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

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.

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

CISE301_Topic8L1KFUPM4 L ecture 28 Lesson 1: Introduction to ODEs

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.

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

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

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

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

CISE301_Topic8L1KFUPM10 Definitions Ordinary differential equation

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

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

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

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

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

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

CISE301_Topic8L1KFUPM17 Solutions of Ordinary Differential Equations Is it unique?

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

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

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

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

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

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.

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