This document discusses boundary value problems and the shooting method and finite difference method for solving them. It begins with an introduction to boundary value problems and contrasts them with initial value problems. It then describes the shooting method, which converts a boundary value problem into an initial value problem by guessing the unknown boundary conditions and iteratively solving the problem. The document provides an example of using the shooting method. Finally, it introduces the finite difference method, which discretizes the boundary value problem domain and approximates derivatives to obtain a system of algebraic equations that can be solved for the unknown values.

1. CISE301_Topic8L8&9 1
CISE301: Numerical Methods
Topic 8
Ordinary Differential Equations (ODEs)
Lecture 28-36
KFUPM
(Term 101)
Section 04
Read 25.1-25.4, 26-2, 27-1

2. CISE301_Topic8L8&9 2
Outline of Topic 8
 Lesson 1: Introduction to ODEs
 Lesson 2: Taylor series methods
 Lesson 3: Midpoint and Heun’s method
 Lessons 4-5: Runge-Kutta methods
 Lesson 6: Solving systems of ODEs
 Lesson 7: Multiple step Methods
 Lesson 8-9: Boundary value Problems

3. CISE301_Topic8L8&9 3
Lecture 35
Lesson 8: Boundary Value Problems

4. CISE301_Topic8L8&9 4
Outlines of Lesson 8
 Boundary Value Problem
 Shooting Method
 Examples

5. CISE301_Topic8L8&9 5
Learning Objectives of Lesson 8
 Grasp the difference between initial value
problems and boundary value problems.
 Appreciate the difficulties involved in solving the
boundary value problems.
 Grasp the concept of the shooting method.
 Use the shooting method to solve boundary
value problems.

6. CISE301_Topic8L8&9 6
Boundary-Value and
Initial Value Problems
Boundary-Value Problems
 The auxiliary conditions are
not at one point of the
independent variable
 More difficult to solve than
initial value problem
5
.
1
)
2
(
,
1
)
0
(
2 2




 
x
x
e
x
x
x t



Initial-Value Problems
 The auxiliary conditions
are at one point of the
independent
variable
5
.
2
)
0
(
,
1
)
0
(
2 2




 
x
x
e
x
x
x t




same different

7. CISE301_Topic8L8&9 7
Shooting Method

8. CISE301_Topic8L8&9 8
The Shooting Method
Target

9. CISE301_Topic8L8&9 9
The Shooting Method
Target

10. CISE301_Topic8L8&9 10
The Shooting Method
Target

11. CISE301_Topic8L8&9 11
Solution of Boundary-Value Problems
Shooting Method for Boundary-Value Problems
1. Guess a value for the auxiliary conditions at one
point of time.
2. Solve the initial value problem using Euler,
Runge-Kutta, …
3. Check if the boundary conditions are satisfied,
otherwise modify the guess and resolve the
problem.
 Use interpolation in updating the guess.
 It is an iterative procedure and can be
efficient in solving the BVP.

12. CISE301_Topic8L8&9 12
Solution of Boundary-Value Problems
Shooting Method
8
.
0
)
1
(
,
2
.
0
)
0
(
2
)
(
2





y
y
x
y
y
y
BVP
solve
to
x
y
Find



Boundary-Value
Problem
Initial-value
Problem
convert
1. Convert the ODE to a system of
first order ODEs.
2. Guess the initial conditions that
are not available.
3. Solve the Initial-value problem.
4. Check if the known boundary
conditions are satisfied.
5. If needed modify the guess and
resolve the problem again.

13. CISE301_Topic8L8&9 13
Example 1
Original BVP
2
)
1
(
,
0
)
0
(
0
4
4





y
y
x
y
y


0 1 x

14. CISE301_Topic8L8&9 14
Example 1
Original BVP
2
)
1
(
,
0
)
0
(
0
4
4





y
y
x
y
y


2. 0
0 1 x

15. CISE301_Topic8L8&9 15
Example 1
Original BVP
2
)
1
(
,
0
)
0
(
0
4
4





y
y
x
y
y


2. 0
0 1 x

16. CISE301_Topic8L8&9 16
Example 1
Original BVP
2
)
1
(
,
0
)
0
(
0
4
4





y
y
x
y
y


to be
determined
2. 0
0 1 x

17. CISE301_Topic8L8&9 17
Example 1
Step1: Convert to a System of First Order ODEs
2
y(1)
have
we
until
)
0
(
y
of
values
different
for
0.01
h
with
RK2
using
solved
be
will
problem
The
?
0
)
0
(
y
)
0
(
y
,
)
4(y
y
y
y
Equations
order
first
of
system
a
to
Convert
2
)
1
(
,
0
)
0
(
0
4
4
2
2
1
1
2
2
1


































x
y
y
x
y
y





18. CISE301_Topic8L8&9 18
Example 1
Guess # 1
0
)
0
(
1
#

y
Guess

-0.7688
0 1 x
2
)
1
(
,
0
)
0
(
0
4
4





y
y
x
y
y



19. CISE301_Topic8L8&9 19
Example 1
Guess # 2
1
)
0
(
2
#

y
Guess

0.99
0 1 x
2
)
1
(
,
0
)
0
(
0
4
4





y
y
x
y
y



20. CISE301_Topic8L8&9 20
Example 1
Interpolation for Guess # 3
2
)
1
(
,
0
)
0
(
0
4
4





y
y
x
y
y


)
0
(
y

Guess y(1)
1 0 -0.7688
2 1 0.9900
0.99
0 1 2 y’(0)
-0.7688
y(1)

21. CISE301_Topic8L8&9 21
Example 1
Interpolation for Guess # 3
2
)
1
(
,
0
)
0
(
0
4
4





y
y
x
y
y


)
0
(
y

Guess y(1)
1 0 -0.7688
2 1 0.9900
0.99
0 1 2 y’(0)
-0.7688
1.5743
2
y(1)
Guess 3

22. CISE301_Topic8L8&9 22
Example 1
Guess # 3
5743
.
1
)
0
(
3
#

y
Guess

2.000
0 1 x
2
)
1
(
,
0
)
0
(
0
4
4





y
y
x
y
y


This is the solution to the
boundary value problem.
y(1)=2.000

23. CISE301_Topic8L8&9 23
Summary of the Shooting Method
1. Guess the unavailable values for the
auxiliary conditions at one point of the
independent variable.
2. Solve the initial value problem.
3. Check if the boundary conditions are
satisfied, otherwise modify the guess and
resolve the problem.
4. Repeat (3) until the boundary conditions
are satisfied.

24. CISE301_Topic8L8&9 24
Properties of the Shooting Method
1. Using interpolation to update the guess often
results in few iterations before reaching the
solution.
2. The method can be cumbersome for high order
BVP because of the need to guess the initial
condition for more than one variable.

25. CISE301_Topic8L8&9 25
Lecture 36
Lesson 9: Discretization Method

26. CISE301_Topic8L8&9 26
Outlines of Lesson 9
 Discretization Method
 Finite Difference Methods for Solving Boundary
Value Problems
 Examples

27. CISE301_Topic8L8&9 27
Learning Objectives of Lesson 9
 Use the finite difference method to solve
BVP.
 Convert linear second order boundary
value problems into linear algebraic
equations.

28. CISE301_Topic8L8&9 28
Solution of Boundary-Value Problems
Finite Difference Method
8
.
0
)
1
(
,
2
.
0
)
0
(
2
)
(
2





y
y
x
y
y
y
BVP
solve
to
x
y
Find



y4=0.8
0 0.25 0.5 0.75 1.0 x
x0 x1 x2 x3 x4
y
y0=0.2
y1=?
y2=?
y3=?
Boundary-Value
Problems
Algebraic
Equations
convert
Find the unknowns y1, y2, y3

29. CISE301_Topic8L8&9 29
Solution of Boundary-Value Problems
Finite Difference Method
 Divide the interval into n sub-intervals.
 The solution of the BVP is converted to
the problem of determining the value of
function at the base points.
 Use finite approximations to replace the
derivatives.
 This approximation results in a set of
algebraic equations.
 Solve the equations to obtain the solution
of the BVP.

30. CISE301_Topic8L8&9 30
Finite Difference Method
Example
8
.
0
)
1
(
,
2
.
0
)
0
(
2 2





y
y
x
y
y
y 


y4=0.8
0 0.25 0.5 0.75 1.0 x
x0 x1 x2 x3 x4
y
y0=0.2
Divide the interval
[0,1 ] into n = 4
intervals
Base points are
x0=0
x1=0.25
x2=.5
x3=0.75
x4=1.0
y1=?
y2=?
y3=?
To be
determined

31. CISE301_Topic8L8&9 31
Finite Difference Method
Example
8
.
0
)
1
(
,
2
.
0
)
0
(
2 2





y
y
x
y
y
y 


2
1
1
2
1
1
2
1
1
2
1
1
2
2
2
2
2
2
Replace
i
i
i
i
i
i
i
i
i
i
i
i
x
y
h
y
y
h
y
y
y
Becomes
x
y
y
y
formula
difference
central
h
y
y
y
formula
difference
central
h
y
y
y
y




























Divide the interval
[0,1 ] into n = 4
intervals
Base points are
x0=0
x1=0.25
x2=.5
x3=0.75
x4=1.0

32. CISE301_Topic8L8&9 32
Second Order BVP
2
1
1
2
2
2
1
4
3
2
1
0
2
2
2
2
)
(
)
(
2
)
(
)
(
)
(
1
,
75
.
0
,
5
.
0
,
25
.
0
,
0
Points
Base
25
.
0
8
.
0
)
1
(
,
2
.
0
)
0
(
2
h
y
y
y
h
h
x
y
x
y
h
x
y
dx
y
d
h
y
y
h
x
y
h
x
y
dx
dy
x
x
x
x
x
h
Let
y
y
with
x
y
dx
dy
dx
y
d
i
i
i
i
i




























33. CISE301_Topic8L8&9 33
Second Order BVP
   
2
1
1
2
1
1
1
4
3
2
1
0
4
3
2
1
0
2
1
2
1
1
2
2
2
16
39
24
8
2
16
8
.
0
?,
?,
?,
,
2
.
0
1
,
75
.
0
,
5
.
0
,
25
.
0
,
0
3
,
2
,
1
2
2
2
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
x
y
y
y
x
y
y
y
y
y
y
y
y
y
y
y
x
x
x
x
x
i
x
y
h
y
y
h
y
y
y
x
y
dx
dy
dx
y
d






































34. CISE301_Topic8L8&9 34
Second Order BVP
0.7436
0.6477,
0.4791,
)
8
.
0
(
24
75
.
0
5
.
0
)
2
.
0
(
16
25
.
0
39
16
0
24
39
16
0
24
39
16
39
24
3
16
39
24
2
16
39
24
1
16
39
24
3
2
1
2
2
2
3
2
1
2
3
2
3
4
2
2
1
2
3
2
1
0
1
2
2
1
1





















































 

y
y
y
Solution
y
y
y
x
y
y
y
i
x
y
y
y
i
x
y
y
y
i
x
y
y
y i
i
i
i

35. CISE301_Topic8L8&9 35
Second Order BVP
   
2
1
1
2
1
1
1
100
99
2
1
0
100
99
2
1
0
2
1
2
1
1
2
2
2
10000
20199
10200
200
2
10000
8
.
0
?,
...
?,
?,
,
2
.
0
1
,
99
.
0
...
02
.
0
,
01
.
0
,
0
100
,...,
2
,
1
2
2
2
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
x
y
y
y
x
y
y
y
y
y
y
y
y
y
y
y
x
x
x
x
x
i
x
y
h
y
y
h
y
y
y
x
y
dx
dy
dx
y
d






































36. CISE301_Topic8L8&9 36

37. CISE301_Topic8L8&9 37
Summary of the Discretiztion Methods
 Select the base points.
 Divide the interval into n sub-intervals.
 Use finite approximations to replace the
derivatives.
 This approximation results in a set of
algebraic equations.
 Solve the equations to obtain the solution
of the BVP.

38. CISE301_Topic8L8&9 38
Remarks
Finite Difference Method :
 Different formulas can be used for
approximating the derivatives.
 Different formulas lead to different
solutions. All of them are approximate
solutions.
 For linear second order cases, this
reduces to tri-diagonal system.

39. CISE301_Topic8L8&9 39
Summary of Topic 8
Solution of ODEs
Lessons 1-3:
• Introduction to ODE, Euler Method,
• Taylor Series methods,
• Midpoint, Heun’s Predictor corrector methods
Lessons 4-5:
• Runge-Kutta Methods (concept & derivation)
• Applications of Runge-Kutta Methods To solve first order ODE
Lessons 6:
•Solving Systems of ODE
Lessons 8-9:
• Boundary Value Problems
• Discretization method
Lesson 7:
Multi-step methods