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

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

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

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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

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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.

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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.

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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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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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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

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Example of Converting a High Order ODE to First Order ODEs

One degree less than the highest order derivative

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Example of Converting a High Order ODE to First Order ODEs

One degree less than the highest order derivative

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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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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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