Introduction to Numerical Analysis I

MATH/CMPSC 455

Ordinary Differential Equations

MODEL PROBLEM

EULER’S METHOD

Example

Example

TAYLOR SERIES METHOD

Idea: keep more terms in the Taylor expansion

A Example (keep second order term)

Example

RUNGE-KUTTA METHODS

A drawback of Taylor Series Method is that it involves derivatives of

Idea: use to express derivatives

2nd order Runge-Kutta Methods:

4th order Runge-Kutta Methods:

BACKWARD EULER METHOD

Backward Euler Method:

Differences:• Implicit• Need to solve an equation (maybe

expensive)

COMPARISON (FROM FPI POINT OF VIEW)

Example:

COMPARISON (FROM STABILITY POINT OF VIEW)

Example:

HOW TO SOLVE THE EXTRA EQUATION

Example: