Upload
lethien
View
215
Download
0
Embed Size (px)
Citation preview
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.6.2 Calculate two iterations of the backward Euler method with the predictor formula for
............................................................................................................58
1.6.3 Calculate two iterations of the backward Euler method with the predictor formula for
................................................................................................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
1.7.6 Calculate two iterations by using the Trapezoidal Method for
..........................................................................................................68
1.7.7 Calculate two iterations by using the Trapezoidal Method for
...................................................................................................69
4
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
1.9.4 Solve the following two point boundary value problem for
................................................................................99
1.9.5 Solve the following two point boundary value problem for
...................................................................101
5
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.9.6 Solve the following two point boundary value problem for
.......................................................................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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
1.10.12 Find approximation of the solution of the following differential equation
by using second order Taylor polynomial for n=2.........................................129
1.10.13 Find approximation of the solution of the following differential equation
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
1.11.6 Find two iteration of the Runge-Kutta method for where
...................................................................143
1.11.7 Find two iteration of the Runge-Kutta method for where
...................................................................144
1.11.8 Find two iteration of the Runge-Kutta method for where
...................................................................145
1.11.9 (**) Forth order Runge-Kutta Method.............................................................................147
8
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1 Ordinary differential equations
1.1 Euler method
1.1.1 Introduction
10
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
For example
11
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
Author
John C. Polking
Department of Mathematics
Rice University
http://math.rice.edu/~dfield/
12
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
http://math.rice.edu/~dfield/matlab8/dfield8.m
13
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
1
2
3
4
5
6
7
8
16
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.1.3 Calculate two iterations of the Euler's method for .
17
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
<<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
18
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.1.4 Calculate two iterations of the Euler's method for .
1.1.5 Calculate two iterations of the Euler's method for .
19
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.1.6 Calculate two iterations of the Euler's method for and n=2.
20
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.1.7 Calculate two iterations of the Euler's method for .
21
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
Then the method is numerically stable if
p=2, dx=0.01
24
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.2.5 Check if the Euler’s method for is numerically stable.
27
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.3 Euler's method for systems of equations
1.3.1 Introduction
28
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.3.3 Calculate two iterations of the Euler's method for .
31
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.3.5 Find two iterations of the Euler's method for
33
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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)
-------------------
34
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
0 20 40 60 80 100 120-6
-4
-2
0
2
4
6
35
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.3.7 Find two iterations of the Euler's method for
37
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.3.8 Find two iterations of the Euler's method for
38
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
Because the norm of the matrix is smaller than 1, then preseted finite difference scheme is numerically stable.
42
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.4.4 Check if the Euler’s method for
is numerically stable by using the matrix norm.
44
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.5 Backward Euler's method
1.5.1 Introduction
49
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
Forward difference
Backward difference
Central difference
Backward difference Forward difference (Euler's method)
50
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.5.2 Calculate two iterations of backward Euler's method for
51
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
% 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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.5.3 Calculate two iterations of backward Euler's method
54
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
0 20 40 60 80 100 1200
1
2
3
4
5
6
7
8
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)----------------
55
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.5.4 Calculate two iterations of backward Euler's method
56
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.5.5 Calculate two iterations of backward Euler's method
57
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.5.6 Calculate two iterations of backward Euler's method
58
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.5.7 Calculate two iterations of the backward Euler's method for
59
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.5.8 Calculate two iterations of backward Euler's method
60
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.6 Backward Euler method with the predictor formula
1.6.1 Introduction
Backward difference
61
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
It is not necessary to solve the equation for .
It is possible to get approximate solution by using the Euler method.
or
62
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.6.2 Calculate two iterations of the backward Euler method with the predictor formula for
Answer
63
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.6.3 Calculate two iterations of the backward Euler method with the predictor formula for
.
64
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.7 Trapezoidal method
1.7.1 Introduction
65
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
67
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
plot(ye)--------------------------------
0 20 40 60 80 100 1200
1
2
3
4
5
6
7
8
68
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.7.3 Calculate two iterations by using the Trapezoidal Method for
69
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.7.4 Calculate two iterations by using the Trapezoidal Method for
70
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.7.5 Calculate two iterations by using the Trapezoidal Method for
71
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.7.6 Calculate two iterations by using the Trapezoidal Method for
73
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.7.7 Calculate two iterations by using the Trapezoidal Method for
74
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.7.8 Calculate two iterations by using the Trapezoidal Method for
75
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.7.9 Calculate two iterations by using the Trapezoidal Method for
77
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.8 The trapezoidal method method with the predictor formula
1.8.1 Introduction
78
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
It is not necessary to solve the equation for .
It is possible to get approximate solution by using the Euler method.
or
79
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.8.2 Calculate two iterations of the trapezoidal method with the predictor formula for
Answer
80
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.1 (*) Multistep methods
Explicit method
Implicit method
81
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.1.1 “Adams” methods
Explicit “Adams–Bashforth” method.
Implicit “Adams-Moulton” method.
Predictor corrector method “Adams–Bashforth- Moulton”
Lagrange interpolation
Consider then
82
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
Constant interpolation (Euler’s method)
Linear interpolation (trapezoidal method)
Quadratic interpolation
Etc.
83
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.1.3 Adams– Moulton 3th order (2 step method)
Let’s us consider
Let us consider .
After the calculations
Example
Solve for .
88
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
Now it is possible to use sample functions and find constants A,B.
After the solution
89
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.1.4 Adams– Moulton 4th order (3 step method)
90
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.9 Finite difference method for two-point boundary value problem
1.9.1 Introduction
Explicit method
Implicit method
Example
92
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
-----------------------------------
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.9.2 Example
Finite difference method for two point boundary value problem.
98
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
For n=3
Matrix form
or shortyly
where
99
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.9.3 Solve the following two-point boundary value problem for n=4, h=0.25
100
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.9.4 Solve the following two-point boundary value problem for
103
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
(*)
Matrix form of the problem
104
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.9.5 Solve the following two point boundary value problem for
105
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.9.6 Solve the following two-point boundary value problem for
107
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.9.7 Solve the following two point boundary value problem for
X (id)
X (value) 0 0.25 0.5 0.75 1
108
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.9.8 Solve the following two-point boundary value problem for
110
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.9.9 Solve the following two-point boundary value problem for
Difference equation
113
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.9.10 (**) Solve the following boundary value problem
116
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.10 Taylor method
1.10.1 Introduction
117
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.10.4 Solve the following differential equation by using Taylor's method.
122
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.10.11 Find approximation of the solution of the following differential equation
by using second order Taylor polynomial
133
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.10.12 Find approximation of the solution of the following differential equation
by using second order Taylor polynomial
134
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.10.13 Find approximation of the solution of the following differential equation
by using second order Taylor polynomial
135
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.10.14 Find approximation of the solution of the following differential equation
by using second order Taylor polynomial for n=2.
136
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.10.15 Find approximation of the solution of the following differential equation
by using second order Taylor polynomial for n=2.
139
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.10.16 Solve the following system of differential equations by using Taylor's method
(*)
141
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.11 Runge-Kutta method
1.11.1 Introduction
Second order
Forth order
---
142
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.11.2 (**) Second order method - derivation
143
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
because
then
Now let's calculate the difference
144
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.11.3 Find two iterations of the Runge-Kutta method for where
.
147
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.11.4 Find two iterations of the Runge-Kutta method for where
.
148
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.11.6 Calculate two iterations of the second order Runge-Kutta method for
150
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.11.7 Find two iterations of the Runge-Kutta method for where
.
151
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.11.8 Find two iteration of the Runge-Kutta method for where
.
152
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.11.9 Find two iteration of the Runge-Kutta method for where
.
153
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.11.11 Solve the following differential equation by using forth order Runge-Kutta method.
156
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
1.12 (*) Multistep methods
157
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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
Andrew Pownuk - http://www.pownuk.com- Math 4329 (Numerical Analysis)
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