e^x – x - 1

EULER

y dx x

dy = +

EULER: TABLE 19.1 h = 0.2

Note: exact solution y = e¹ – 1 – 1 = 0.718 Approx. is 0.489

error 0.229 ® significant

A priori, one would want to create methods which use more terms of the Taylor series

where

IMPROVED EULER (HEUN'S METHOD)

( )

f f

f

x y y

f x

y f x f

y

x + ×

=

¶

× ¶

¶ + ¶

¶

= ¶ ,

'

( ) ( ) ( ) ( )

( ) ( ) ( )

L

+ +

+

=

+ +

+

= +

y , x ' 2 f y h

x, hf x

y

x

"

2 y x h

hy' x

y h

x y

2 2

Obtaining the derivatives of f is time consuming and cumbersome.

Instead an another approach is preferred where intermediate values are introduced to increase accuracy.

Uses Predictor - Corrector Method Predictor:

Corrector:

IMPROVED EULER METHOD

(

)

n 1

n

y h f x , y y

= +

( ) ( )

[

]

+₁ = _n + _{n n} + _{n 1 n 1}

n f x ,y f x ,y

2 y h

y

Improved Euler is called a predictor-corrector method, i.e., one step predicts, one step predicts.

Example: Consider:

IMPROVED EULER METHOD

( ) ^{0 0}

y

y dx x

dy

= +

=

IMPROVED EULER METHOD (cont)

( ) ( )

[ ]

[ ]

n+1 n 2 n n n+1 1

n 2 n n n n n n

n 2 n n n n n n

n 2 n n

n

y = y x + y + x + y , y = x + y = y x + y + x + h + y + h x + y

= y x + y + x + y + 0.2x + 0.2y , h = 0.2 = y 2.2x + 2.2y + 0.2

= y

h

n h

h

h

*

+ ¢

é ù

+ ë û

+ éë ùû

+ + +

( )

( )

n n

n n n

n+1 n n n

0.22x + 0.22y + 0.02

= y 0.22 x + y + 0.02 Improved Euler y = y 0.2 x + y Euler

+ +

ILLUSTRATION – TABLE 19.3

( )

( ) ( )

Define:

y = y h f x , y

y = y f x , y + f x , y

n n n

n n n n+1

n

n h

n +

*

+ *+

+

+ é

ëê

ù ûú

1

1 2 1

ILLUSTRATION – TABLE 19.3 (cont)

( ) ( )

[ ]

[ ]

( )

y = y x + y + x + h + y + h x + y = y .1 2.2x + 2.2y + .2

= y 0.22 x + y + .02

n+1 n n n n n n n

n n n

n n n

+ + +

h 2

0 0

0

RUNGE-KUTTA

( )

( )

( )

0 0

n+1 0

n+1

1 n n

2 n n 1

3 n n 2

= f x, y ; x = x , y = y x = x + n + 1

Find y :

= h f x , y

1 1

= h f x + h, y +

2 2

1 1

= h f x + h, y +

2 2

y

at h

Define k

k k

k k

¢

æ ö

ç ÷

è ø

æ è

( )

( )

4 n n 3

n+1 n 1 2 3 4

= h f x + h, y +

y = y + 1 + 2 + 2 +

k k

k k k k

ç ö÷

ø

RUNGE-KUTTA (Cont.)

( )

( )

( )

1 n n

2 n n 1

3 n n 2

4

Again = x + y

= 0.2, 5 steps, from 0 to 1 = 0.2 x , y

= 0.2 x + 0.1 + y + 0.5 = 0.1 x + 0.1 + y + 0.5 = 0

y h k

k k

k k

k

¢

( )

( )

( )

n n 3

n+1 n n n

5 n

n+1 n n n

.2 x + 0.2 + y + After simplification:

y = y + 0.2214 x , y + 0.0214 at n=5: x =1

y = 0.718215 very close y = y + 0.22 x , y + 0.02 Euler: 0.48

k

®

9 IE: 0.7027 RK: 0.718215 True : 0.718282

MULTI-STEP METHOD (Optional)

Multi-step method are methods that use information from more than one of the preceding steps, thereby increasing accuracy.

ADAMS-BASFORTH METHOD (Optional)

(

)

( )

f

y¢ = =

Consider

where f such that there is a unique solution

IDEA!! Replace f(x, y(x)) by an interpolation polynomial p(x).

( ) ( ] ( ) ( )

( )

( )

ò ò

+ + +

=

-

=

¢ = ₊

1 n

n

1 n n 1

n

n

x

x

n 1

n x

x x

x

dx x

y , x f

x y x

y y

dx x

y

The method is explained by taking a polynomial of order 3, p₃(x) such that

ADAMS-BASFORTH METHOD (Optional)

ò ( )

+

+

+

=

1 n

n

x

x n

1

n

y p x dx

y

( )

( )

( )

(

)

3 n

2 n 2

n 2

n

1 n 1

n 1

n

n n

n

y , x

f f

y , x

f f

y , x

f f

y , x f f

- -

-

- -

-

- -

-

=

=

=

=

p₃(x) can for instance be given by the Newton-backward difference formula (p807)

where

ADAMS-BASFORTH METHOD (Optional)

( ) ( )

( )( )

n 2 n

n

3 r r 1 r 2 f

6 f 1

2 1 r f r

r f

x

p + Ñ + + + Ñ

+ Ñ

+

=

1 n 1 k n

1 k n

k

1 n n

n

f f

f

f f

f

- - -

-

Ñ - Ñ

= Ñ

-

= Ñ

ADAMS-BASFORTH METHOD (Optional)

( )

3 n 2

n 1

n

2 n 1

n 1

n 2

1 n 2 n

2 n

3

2 n 1

n n

2 n 1

n 1

n n

1 n n

n 2

1 n n

n

f f

2 f

f f

f

f f

f

f f

2 f

f f

f f

f f

f

f f

f

- -

-

- -

-

-

- -

- -

-

- -

+ -

=

Ñ - Ñ

= Ñ

Ñ - Ñ

= Ñ

+ -

= -

- -

=

Ñ - Ñ

= Ñ

-

=

Ñ

ADAMS-BASFORTH METHOD (Optional)

( ) ( ) ( )

( )

( )( )( )

h x r x

f f

2 f

f f

2 f

2 r 1 r 6 r 1

f f

2 2 f

1 r f r

f r f

x p

n

3 n 2

n 1

n 2

n 1

n n

2 n 1

n n

1 n n

n 3

= -

- +

- +

- +

+ +

+ + -

+ -

+

=

- -

- -

-

- -

-

where r varies between 0 and 1 h dr = dx

ADAMS-BASFORTH METHOD (Optional)

( ) [ ]

( )

( )

( ) _ò ( )

ò ò

ò ò ò

+ +

- +

- +

+ + -

+

- +

=

=

- -

-

- -

-

+

1

0

2 3

n 2

n 1

n n

1

0

2 2

n 1

n n

1

0 1 n n

1

0 n 1

0 x

x

3

dr 2 r 3 r

6 r f 1

f 3 f

3 f

h

2 dr r f r

f 2 f

h

dr r f

f h

dr f

h

dr h dx

x p

1 n

n

L

ADAMS-BASFORTH METHOD (Optional)

( )

( ] ( )

( )

( )

úú û ù çç

è

æ + +

- +

- +

úú û ù çç

è

æ + +

- +

úú û ù çç è - æ

+

=

=

- -

-

- -

-

ò

+

2 2 r 3

3r 4

r 6 f 1

f 3 f

3 f

h

4 r 6 f r

f 2 f

h

2 f r

f h r

f h

? dx x p

2 3

4 3

n 2

n 1

n n

1

0 2 3

2 n 1

n n

1

0 2 1

n n

1 n 0 x

x

3

1 n

n

ADAMS-BASFORTH METHOD (Optional)

( ) ( )

( )

( )

+L úû ù êë

é ÷

ø ç ö

è æ +

÷- ø ç ö

è æ + -

- +

÷ø ç ö

è

æ + + + + + +

=

÷ø ç ö

è

æ + + -

+ -

+

÷ø ç ö

è æ + +

- +

- +

=

-

- -

-

- -

ò

+

4 2 1 6 3 4

1 6

2 1 2 f 1

h

24 4 24

4 24

1 4

1 6

1 2

1 1 f h

1 4 1

1 6 f 1

f 3 f

3 f

h

4 1 6

f 1 f

2 f

h

f 2 f

f h h dx

x p

1 n n

3 n 2

n 1

n n

2 n 1

n n

1 n n

n x

x

3

1 n

n

ADAMS-BASFORTH METHOD (Optional)

( )

÷ø ç ö

è

æ - - +

÷ø ç ö

è

æ + + + +

÷ø ç ö

è

æ - - - - - +

+ + +

= +

úû ù êë

é ÷

ø ç ö

è æ + -

ú + û ù êë

é ÷

ø ç ö

è æ + +

+ +

=

- - -

-

ò

+

24 8 f 1

h

24

24 3

6 f 4

h

24

24 3

12 8

f 12 h

f 24 h

1 6 4 12 24

4 2 1 6 f 1

h 4 2

1 6 3 4 1 6 f 1

h dx

x p

3 n

2 n

1 n

n

3 n 2

n x

x

3

1 n

n

L

ADAMS-BASFORTH METHOD (Optional)

( )

3 n 2

n 1

n n

n 1

n

3 n 2

n 1

n n

x

x

3

f 24 h f 9

24 h f 37

24 h f 59

24 h y 55

y

24 f 9

24 h f 37

24 h f 59

h f

24 h dx 55

x p

1 n

n

- -

- +

- -

-

- +

-

= -

÷ø ç ö

è + æ -

÷ø ç ö

è + æ

÷ø ç ö

è + æ -

ò

+