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
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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
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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.
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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
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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
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