Ordinary differential equations · Web view1.9Finite difference method for two point boundary value problem86 1.9.1Introduction86 1.9.2Solve the following two point boundary value

Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)

Table of Contents

1 Ordinary differential equations.........................................................................................................10

1.1 Euler method.............................................................................................................................10

1.1.1 Introduction.......................................................................................................................10

1.1.2 Example .........................................................................................14

1.1.3 Calculate two iterations of the Euler's method for ........................16

1.1.4 Calculate two iterations of the Euler's method for ................18

1.1.5 Calculate two iterations of the Euler's method for .........18

1.2 Euler method – numerical stability............................................................................................20

1.2.1 Introduction.......................................................................................................................20

1.2.2 Check numerical stability of the Euler’s method for the following differential equation

where ...............................................................................................................21

1.2.3 Check if the Euler’s method for is numerically stable.................23

1.2.4 Check if the Euler’s method for is numerically stable.........23

1.3 Euler's method for systems of equations...................................................................................25

1.3.1 Introduction.......................................................................................................................25

1


1.3.2 Example , ......................................................................27

1.3.3 Calculate two iterations of the Euler's method for .......................28

1.3.4 Example ..................................................................................................29

1.3.5 Find two iterations of the Euler's method for ...................30

1.3.6 Find two iterations of the Euler's method for ...............34

1.3.7 Find two iterations of the Euler's method for35

1.4 Euler method – numerical stability............................................................................................37

1.4.1 Introduction.......................................................................................................................37

2


1.4.2 Check if the Euler’s method for is numerically stable.............................................................................................................................38

1.4.3 Check if the Euler’s method for is numerically stable by using the matrix norm.....................................................................................40

1.4.4 Check if the Euler’s method for is numerically stable by using a spectral radious..................................................................................43

1.5 Backward Euler's method..........................................................................................................45

1.5.1 Introduction.......................................................................................................................45

1.5.2 Calculate two iterations of backward Euler's method ...................47

1.5.3 Calculate two iterations of backward Euler's method ...50

1.5.4 Calculate two iterations of backward Euler's method ...52

1.5.5 Calculate two iterations of backward Euler's method . . .53

1.5.6 Calculate two iterations of the backward Euler's method for

.........................................................................................................54

1.6 Backward Euler method with the predictor formula.................................................................56

1.6.1 Introduction.......................................................................................................................56

3


1.6.2 Calculate two iterations of the backward Euler method with the predictor formula for

............................................................................................................58


................................................................................................59

1.7 Trapezoidal method...................................................................................................................60

1.7.1 Introduction.......................................................................................................................60

1.7.2 Example .........................................................................................62

1.7.3 Calculate two iterations by using the Trapezoidal Method for 64

1.7.4 Calculate two iterations by using the Trapezoidal Method for 65

1.7.5 Calculate two iterations by using the Trapezoidal Method for

.....................................................................................................66


..........................................................................................................68


...................................................................................................69

4


1.8 The trapezoidal method method with the predictor formula....................................................71

1.8.1 Introduction.......................................................................................................................71

1.8.2 Calculate two iterations of the trapezoidal method with the predictor formula for

............................................................................................................73

1.9 Finite difference method for two point boundary value problem.............................................86

1.9.1 Introduction.......................................................................................................................86

1.9.2 Solve the following two point boundary value problem for n=4, h=0.25 92

1.9.3 Solve the following two point boundary value problem for

......................................................................................97


................................................................................99


...................................................................101

5



.......................................................................103

1.9.7 olve the following boundary value problem

(**).................................................................109

1.10 Taylor method.........................................................................................................................110

1.10.1 Introduction.....................................................................................................................110

1.10.2 Example ..........................................................................................................113

1.10.3 Solve the following differential equation by using Taylor's method.. . .115

1.10.4 Example ........................................................................................117

1.10.5 Example ..........................................................................................................119

1.10.6 Example ..........................................................................................................120

1.10.7 Example .........................................................................................................122

6


1.10.8 Example ......................................................................................124

1.10.9 Find approximation of the solution of the following differential equation

by using second order Taylor polynomial.....................................................126

1.10.10 Find approximation of the solution of the following differential equation by using second order Taylor polynomial........................................................................................127

1.10.11 Find approximation of the solution of the following differential equation by using second order Taylor polynomial........................................................................................128


by using second order Taylor polynomial for n=2.........................................129


by using second order Taylor polynomial for n=2.....................131

1.10.14 Solve the following system of differential equations by using Taylor's method

(*).............................................................................................................................133

1.11 Runge-Kutta method................................................................................................................134

1.11.1 Introduction.....................................................................................................................134

7


1.11.2 (**) Second order method - derivation............................................................................135

1.11.3 Find two iteration of the Runge-Kutta method for where

...................................................................139

1.11.4 Example .....................................................................................141

1.11.5 Calculate two iterations of the second order Runge-Kutta method for

...................................................................................................................142


...................................................................143


...................................................................144


...................................................................145

1.11.9 (**) Forth order Runge-Kutta Method.............................................................................147

8


1.11.10 Solve the following differential equation by using forth order Runge-Kutta method.......................................................................................................................148

1.12 (*) Multistep methods.............................................................................................................149

1.13 Review.....................................................................................................................................151

1.13.1 Summer 2015...................................................................................................................151

1.13.2 Summer 2014...................................................................................................................152

9


1 Ordinary differential equations

1.1 Euler method

1.1.1 Introduction

10


For example

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https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CAcQjRxqFQoTCNfW6-bCmskCFYPsJgodtqMAqQ&url=https://en.wikipedia.org/wiki/Euler_method&psig=AFQjCNGeZHE8_t-bPj8CYgzRJwzffXtWDA&ust=1447954985485670


Author

John C. Polking

Department of Mathematics

Rice University

http://math.rice.edu/~dfield/

12

http://math.rice.edu/~dfield/


http://math.rice.edu/~dfield/matlab8/dfield8.m

13

http://math.rice.edu/~dfield/matlab8/dfield8.m


14


1.1.2 Example for n=2

15


Matlab code

n=100;y=zeros(n+1);ye=zeros(n+1);y(1)=1;ye(1)=1;dx=0.01;x0=0;for i=1:n x=x0+(i+1)*dx; ye(i+1)=exp(2*x); y(i+1)=y(i)+2*y(i)*dx;endplot(y)holdplot(ye)

0 20 40 60 80 100 1200

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2

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8

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1.1.3 Calculate two iterations of the Euler's method for .

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<<VectorFieldPlots`;St=StreamPlot[{2y,1},{x,-2,2},{y,-2,2}];Show[St,Frame->True]

2 1 0 1 2

2

1

0

1

2

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1.1.6 Calculate two iterations of the Euler's method for and n=2.

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1.2 Euler method – numerical stability

1.2.1 Introduction

Let us consider the Euler method for the autonomus diffential equation .

Euler method for presented problem can be written as

Presented equation can be viewed as the fixed point iteration .

Theorem (Contraction mapping theorem)

Assume that and are continuous for and assume that satisfies the theorem, and

Then

1) There is a unique solution of the equation (i.e. ).

2) For any initial estimate of in , the iterates will converge to .

3)

4) for close to we have .

22


1.2.2 Check numerical stability of the Euler’s method for the following differential equation

where .Answer

Euler’s method for the equation lead to the following finite difference equation.

Presented FDM equation can be related with the following fixed point equation

Fixed point iterations converge if

For

23


Then the method is numerically stable if

p=2, dx=0.01

24


dx=1, p=2

25


1.2.3 Check if the Euler’s method for is numerically stable.Answer

Because then the Euler’s method is numerically stable.

1.2.4 Check if the Euler’s method for is numerically stable.

26


1.2.5 Check if the Euler’s method for is numerically stable.

27


1.3 Euler's method for systems of equations

1.3.1 Introduction

28


29


1.3.2 Example ,

30



31


1.3.4 Example

32


1.3.5 Find two iterations of the Euler's method for

33


Matlab code

-------------------

n=100;

t=zeros(n+1);

y=zeros(n+1);

v=zeros(n+1);

ye=zeros(n+1);

y(1)=0;

v(1)=2;

t(1)=0;

dt=0.1;

for i=1:n

t(i+1)=0+(i+2)*dt;

ye(i+1)=sin(2*t(i+1));

y(i+1)=y(i)+v(i)*dt;

v(i+1)=v(i)-4*y(i)*dt;

end

plot(y)

hold

plot(ye)

-------------------

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0 20 40 60 80 100 120-6

-4

-2

0

2

4

6

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1.3.6 Example

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1.4 Euler method – numerical stability

1.4.1 Introduction

Let us consider the Euler method for the autonomus diffential equation .

Euler method for presented problem can be written as

Presented equation can be viewed as the fixed point iteration .

Theorem (Contraction mapping theorem)

Let where and and

, then

the iterative process converges to the unique fixed point and .

In particular for the linear equation fixed point exist if

40


1.4.2 Check of the Euler’s method for is numerically stable.

Solution

Euler’s method for the system

Matrix of the system

Matrix norm (maximum absolute row sum of the matrix)

For presented matrix

41


Because the norm of the matrix is smaller than 1, then preseted finite difference scheme is numerically stable.

42


1.4.3 Check if the Euler’s method for is numerically stable by using the matrix norm.

Norm of the matrix

Then the Euler’s method is numerically stable.

43


1.4.4 Check if the Euler’s method for

is numerically stable by using the matrix norm.

44


45


46


1.4.5 (*) Check if the Euler’s method for is numerically stable by using a spectral radious.

Solution

Euler’s method for the system

Matrix of the system

Eigenvalues of the matrix

For

47


Spectral radious of the matrix

Because the spectral radious of the matrix is smaller than , then the matrix the finite difference scheme is numerically stable.

1.4.6 Check if the Euler’s method for

is numerically stable by using a spectral radious.

48


1.5 Backward Euler's method

1.5.1 Introduction

49


Forward difference

Backward difference

Central difference

Backward difference Forward difference (Euler's method)

50


1.5.2 Calculate two iterations of backward Euler's method for

51


% Octave script

n=150;

x=zeros(n+1);

y=zeros(n+1);

ye=zeros(n+1);

y1=zeros(n+1);

x(1)=0;

y(1)=1;

ye(1)=1;

y1(1)=1;

dx=0.01;

x0=0;

for i=1:n

xc=x0+i*dx;

x(i+1)=xc;

ye(i+1)=exp(2*xc);

y(i+1)=y(i)+2*y(i)*dx;

y1(i+1)=y1(i)/(1-2*dx);

end

plot(x,ye,"color", "b",x,y1,"color", "g",x,y,"color", "r")

52


53


1.5.3 Calculate two iterations of backward Euler's method

54


0 20 40 60 80 100 1200

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Matlab code

----------------

n=100;yb=zeros(n+1);y(1)=1;ye(1)=1;yb(1)=1;dx=0.01;x0=0;for i=1:n x=x0+(i+1)*dx; yb(i+1)=yb(i)/(1-2*dx);endplot(yb)----------------

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56



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58


1.5.7 Calculate two iterations of the backward Euler's method for

59



60


1.6 Backward Euler method with the predictor formula

1.6.1 Introduction

Backward difference

61


It is not necessary to solve the equation for .

It is possible to get approximate solution by using the Euler method.

or

62



Answer

63



.

64


1.7 Trapezoidal method

1.7.1 Introduction

65


66


1.7.2 Example

Matlab code

--------------------------------

n=100;y=zeros(n+1);yt=zeros(n+1);ye=zeros(n+1);y(1)=1;

ye(1)=1;yt(1)=1;dx=0.01;x0=0;for i=1:n x=x0+(i+1)*dx; ye(i+1)=exp(2*x); yt(i+1)=(2*yt(i)+2*yt(i)*dx)/(2-2*dx);endplot(yt)hold

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plot(ye)--------------------------------

0 20 40 60 80 100 1200

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2

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1.8 The trapezoidal method method with the predictor formula

1.8.1 Introduction

78


It is not necessary to solve the equation for .

It is possible to get approximate solution by using the Euler method.

or

79


1.8.2 Calculate two iterations of the trapezoidal method with the predictor formula for

Answer

80


1.1 (*) Multistep methods

Explicit method

Implicit method

81


1.1.1 “Adams” methods

Explicit “Adams–Bashforth” method.

Implicit “Adams-Moulton” method.

Predictor corrector method “Adams–Bashforth- Moulton”

Lagrange interpolation

Consider then

82


Constant interpolation (Euler’s method)

Linear interpolation (trapezoidal method)

Quadratic interpolation

Etc.

83


1.8.2.1ExampleVerify the scheme:

for the ODE .

Answer

Let’s us consider three points

And appropriate second order interpolation polynomial in a Lagrange’ form.

If then , consequently the solution of the equation is

close to the solution of the equation , in particular

84


Let , then

85


1.1.2 Adams–Bashforth of order 2 (two step method).

Interpolate on by using the linear interpolation

then

Local truncation error global truncation error .

Now it is possible to use sample functions and find constants A,B.

86


Example

First step we can find by using the modified Euler’s method.

Second step we can find by using the Adams–Bashforth method.

87


1.1.3 Adams– Moulton 3th order (2 step method)

Let’s us consider

Let us consider .

After the calculations

Example

Solve for .

88


Now it is possible to use sample functions and find constants A,B.

After the solution

89


1.1.4 Adams– Moulton 4th order (3 step method)

90


1.1.5 Comparison m-step Adams–Bashforth and m-1 Adams– Moulton

In the multistep methods it is necessary to use additional methods to calculate the first iterations.

The order of the methods must be the same as the original method.

Not self-storging????

(1) m-step Adams–Bashforth and m-1 Adams– Moulton

4-th order Runge Kutta – more functions evaluations

Adams–Bashforth – one function evaluation at the time (we can use previous function values).

1.1.6 Predictor corrector method Adams–Bashforth-Moulton

Forth-order Adams–Bashforth (predictor)

Adams–Moulton (corrector)

91


1.9 Finite difference method for two-point boundary value problem

1.9.1 Introduction

Explicit method

Implicit method

Example

92


93


Example

Exact solution

94


95


n=4,

96


-----------------------------------

97

h 0.25x

0

0.25

0.5

0.75

1

A

1

1

h2

0

0

0

0

2

h2

1

h2

0

0

0

1

h2

2

h2

1

h2

0

0

0

1

h2

2

h2

0

0

0

0

1

h2

1

b

0

0.25

0.5

0.75

0

u A 1 b u

0

0.039

0.063

0.055

0

ue x( )x3

6x6

0 0.2 0.4 0.6 0.8 10.08

0.06

0.04

0.02

0

u

x

ue x( )

0

0.039

0.063

0.055

0


1.9.2 Example

Finite difference method for two point boundary value problem.

98


For n=3

Matrix form

or shortyly

where

99


1.9.3 Solve the following two-point boundary value problem for n=4, h=0.25

100


101


1 2 3 4 5

0.5

1.0

1.5

2.0

102


1.9.4 Solve the following two-point boundary value problem for

103


(*)

Matrix form of the problem

104



105


Matrix form (*)

106



107



X (id)

X (value) 0 0.25 0.5 0.75 1

108


U(id)

U(value) 1 ? ? ? 2

109



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111


112



Difference equation

113


114


115


1.9.10 (**) Solve the following boundary value problem

116


1.10 Taylor method

1.10.1 Introduction

117


In general

118


1.10.2 Example

119


1.10.3 Example

120


121


1.10.4 Solve the following differential equation by using Taylor's method.

122


123


1.10.5 Example

124


1.10.6 Example

125


1.10.7 Example

126


1.10.8 Example

127


128


1.10.9 Example

129


130


1.10.10 Example

131


132



by using second order Taylor polynomial

133




134




135



by using second order Taylor polynomial for n=2.

136


137


138



by using second order Taylor polynomial for n=2.

139


140


1.10.16 Solve the following system of differential equations by using Taylor's method

(*)

141


1.11 Runge-Kutta method

1.11.1 Introduction

Second order

Forth order

---

142


1.11.2 (**) Second order method - derivation

143


because

then

Now let's calculate the difference

144


145


For we have

146


1.11.3 Find two iterations of the Runge-Kutta method for where

.

147



.

148


1.11.5 Example

149


1.11.6 Calculate two iterations of the second order Runge-Kutta method for

150



.

151



.

152



.

153


154


1.11.10 (**) Forth order Runge-Kutta Method

Let an initial value problem be specified as follows.

Then, the RK4 method for this problem is given by the following equations:

where yn + 1 is the RK4 approximation of y(tn + 1), and

155


1.11.11 Solve the following differential equation by using forth order Runge-Kutta method.

156


1.12 (*) Multistep methods

157


158


1.13 Review

1.13.1 Summer 2017

Problem 1.2.4 (Numerical stability)

Problem 1.5.7 (The Backward Euler’s Method)

Problem 1.7.4 (Trapezoidal method)

Problem 1.9.8 (Finite difference method for two-point boundary value problem)

159


1.13.2 Summer 2014

Section 1.3

Problem 1.3.5

Problem 1.3.7

Section 1.4

Problem 1.4.2

Section 1.5

Problem 1.5.3

Problem 1.5.5

Section 1.6

Problem 1.6.2

Section 1.7

Problem 1.7.5

Problem 1.7.6

Section 1.8

Problem 1.8.2

160


Section 1.9

Problem 1.9.5

Problem 1.9.7

Section 1.10

Problem 1.10.5

Problem 1.10.7

Section 1.11

Problem 1.11.7

Problem 1.11.8

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Documents

Ordinary differential equations · Web view1.9Finite difference method for two point boundary value problem86 1.9.1Introduction86 1.9.2Solve the following two point boundary value