Partial Differential Equations with MATLAB

Introduction 25

(The last two boundary conditions are encountered in connection with

the Euler–Bernoulli beam PDE, for example.)

39. Decide whether the given function is aproductfunction, that is, if it
    canbewrittenintheformf(x)g(y) for functionsfandg.Ifitisnot,
    justify your answer.

a)u(x, y)=exy

b) u(x, y)=e^2 x−^3 y

c) u(x, y)=y^2 −xy+1

d) u(x, y)=sin(x+y)

1.7 EigenvalueProblems

In Section 1.3, we discussed the heat equation, subject to initial and boundary
conditions. Suppose, for instance, we’re solving the heat equationut=uxx
on the interval 0≤x≤1. Suppose, further, that the equation is subject to
the boundary conditions

u(0,t)=u(1,t)=0

fort>0. We first separate the PDE, resulting in the ODEs

X′′+λX=0 and T′+λT=0.

Then, as in Exercise 38 in the previous section, we separate the boundary
conditions, as follows:

u(0,t)=X(0)T(t) = 0 for allt> 0 ⇒X(0) = 0,orT(t) = 0 for allt> 0.

So we have two types of product solutions of the PDE which satisfy the
left boundary condition: those which satisfyX(0) = 0 and those for which
T(t)≡0, that is, those for whichT(t) is the zero-function. But the latter
gives us the zero-solution (which we already know is a solution). So the only
nontrivial product solutions which satisfy the left boundary condition are
those which satisfyX(0) = 0.
Similarly, the only nontrivial product solutions which satisfy the right bound-
ary condition will satisfyX(1) = 0. So, we actually need to solve the system

X′′+λX=0 T′+λT=0

X(0) =X(1) = 0.

TheX-system looks like the problems in Section 1.3, Exercises 1–5. Essen-
tially, then, it is anODE boundary-value problem. However, it differs from