CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36 KFUPM Read 25.1-25.4, 26-2, 27-1 CISE301_Topic8L1 KFUPM

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. CISE301_Topic8L1 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 CISE301_Topic8L1 KFUPM

Lecture 28 Lesson 1: Introduction to ODEs CISE301_Topic8L1 KFUPM

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. CISE301_Topic8L1 KFUPM

Derivatives Derivatives Partial Derivatives Ordinary Derivatives v is a function of one independent variable Partial Derivatives u is a function of more than one independent variable CISE301_Topic8L1 KFUPM

Differential Equations Ordinary Differential Equations involve one or more Ordinary derivatives of unknown functions Partial Differential Equations involve one or more partial derivatives of unknown functions CISE301_Topic8L1 KFUPM

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 CISE301_Topic8L1 KFUPM

Example of ODE: Model of Falling Parachutist The velocity of a falling parachutist is given by: CISE301_Topic8L1 KFUPM

Ordinary differential equation Definitions Ordinary differential equation CISE301_Topic8L1 KFUPM

(Dependent variable) unknown function to be determined Definitions (Cont.) (Dependent variable) unknown function to be determined CISE301_Topic8L1 KFUPM

Definitions (Cont.) (independent variable) the variable with respect to which other variables are differentiated CISE301_Topic8L1 KFUPM

Order of a Differential Equation The order of an ordinary differential equations is the order of the highest order derivative. First order ODE Second order ODE Second order ODE CISE301_Topic8L1 KFUPM

Solution of a Differential Equation A solution to a differential equation is a function that satisfies the equation. CISE301_Topic8L1 KFUPM

Linear ODE Linear ODE Non-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 CISE301_Topic8L1 KFUPM

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 CISE301_Topic8L1 KFUPM

Solutions of Ordinary Differential Equations Is it unique? CISE301_Topic8L1 KFUPM

Uniqueness of a Solution In order to uniquely specify a solution to an nth order differential equation we need n conditions. Second order ODE Two conditions are needed to uniquely specify the solution CISE301_Topic8L1 KFUPM

Auxiliary Conditions 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 CISE301_Topic8L1 KFUPM

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 CISE301_Topic8L1 KFUPM

Classification of ODEs ODEs can be classified in different ways: Order First order ODE Second order ODE Nth order ODE Linearity Linear ODE Nonlinear ODE Auxiliary conditions Initial value problems Boundary value problems CISE301_Topic8L1 KFUPM

Analytical Solutions Analytical Solutions to ODEs are available for linear ODEs and special classes of nonlinear differential equations. CISE301_Topic8L1 KFUPM

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. CISE301_Topic8L1 KFUPM

Classification of the Methods Numerical Methods for Solving ODE Single-Step Methods Estimates of the solution at a particular step are entirely based on information on the previous step Multiple-Step Methods Estimates of the solution at a particular step are based on information on more than one step CISE301_Topic8L1 KFUPM