Ordinary Differential Equations - Numerical Methods - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Ordinary Differential Equations

Ordinary Differential Equations

• A differential equation defines a relationship

between an unknown function and one or

more of its derivatives

• Physical problems using differential equations
  • electrical circuits
  • heat transfer
  • motion

Ordinary Differential Equations

• A second order differential equation would

have the form:

d y dx

f x y

dy dx

2 2 ^

 ^

 

, ,

does not necessarily have to include all of these variables

Ordinary Differential Equations

• An ordinary differential equation is one

with a single independent variable.

• Thus, the previous two equations are

ordinary differential equations

• The following is not:

dy dx

f x x y 1

 1 , 2 ,

Ordinary Differential Equations

• An ordinary differential equation of order n

requires that n conditions be specified.

• Boundary conditions
• Initial conditions

Ordinary Differential Equations

• An ordinary differential equation of order n

requires that n conditions be specified.

• Boundary conditions
• Initial conditions

consider this beam where the deflection is zero at the boundaries x= 0 and x = L These are boundary conditions

Ordinary Differential Equations

• At best, only a few differential equations can

be solved analytically in a closed form.

• Solutions of most practical engineering

problems involving differential equations

require the use of numerical methods.

Scope of Lectures on ODE

• One Step Methods
  • Euler’s Method
  • Heun’s Method
  • Improved Polygon
  • Runge Kutta
  • Systems of ODE
• Adaptive step size control

Specific Study Objectives

• Understand the visual representation of Euler’s,

Heun’s and the improved polygon methods.

• Understand the difference between local and

global truncation errors

• Know the general form of the Runge-Kutta

methods.

• Understand the derivation of the second-order RK

method and how it relates to the Taylor series

expansion.

Specific Study Objectives

• Realize that there are an infinite number of

possible versions for second- and higher-order RK

methods

• Know how to apply any of the RK methods to

systems of equations

• Understand the difference between initial value

and boundary value problems

y

x C

for y

C

then C for y

C

and C

 



 

 

 



4 3 0 1

1

4 0 3 1 0 2

2

4 0 3 2

3

3

3

What we see are different values of C for the two different initial conditions.

The resulting equations are:

y

x

y x

 

 

4 3

1

4 3

2

3

3

y

x

y(0)=

y(0)=

y(0)=

y(0)=

Euler’s Method

• The first derivative provides a direct estimate

of the slope at xi

• The equation is applied iteratively, or one step

at a time, over small distance in order to

reduce the error

• Hence this is often referred to as Euler’s One-

Step Method

EXAMPLE

2 4 x dx

dy 

For the initial condition y(1)=1, determine

y for h = 0.1 analytically and usingEuler’s

method given: