Danny's VBA Scripts

Elliptic Parabolic and Hyperbolic Partial Differential Equations

• Project Management Template Kit

A general form of the partial differential equation (up to the second order) is d2F{x,y) , ud2F(x,y) J2F(x,y) dF(x,y) 8F(x,y)

dx2 dxdy dy2 dx dy

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 c, 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.

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.

8x2 dy2

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 dT d2 T

dt dx2

describing the temperature T = F(x,t) at position jc 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,

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.

0 0