Métodos Numéricos

Definición, Clasificación y

Solución de

Ecuaciones Diferenciales

1

Ecuaciones Diferenciales

Ordinarias (EDO)Lesson 1: Introduction to ODE

Objectives

� Solve Ordinary differential equation (ODE) problems.

� Appreciate the importance of numerical method in solving ODE.

2

method in solving ODE.

� Assess the reliability of the different techniques.

� Select the appropriate method for any particular problem.

Computer

� Develop programs to solve ODE.

� Use software packages to find the solution of ODE

3

Solution of ODE

Lesson 1:

4

Introduction to Ordinary Differential Equations (ODE)

Learning Objectives of Lesson 1

� Recall basic definitions of ODE,� order,

� linearity

� initial conditions,

5

� solution,

� Classify ODE based on( order, linearity, conditions)

� Classify the solution methods

Partial DerivativesOrdinary Derivatives

Derivatives

Derivatives

6

u is a function of more than one

independent variable

v is a function of one independent variable

dt

dvy

u

∂∂

Partial Differential EquationsOrdinary Differential Equations

Differential EquationsDifferentialEquations

∂∂

SE301_Topic 8 (c)Al-Amer 2006 7

involve one or more partial derivatives of unknown functions

involve one or more

Ordinary derivatives of

unknown functions

162

2

=+ tvdt

vd 02

2

2

2

=∂∂−

∂∂

x

u

y

u

Ordinary Differential EquationsOrdinary Differential Equations

:Examples

Ordinary Differential Equations (ODE) involve one or more ordinary derivatives of unknown functions with respect to one independent variable

8

)cos()(2)(

5)(

)()(

2

2

ttxdt

tdx

dt

txd

etvdt

tdv t

=+−

=− x(t): unknown function

t: independent variable

Example of ODE:

Model of falling parachutist

The velocity of a falling parachutist is given by

vM

c

td

vd8.9 −=

9

velocityv

tcoefficiendragc

massM

vMtd

:

:

:

8.9 −=

Definitions

vcdv −= 8.9

Ordinary differential equation

10

vMdt

−= 8.9

vcvd −= 8.9 (Dependent

11

vMtd

−= 8.9 (Dependent variable) unknown

function to be determined

vcvd −= 8.9

12

vMtd

−= 8.9

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

Order of a differential equationOrder of a differential equation

)()(

:

=− etxtdx

Examples

t

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

First order ODE

13

1)(2)()(

)cos()(2)(

5)(

)(

4

3

2

2

2

2

=+−

=+−

=−

txdt

tdx

dt

txd

ttxdt

tdx

dt

txd

etxdt

Second order ODE

First order ODE

Second order ODE

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

Solution of a differential equationSolution of a differential equation

0)()(

:

=+ txtdx

Example

:Proof

)( = −tetxSolution

14

0)()( =+ tx

dt

tdx

0)()(

)(

:Proof

=+−=+

−=

−−

−

tt

t

eetxdt

tdx

edt

tdx

Linear ODELinear ODE

)(

:

tdx

Examples

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

15

1)()()(

)cos()(2)(

5)(

)()(

3

2

2

22

2

=+−

=+−

=−

txdt

tdx

dt

txd

ttxtdt

tdx

dt

txd

etxdt

tdx t

Linear ODE

Linear ODE

Non-linear ODE

Nonlinear ODENonlinear ODE

:ODEnonlinear of Examples

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

16

1)()()(

2)()(

5)(

1))(cos()(

2

2

2

2

=+−

=−

=−

txdt

tdx

dt

txd

txdt

tdx

dt

txd

txdt

tdx

Solutions of Ordinary Differential Solutions of Ordinary Differential

EquationsEquations

0)(4)(

ODE thetosolution a is

)2cos()(

2

=+

=

txtxd

ttx

17

0)(42

=+ txdt

Is it unique?

solutions are constant) real a is (where

)2cos()( form theof functions All

c

cttx +=

Uniqueness of a solutionUniqueness of a solution

xyd )(2

In order to uniquely specify a solution to an n th order differential equation we need n conditions

18

by

ay

xydt

xyd

==

=+

)0(

)0(

0)(4)(

2

2

ɺ

Second order ODE

Two conditions are needed to uniquely specify the solution

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

� The auxiliary conditions are at one point of the

independent variable

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� More difficult to solve than initial value problem

5.1)2(,1)0(

2 2

===++ −

xx

exxx tɺɺɺ

variable

5.2)0(,1)0(

2 2

===++ −

xx

exxx t

ɺ

ɺɺɺ

same different

Classification of ODE

ODE can be classified in different ways

� Order� First order ODE

� Second order ODE

� Nth order ODE

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� Nth order ODE

� Linearity� Linear ODE

� Nonlinear ODE

� Auxiliary conditions� Initial value problems

� Boundary value problems

Analytical Solutions

� Analytical Solutions to ODE are available for linear ODE and special classes of nonlinear differential equations.

22

Numerical Solutions

� Numerical method are used to obtain a graph or a table of the unknown function

� Most of the Numerical methods used to solve ODE are based directly (or

23

solve ODE are based directly (or indirectly) on truncated Taylor series expansion

Dsolve

Resolucion de EDOs con solución exacta en

MATLAB

(simbólico)

Resolución de EDOs usando

métodos numéricos

Numerical Methods

for solving ODE

Multiple-Step MethodsSingle-Step Methods

SE301_Topic 8 (c)Al-Amer 2006 24

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

� Taylor series methods

� Midpoint and Heun’s method

� Runge-Kutta methods

� Multiple step Methods

Resolución de EDOs usando

métodos numéricos

SE301_Topic 8 (c)Al-Amer 2006 25

� Multiple step Methods

� Boundary value Problems