Partial Differential Equations

This chapter presents a brief introduction to some techniques available for approximating the solution to partial differential equations (PDEs) of the form

A ∂2

∂ x2 u(x, y) +B

∂ x ∂ y u(x, y) + C

∂ y2 u(x, y) = f

( x, y, u,

∂u

∂ x , ∂u

∂ y

Eqns. (15.1) are classified into three categories depending on the values of the coefficients A, B, and C:

If B2 − 4AC > 0, the equation is hyperbolic. If B2 − 4AC = 0, the equation is parabolic. If B2 − 4AC < 0, the equation is elliptic.