CISE301
Download
1 / 28

CISE301 : Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36 - PowerPoint PPT Presentation


  • 140 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' CISE301 : Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36' - dorie


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

CISE301: Numerical MethodsTopic 8Ordinary Differential Equations (ODEs)Lecture 28-36

KFUPM

Read 25.1-25.4, 26-2, 27-1

KFUPM


Outline of topic 8
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


L ecture 33 lesson 6 solving systems of odes

Lecture 33Lesson 6: Solving Systems of ODEs

KFUPM


Learning objectives of lesson 6
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


Outlines of lesson 6
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


Solving a system of first order odes
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


Euler method for solving a system of first order odes
Euler Method for Solving a System of First Order ODEs

Recall Euler method for solving a first order ODE:

KFUPM


Example euler method
Example - Euler Method

Euler method to solve a system of n first order ODEs.

KFUPM


Solving a system of n first order odes
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


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

KFUPM


Example rk2 method for solving a system of first order odes
Example :RK2 method for solving a system of first order ODEs

KFUPM


Example rk2 method for solving a system of first order odes1
Example :RK2 method for solving a system of first order ODEs

KFUPM


Methods for solving a system of first order odes
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


High order odes
High Order ODEs

  • How to solve a second order ODE?

  • How to solve high order ODEs?

KFUPM


The general approach to solve odes
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


Conversion procedure
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


Remarks on the conversion procedure
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


Example of converting a high order ode to first order odes
Example of Converting a High Order ODE to First Order ODEs

One degree less than the highest order derivative

KFUPM



Example of converting a high order ode to first order odes2
Example of Converting a High Order ODE to First Order ODEs

One degree less than the highest order derivative

KFUPM



Conversion procedure for systems of high order odes
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


Example of converting a high order ode to first order odes4
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



Solution of a second order ode
Solution of a Second Order ODE

  • Solve the equation using Euler method. Use h=0.1

KFUPM



Summary
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


Remaining lessons in topic 8
Remaining Lessons in Topic 8

Solution of ODEs

Lesson 7:

Multi-step methods

Lessons 8-9:

Boundary Value Problems

KFUPM


ad