Introduction

A Partial Differential Equation (PDE) is an equation relating a function of two or more independent variables and its partial derivatives. Consider the general equation

$$a{\partial^{2}u\over \partial x^{2}} + b{\partial^{2}u\over... ...\over \partial x} + e{\partial u \over \partial t} + fu = g,$$ (1.1)

where $a$, $b$, $c$, $d$, $e$ and $f$ may be functions of $x$, $t$ and even $u$. The order of the equation is given by the order of the highest derivative. Thus, if one of the functions, $a$, $b$ or $c$ are non-zero, then the PDE is second order. If $a = b = c = 0$ but $d$ or $e$ are non-zero, then the PDE is first order.

If the functions $a$, $b$, $c$, $d$, $e$ and $f$ do not depend on the dependent variable $u$, then Equation (1.1) is linear otherwise it is non-linear. We are mainly concerned with linear PDEs in this course.

If $g = 0$, then Equation (1.1) is honogeneous otherwise it is inhomogeneous.

Example 1. .1First order Partial Differential Equations.

1. 
   
   $${\partial u\over \partial x} - {\partial u\over \partial t} = 0,$$
   
   
           linear PDE.

2. 
   
   $$u{\partial u\over \partial x} - {\partial u\over \partial t} = 0,$$
   
   
           non-linear PDE.

3. 
   
   $$e^{x}{\partial u\over \partial x} + 4{\partial u\over \partial t} = t,$$
   
   
           linear, inhomogeneous PDE.

Example 1. .2Second order Partial Differential Equations.

1. 
   
   $${\partial^{2} u\over \partial x^{2}} + 4{\partial u\over \partial t} = 0,$$
   
   
           linear PDE.

2. 
   
   $${1\over x}{\partial \partial x}\left (x{\partial u\over \pa... ... x}\right ) - {\partial^{2} u\over \partial t^{2}} = x^{2},$$
   
   
           linear, inhomogeneous PDE.

3. 
   
   $${\partial^{2} u\over \partial x^{2}} - e^{2x}{\partial^{2} u\over \partial t^{2}} = u^{3},$$
   
   
           non-linear PDE.