Numerical Methods

11

Partial Differential Equations

Partial differential equations appear in the description of physical processes in applied sciences and engineering. A differential equation that involves more than one independent variable is called a partial differential equation. We restrict ourselves to second order partial differential equations. The general second order linear partial differential equation is of the form

 

Au_xx + Bu_xy + Cu_yy + Du_x + Eu_y+ Fu = G,

where A, B, C, D, E, F, G are all functions of x and y. Equations of the above form can be classified into three types:

1. If B² – 4AC <; 0 at a point in the (x, y) plane, then the equation is called elliptic. For example, the equation u_xx + u_yy = 0, known as Laplace equation, is elliptic.
2. If B