University Mathematics Handbook (2015)

XIV. Partial Differential Equations (PDE)

Chapter 1. Introduction

a.  A PDE is an equation relating an unknown multi-variable function and its partial differentials.

b.  If the highest order of differentiation of a PDE is , it is an -th order PDE.

c.  The general form of a PDE with 2 variables, is

1.  First order:

2.  Second order:

when is an unknown function and is a given function.

d.  Function is a solution or an integral of -th order PDE in domain if is times partially differentiable in , and, substituting it in the equation, one attain an identity in .

Example: The solution of is:

in domain

in domain

in domain .

e.  A PDE solution is called an integral surface.

f.  A general solution of a PDE includes all solutions of the equation.

Example: A general solution of is

where and are arbitrary differentiable functions.

g.   is a linear PDE if and (see 4.1).