Partial Differential Equations with MATLAB

1 Introduction

1.1 WhatarePartialDifferentialEquations?

Roughly speaking, apartial differential equation (PDE)is similar to anor-
dinary differential equation (ODE), except that the dependent variable is a
function of not just one, but of several independent variables. Let’s be more
precise. Given a functionu=u(x 1 ,x 2 ,...,xn), apartial differential equa-
tion (PDE)inuis an equation which relates any of the partial derivatives
ofuto each other and/or to any of the variablesx 1 ,x 2 ,...,xnandu.
Before doing some examples, we introduce a bit of notation: Instead of the
somewhat unwieldy∂u∂x, ∂

(^3) u
∂x^2 ∂yand the like, we will use subscripts whenever
possible. We write
ux=
∂u
∂x

.

For higher order derivatives, we read the subscripts from left to right. So, for
example,

uxy=

∂

∂y

(

∂u

∂x

)

=

∂^2 u

∂y∂x

.

However, for all practical purposes, the order of differentiation will not matter
to us. So, for example, we’ll have

uxzyx=uzxxy=uyxzx,etc.

Examples

1.ux+u= 0 is a PDE inu=u(x, y). However, it also could be a PDE

inu=u(x, y, z). In general, we only know the number of independent

variables from the context.

2. 2ux+3uz= 0 is a PDE inu(x, z), although, more likely, it is a PDE in
   u(x, y, z).

3.uxuyy−xy^3 u=euis a PDE inu(x, y).

4.z^2 uxxy−xcosyuyy+uy−e^3 zu=tany^2 zis a PDE inu(x, y, z).

5.u^5 xxz+uxxyz=uzzzis a PDE inu(x, y, z).

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