Page 286 - Excel for Scientists and Engineers: Numerical Methods

Chapter 12

Partial

Differential Equations

For a function F(x,y) that depends on more than one independent variable,
the partial derivative of the function with respect to a particular variable is the
derivative of the function with respect to that variable while holding the other
variables constant. For a function of two independent variables x and y, the
partial derivatives are B(x,y)/& (y held constant) and B(XJ)/+ (x held
constant). There are three second-order partial derivatives for the function
F(x,y): 8F(x,y)l&’, 8F(x,y)/&+ and ~?F(x,y)/$~.
Many physical systems are described by equations involving partial
differential equations (PDEs). In this chapter, discussion will be limited to linear
second-order partial differential equations in two independent variables. Typical
examples include the variation of a property in two spatial dimensions, or the
variation of a property as a function of time and distance.

Elliptic, Parabolic and Hyperbolic

Partial Differential Equations
A general form of the partial differential equation (up to the second order) is

(12-1)
where the coefficients a . . .fare functions of x and y. Of course, a particular
differential equation may be much simpler than equation 12-1. Depending on the
values of the coefficients a, b and cy a partial differential equation is classified as
elliptic, parabolic, or hyperbolic. A partial differential equation is elliptic if b2 -
4ac < 0, parabolic if b2 - 4ac = 0, hyperbolic if b2 - 4ac > 0.

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