SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36 KFUPM (Term 101) Section 04 Read 25.1-25.4, 26-2, 27-1 CISE301_Topic8L6

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_Topic8L6

Lecture 33 Lesson 6: Solving Systems of ODEs CISE301_Topic8L6

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. CISE301_Topic8L6

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. CISE301_Topic8L6

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. CISE301_Topic8L6

Euler Method for Solving a System of First Order ODEs Recall Euler method for solving a first order ODE: CISE301_Topic8L6

Example - Euler Method Euler method to solve a system of n first order ODEs. CISE301_Topic8L6

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. CISE301_Topic8L6

Example : Euler method for solving a system of first order ODEs. CISE301_Topic8L6

Example : RK2 method for solving a system of first order ODEs CISE301_Topic8L6

Example : RK2 method for solving a system of first order ODEs CISE301_Topic8L6

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. CISE301_Topic8L6

High Order ODEs How do solve a second order ODE? How do solve high order ODEs? CISE301_Topic8L6

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 CISE301_Topic8L6

System of first order ODEs Conversion Procedure Convert Solve High order ODE System of first order ODEs 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. CISE301_Topic8L6

Remarks on the Conversion Procedure Convert Solve High order ODE System of first order ODE 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. CISE301_Topic8L6

Example of Converting a High Order ODE to First Order ODEs One degree less than the highest order derivative CISE301_Topic8L6

Example of Converting a High Order ODE to First Order ODEs CISE301_Topic8L6

Example of Converting a High Order ODE to First Order ODEs One degree less than the highest order derivative CISE301_Topic8L6

Example of Converting a High Order ODE to First Order ODEs CISE301_Topic8L6

Conversion Procedure for Systems of High Order ODEs Convert Solve System of high order ODEs System of first order ODE 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. CISE301_Topic8L6

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 CISE301_Topic8L6

Example of Converting a High Order ODE to First Order ODEs CISE301_Topic8L6

Solution of a Second Order ODE Solve the equation using Euler method. Use h=0.1 CISE301_Topic8L6

Solution of a Second Order ODE CISE301_Topic8L6

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. CISE301_Topic8L6

Remaining Lessons in Topic 8 Solution of ODEs Lesson 7: Multi-step methods Lessons 8-9: Boundary Value Problems CISE301_Topic8L6