CISE301 : Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36

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CISE301 : 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

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CISE301: Numerical MethodsTopic 8Ordinary Differential Equations (ODEs)Lecture 28-36

KFUPM

KFUPM

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

KFUPM

### Lecture 33Lesson 6: Solving Systems of ODEs

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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.

KFUPM

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.

KFUPM

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.

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Euler Method for Solving a System of First Order ODEs

Recall Euler method for solving a first order ODE:

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Example - Euler Method

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

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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.

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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.

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High Order ODEs
• How to solve a second order ODE?
• How to solve high order ODEs?

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The General Approach to Solve ODEs

Convert

Solve

High order ODE

System of first order ODEs

Convert

Solve

Second order ODE

Two first order ODEs

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Conversion Procedure

Convert

Solve

• 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.

• 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.
• Express the equations in a matrix form.

High order ODE

System of first order ODEs

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Remarks on the Conversion Procedure

Convert

Solve

• Any nth order ODE is converted to a system of n first order ODEs.
• 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.
• Use a table to make the conversion easier.

High order ODE

System of first order ODE

KFUPM

Example of Converting a High Order ODE to First Order ODEs

One degree less than the highest order derivative

KFUPM

Example of Converting a High Order ODE to First Order ODEs

One degree less than the highest order derivative

KFUPM

Conversion Procedure for Systems of High Order ODEs

Convert

Solve

• 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.

• 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.
• Express the equations in a matrix form.

System of high order ODEs

System of first order ODE

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Example of Converting a High Order ODE to First Order ODEs

One degree less than the highest order derivative

One degree less than the highest order derivative

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Solution of a Second Order ODE
• Solve the equation using Euler method. Use h=0.1

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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.

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Remaining Lessons in Topic 8

Solution of ODEs

Lesson 7:

Multi-step methods

Lessons 8-9:

Boundary Value Problems

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