Page 274 - Applied Numerical Methods Using MATLAB

6

ORDINARY DIFFERENTIAL

EQUATIONS

Differential equations are mathematical descriptions of how the variables and
their derivatives (rates of change) with respect to one or more independent
variable affect each other in a dynamical way. Their solutions show us how
the dependent variable(s) will change with the independent variable(s). Many
problems in natural sciences and engineering fields are formulated into a scalar
differential equation or a vector differential equation—that is, a system of dif-
ferential equations.
In this chapter, we look into several methods of obtaining the numerical solu-
tions to ordinary differential equations (ODEs) in which all dependent variables
(x) depend on a single independent variable (t). First, the initial value problems
(IVPs) will be handled with several methods including Runge–Kutta method and
predictor–corrector methods in Sections 6.1 to 6.5. The final section (Section 6.6)
will introduce the shooting method and the finite difference method for solving
the two-point boundary value problem (BVP). ODEs are called an IVP if the
values x(t 0 ) of dependent variables are given at the initial point t 0 of the inde-
pendent variable, while they are called a BVP if the values x(t 0 )/ x(t f ) are given
at the initial/final points t 0 and t f .

6.1 EULER’S METHOD

When talking about the numerical solutions to ODEs, everyone starts with the
Euler’s method, since it is easy to understand and simple to program. Even though
its low accuracy keeps it from being widely used for solving ODEs, it gives us a


Applied Numerical Methods Using MATLAB , by Yang, Cao, Chung, and Morris
Copyright  2005 John Wiley & Sons, I nc., ISBN 0-471-69833-4

263