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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