1. CISE301_Topic8L1KFUPM1 CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36 KFUPM Read 25.1-25.4, 26-2, 27-1

2. CISE301_Topic8L1KFUPM2 Objectives of Topic 8  Solve Ordinary Differential Equations (ODEs).  Appreciate the importance of numerical methods in solving ODEs.  Assess the reliability of the different techniques.  Select the appropriate method for any particular problem.

3. CISE301_Topic8L1KFUPM3 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

4. CISE301_Topic8L1KFUPM4 L ecture 28 Lesson 1: Introduction to ODEs

5. CISE301_Topic8L1KFUPM5 Learning Objectives of Lesson 1  Recall basic definitions of ODEs: Order Linearity Initial conditions Solution  Classify ODEs based on: Order, linearity, and conditions.  Classify the solution methods.

6. CISE301_Topic8L1KFUPM6 Partial Derivatives u is a function of more than one independent variable Ordinary Derivatives v is a function of one independent variable Derivatives

7. CISE301_Topic8L1KFUPM7 Partial Differential Equations involve one or more partial derivatives of unknown functions Ordinary Differential Equations involve one or more Ordinary derivatives of unknown functions Differential Equations Differential Equations

8. CISE301_Topic8L1KFUPM8 Ordinary Differential Equations Ordinary Differential Equations (ODEs) involve one or more ordinary derivatives of unknown functions with respect to one independent variable x(t): unknown function t: independent variable

9. CISE301_Topic8L1KFUPM9 Example of ODE: Model of Falling Parachutist The velocity of a falling parachutist is given by:

10. CISE301_Topic8L1KFUPM10 Definitions Ordinary differential equation

11. CISE301_Topic8L1KFUPM11 Definitions (Cont.) (Dependent variable) unknown function to be determined

12. CISE301_Topic8L1KFUPM12 Definitions (Cont.) (independent variable) the variable with respect to which other variables are differentiated

13. CISE301_Topic8L1KFUPM13 Order of a Differential Equation The order of an ordinary differential equations is the order of the highest order derivative. Second order ODE First order ODE Second order ODE

14. CISE301_Topic8L1KFUPM14 A solution to a differential equation is a function that satisfies the equation. Solution of a Differential Equation

15. CISE301_Topic8L1KFUPM15 Linear ODE An ODE is linear if The unknown function and its derivatives appear to power one No product of the unknown function and/or its derivatives Linear ODE Non-linear ODE

16. CISE301_Topic8L1KFUPM16 Nonlinear ODE An ODE is linear if The unknown function and its derivatives appear to power one No product of the unknown function and/or its derivatives

17. CISE301_Topic8L1KFUPM17 Solutions of Ordinary Differential Equations Is it unique?

18. CISE301_Topic8L1KFUPM18 Uniqueness of a Solution In order to uniquely specify a solution to an n th order differential equation we need n conditions. Second order ODE Two conditions are needed to uniquely specify the solution

19. CISE301_Topic8L1KFUPM19 Auxiliary Conditions Boundary Conditions  The conditions are not at one point of the independent variable Initial Conditions  All conditions are at one point of the independent variable Auxiliary Conditions

20. CISE301_Topic8L1KFUPM20 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 problems Initial-Value Problems  The auxiliary conditions are at one point of the independent variable same different

21. CISE301_Topic8L1KFUPM21 Classification of ODEs ODEs can be classified in different ways:  Order First order ODE Second order ODE N th order ODE  Linearity Linear ODE Nonlinear ODE  Auxiliary conditions Initial value problems Boundary value problems

22. CISE301_Topic8L1KFUPM22 Analytical Solutions  Analytical Solutions to ODEs are available for linear ODEs and special classes of nonlinear differential equations.

23. CISE301_Topic8L1KFUPM23 Numerical Solutions  Numerical methods are used to obtain a graph or a table of the unknown function.  Most of the Numerical methods used to solve ODEs are based directly (or indirectly) on the truncated Taylor series expansion.

24. CISE301_Topic8L1KFUPM24 Classification of the Methods Numerical Methods for Solving ODE Multiple-Step Methods Estimates of the solution at a particular step are based on information on more than one step Single-Step Methods Estimates of the solution at a particular step are entirely based on information on the previous step