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

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CISE301: Numerical Methods

Topic 8 Ordinary Differential

Equations (ODEs)Lecture 28-36

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

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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.

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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

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Lecture 28Lesson 1: Introduction

to ODEs

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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.

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Partial Derivatives

u is a function of more than one

independent variable

Ordinary Derivatives

v is a function of one independent variable

Derivatives Derivatives

dtdv

yu

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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 DifferentialEquations

162

2

tvdtvd 02

2

2

2

xu

yu

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Ordinary Differential Equations

)cos()(2)(5)(

)()(:

2

2

ttxdttdx

dttxd

etvdttdv

Examples

t

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

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Example of ODE:Model of Falling Parachutist

The velocity of a falling parachutist is given by:

velocityvtcoefficiendragc

massM

vMc

tdvd

::

:

8.9

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Definitions

vMc

dtdv

8.9

Ordinary differential equation

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Definitions (Cont.)

vMc

tdvd

8.9 (Dependent variable) unknown

function to be determined

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Definitions (Cont.)

vMc

tdvd

8.9

(independent variable) the variable with respect to which other variables are differentiated

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Order of a Differential Equation

1)(2)()(

)cos()(2)(5)(

)()(:

43

2

2

2

2

txdttdx

dttxd

ttxdttdx

dttxd

etxdttdx

Examples

t

The order of an ordinary differential equations is the order of the highest order derivative.

Second order ODE

First order ODE

Second order ODE

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A solution to a differential equation is a function that satisfies the equation.

Solution of a Differential Equation

0)()(:

txdttdx

Example

0)()(

)(:Proof

)(

tt

t

t

eetxdttdx

edttdx

etxSolution

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Linear ODE

1)()()(

)cos()(2)(5)(

)()(:

3

2

2

22

2

txdttdx

dttxd

ttxtdttdx

dttxd

etxdttdx

Examples

t

An ODE is linear ifThe unknown function and its derivatives appear to power one

No product of the unknown function and/or its derivatives

Linear ODE

Linear ODENon-linear ODE

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Nonlinear ODE

1)()()(

2)()(5)(

1))(cos()(:ODEnonlinear of Examples

2

2

2

2

txdttdx

dttxd

txdttdx

dttxd

txdttdx

An ODE is linear ifThe unknown function and its derivatives appear to power one

No product of the unknown function and/or its derivatives

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Solutions of Ordinary Differential Equations

0)(4)(

ODE thetosolution a is)2cos()(

2

2

txdttxd

ttx

Is it unique?

solutions. are constant) real a is (where)2cos()( form theof functions All

ccttx

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Uniqueness of a Solution

byay

xydtxyd

)0()0(

0)(4)(2

2

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

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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

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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

5.1)2(,1)0(2 2

xxexxx t

Initial-Value Problems

The auxiliary conditions are at one point of the independent variable

5.2)0(,1)0(2 2

xxexxx t

same different

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Classification of ODEsODEs 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

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Analytical Solutions Analytical Solutions to ODEs are available

for linear ODEs and special classes of nonlinear differential equations.

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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.

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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