SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36

1. SE301: Numerical MethodsTopic 8Ordinary Differential Equations (ODEs)Lecture 28-36 KFUPM Read 25.1-25.4, 26-2, 27-1 KFUPM

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 KFUPM

3. Lecture 33Lesson 6: Solving Systems of ODEs KFUPM

4. 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

5. 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

6. 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. KFUPM

7. Euler Method for Solving a System of First Order ODEs Recall Euler method for solving a first order ODE: KFUPM

8. Example - Euler Method Euler method to solve a system of n first order ODEs. KFUPM

9. 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. KFUPM

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

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

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

13. 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. KFUPM

14. High Order ODEs • How do solve a second order ODE? • How do solve high order ODEs? KFUPM

15. 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 KFUPM

16. 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 KFUPM

17. 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

18. Example of Converting a High Order ODE to First Order ODEs One degree less than the highest order derivative KFUPM

19. Example of Converting a High Order ODE to First Order ODEs KFUPM

20. Example of Converting a High Order ODE to First Order ODEs One degree less than the highest order derivative KFUPM

21. Example of Converting a High Order ODE to First Order ODEs KFUPM

22. 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 KFUPM

23. 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 KFUPM

24. Example of Converting a High Order ODE to First Order ODEs KFUPM

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

26. Solution of a Second Order ODE KFUPM

27. 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. KFUPM

28. Remaining Lessons in Topic 8 Solution of ODEs Lesson 7: Multi-step methods Lessons 8-9: Boundary Value Problems KFUPM