se301: numerical methods topic 8 ordinary differential equations (odes) lecture 28-36

CISE301_Topic8L6 KFUPM 1

SE301: Numerical Methods

Topic 8 Ordinary Differential

Equations (ODEs)Lecture 28-36

KFUPM

Read 25.1-25.4, 26-2, 27-1


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


Lecture 33Lesson 6: Solving Systems

of ODEs


Learning Objectives of Lesson 6

Convert a single (or a system of) high order ODE to a system of first order ODEs.

Use the methods discussed earlier in this topic to solve systems of first order ODEs.


Outlines of Lesson 6 Solution of a system of first order ODEs. Conversion of a high order ODE to a system of

first order ODEs. Conversion of a system of high order ODEs to a

system of first order ODEs. Use different methods to solve systems of first

order ODEs. Use different methods to solve high order ODEs. Use different methods to solve systems of high

order ODEs.


Solving a System of First Order ODEs Methods discussed earlier such as Euler,

Runge-Kutta,… are used to solve first order ordinary differential equations.

The same formulas will be used to solve a system of first order ODEs. In this case, the differential equation is a

vector equation and the dependent variable is a vector variable.


Euler Method for Solving a System of First Order ODEsRecall Euler method for solving a first order ODE:

)2),2(()2()3(

)),(()()2(

)),(()()(

:

)(),,()(

hahayfhhayhay

hahayfhhayhay

aayfhayhay

MethodEuler

yayxyfdx

xdyGiven a


Example - Euler MethodEuler method to solve a system of n first order ODEs.

)2),2(()2()3(

)),(()()2(

)),(()()(

:

)(

...

)(

)(

)(,

),(

...

),(

),(

),()( 2

1

2

1

hahaYFhhaYhaY

hahaYFhhaYhaY

aaYFhaYhaY

MethodEuler

ay

ay

ay

aY

xYf

xYf

xYf

xYFdx

xYdGiven

nn


Solving a System of n First Order ODEs Exactly the same

formula is used but the scalar variables and functions are replaced by vector variables and vector values functions.

Y is a vector of length n.

F(Y,x) is a vector valued function.

)2),2(()2()3(

)),(()()2(

)),(()()(

),(

),(

...

),(

),(

...

)(

1

)(

...

)(

)(

)(

2

1

2

1

2

1

hahaYFhhaYhaY

hahaYFhhaYhaY

aaYFhaYhaY

xYF

xYf

xYf

xYf

dx

yd

dx

yddx

yd

dx

xYd

vectornisY

xy

xy

xy

xY

nn

n


Example :Euler method for solving a system of first order ODEs.

39.1

78.0

)9.01(1.2.1

12.09.0

)1.0(1

)1.0(1.0

)1.0(

)1.0(

)2.0(

)2.0(

)),(()()20(

:2

2.1

9.0

)11(1.01

1.01

)0(1

)0(1.0

)0(

)0(

)1.0(

)1.0(

)0),0(()0()0(

:1

1.0

1

1

)0(

)0()0(),,(

1)(

)(

1

2

2

1

2

1

1

2

2

1

2

1

2

1

1

2

2

1

y

y

y

y

y

y

hhYFhhYhY

STEP

y

y

y

y

y

y

YFhYhY

STEP

hwithMethodEulerofstepsTwo

y

yYxYF

y

y

xy

xy


Example :RK2 method for solving a system of first order ODEs

195.1

89.0

19.0

12.0

2.0

1.0

2

1

1

1

)1.0(

)1.0(

)21(5.0)0()0(

19.0

12.0

)1.0)0((1

2.0)0(1.0)0,1)0((2

2.0

1.0

)0(1

)0(1.0)0),0((1

:1

1.0MethodKuttaRungeorder second

1

1

)0(

)0()0(),,(

1)(

)(

2

1

1

2

1

2

2

1

1

2

2

1

y

y

KKYhY

y

yhKYFhK

y

yYFhK

STEP

hwithofstepsTwo

y

yYxYF

y

y

xy

xy


Example :RK2 method for solving a system of first order ODEs

3780.1

7611.0

1771.0

1384.0

1890.0

1195.0

2

1

195.1

89.0

)2.0(

)2.0(

)21(5.0)1.0()1.0(

1771.0

1384.0

)1195.0)1.0((1

189.0)1.0(1.0)1.0,1)1.0((2

1890.0

1195.0

)1.0(1

)1.0(1.0)1.0),1.0((1

:2

1

1

)0(

)0()0(),,(

1)(

)(

2

1

1

2

1

2

2

1

1

2

2

1

y

y

KKYhY

y

yhKYFhK

y

yYFhK

STEP

y

yYxYF

y

y

xy

xy


Methods for Solving a System of First Order ODEs

We have extended Euler and RK2 methods to solve systems of first order ODEs.

Other methods used to solve first order ODE can be easily extended to solve systems of first order ODEs.


High Order ODEs How do solve a second order ODE?

How do solve high order ODEs?

163 xxx


The General Approach to Solve ODEs

4)0(;1)0(

163

xx

xxx

Convert

1

4)0(

,631 12

2

2

1

Z

zz

z

z

z

Solve

High order ODE Convert System of first order ODEs Solve

Second order ODE Two first order ODEs


Conversion Procedure High order ODE Convert System of first order ODEs Solve

1. Select the dependent variables One way is to take the original dependent

variable and its derivatives up to one degree less than the highest order derivative.

2. Write the Differential Equations in terms of the new variables. The equations come from the way the new variables are defined or from the original equation.

3. Express the equations in a matrix form.


Remarks on the Conversion Procedure

High order ODE Convert System of first order ODE Solve

1. Any nth order ODE is converted to a system of n first order ODEs.

2. There are an infinite number of ways to select the new variables. As a result, for each high order ODE there are an infinite number of set of equivalent first order systems of ODEs.

3. Use a table to make the conversion easier.


Example of Converting a High Order ODE to First Order ODEs

xz

xz

xxxxx

2

1

) variables twoneed WeODEorder (Second

variablesofset new aSelect 1.

ODEsorder first of system a to

4)0(;1)0(,163Convert

One degree less than the highest order derivative



1

4)0(,

631

6311

4

cond. name name

Equation Initial new old

12

2

2

1

1222

211

Zzz

z

z

z

zzzzx

zzzx



xz

xz

xz

xxx

xxxx

3

2

1

them)of (3 variablesofset new aSelect 1.

4)0(;1)0(,9)0(

0872

Convert




9

1

4

)0(,

872

8729

1

4

cond. name name


123

3

2

3

2

1

12333

322

211

Z

zzz

z

z

z

z

z

zzzzzx

zzzx

zzzx


Conversion Procedure for Systems of High Order ODEs

System of high order ODEs Convert System of first order ODE Solve

1. Select the dependent variables Take the original dependent variables and their

derivatives up to one degree less than the highest order derivative for each variable.

2. Write the Differential Equations in terms of the new variables. The equations come from the way the new variables are defined or from the original equation.

3. Express the equations in a matrix form.



yz

yz

xz

xz

xz

yyxxx

xxyy

yxxx

5

4

3

2

1

) variables2)((3 variablesofset new aSelect 1.

3)0(;1)0(;9)0(;2)0(;4)0(

22

0825

Convert





41255

544

42333

322

211

223

1

8259

2

4

cond. name name


zzzzzy

zzzy

zzzzzx

zzzx

zzzx


Solution of a Second Order ODE

Solve the equation using Euler method. Use h=0.1

2

1)0(,

822)(

:as expressed isequation order second The

,: variablesofset new aSelect

2)0(;1)0(

282

12

2

2

1

21

Zzz

z

z

zZFZ

xzxz

xx

xxx


Solution of a Second Order ODE

2.2

58.0

)8.0(8)2.2(22

2.21.0

2.2

8.0

))1.0(()1.0()2.0(

2.2

8.0

)1(8)2(22

21.0

2

1

))0(()0()1.00(

1.0,2

1)0(,

822)(

12

2

ZhFZZ

ZhFZZ

hZzz

zZF


Summary Formulas used in solving a first order ODE

are used to solve systems of first order ODEs. Instead of scalar variables and functions, we

have vector variables and vector functions.

High order ODEs are converted to a set of first order ODEs.


Remaining Lessons in Topic 8

Solution of ODEs

Lessons 8-9:Boundary Value Problems

Lesson 7:Multi-step methods

se301: numerical methods topic 8 ordinary differential equations (odes) lecture 28-36

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