Page 287 - Excel for Scientists and Engineers: Numerical Methods

264 EXCEL: NUMERICAL METHODS

In many physical models, x represents space and y represents time. The
partial differential equation known as Laplace's equation (equation 12-2) is an
example of an elliptic partial differential equation.

( 12-2)

Elliptic equations are often used to describe the steady-state value of a function
in two dimensions. Parabolic partial differential equations are often used to
describe how a quantity varies with respect to both distance and time. The one-
dimensional thermal diffusion equation
( 12-3)

describing the temperature T = F(x,t) at position x and time t in a material with
thermal diffusion coefficient K is an example of a parabolic equation (a = b = 0, c
= K, thus b2 - 4ac = 0). A similar equation, Fick's Second Law, describes the
diffusion of molecules or ions in solution, diffusion of dopant atoms into a
semiconductor, and so on.
Hyperbolic partial differential equations, involving the second derivative
with respect to time, are used to describe oscillatory systems. The wave equation
in one dimension,

(1 2-4)

describes the vibration of a violin string. Equation 12-4 is an example of a
hyperbolic partial differential equation (a = -k, b = 0, c = 1, thus b2 - 4ac = 4k).
Other applications include the vibration of structural members or the
transmission of sound waves.
In the previous chapter, some general methods were described that could be
applied to any system of ordinary differential equations. In contrast, different
methods of solution are required in order to solve partial differential equations of
these three different types. The following sections will illustrate the different
methods for solving elliptic, parabolic and hyperbolic partial differential
equations.

Elliptic Partial Differential Equations

Elliptic equations describe the value of a function in two spatial dimensions.
Elliptic partial differential equations have boundary conditions which are
specified around a closed boundary, while hyperbolic and parabolic partial
differential equations have at least one open boundary. Since the values are