Partial Differential Equations with MATLAB

4 An Introduction to Partial Differential Equations with MATLAB©R

Very important in the categorization of PDEs, as it is for ODEs, is a PDE’s
order. We define theorderof a PDE to be the order of the highest derivative
which appears in the equation. So, for example, the orders of the PDEs in
Examples 1–5 are, respectively, 1, 1, 2, 3 and 4.
As with ODEs, the dependent variable in a PDE generally isunknownand
we wish tosolvefor it. Asolutionof a PDE, then, is any functionuwhich
satisfies the PDE identically, that is, for all possible values of the independent
variables.

Examples

6.u(x, y)=e−x,u(x, y)=5e−x,u(x, y)=ye−xandu(x, y)=y^3 cosye−x

all are solutions ofux+u= 0, since, in each case,ux=−u.

7.u(x, y, z)=y^2 andu(x, y, z)=y^5 cosyare both solutions of 2ux+

3 uz=0. Infact,soisu(x, y, z)=f(y) for any (wellalmostany—see

below) functionf.Also,u(x, y, z)=e^3 x−^2 zandu(x, y, z)=ye^3 x−^2 zare

solutions.

8.u(x, y)=cx+dis a solution of

u^5 xxz+uxxyz=uzzz

for any choice of the constantscandd.

Frequently, we will seek functions which are solutions of a given PDE in
some restricted region. Also, in order to ensure that there is never a problem
with the order of differentiation,we will require any solutionuof annth-order
PDE to have the property that all of thenthpartial derivatives ofuexist and
are continuous.
We conclude this section with a list of important PDEs which arise from
physical problems (most of which we will study in some detail):

ut+cux=0 convection(oradvectionortransport)

equation

ut+uux=0 Burger’s equation(from the study

of the dynamics of gases)

u^2 x+u^2 y=1 eikonal equation(from optics)

ut=α^2 uxx heat equation(in one space variable)

utt=c^2 (uxx+uyy) wave equation(in two space variables)

uxx+uyy+uzz Schr ̈odinger’s equation(time independent,

+[E−V(x, y, z)]u= 0 in three space variables; from

quantum mechanics)

urr+^1 rur+r^12 uθθ=0 Laplace’s equation(in polar coordinates)

utt+α^4 uxxxx=0 Euler–Bernoulli beam equation

In Appendix D we provide a complete list of the PDEs covered in this text,
along with many other important PDEs, organized by application.