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376 Chapter 6 Laplace Transform

6.3 Partial Differential Equations

In applying the Laplace transform to partial differential equations, we treat
variables other thantas parameters. Thus, the transform of a functionu(x,t)
is defined by

L

(

u(x,t)

)

=

∫∞

0

e−stu(x,t)dt=U(x,s).

For instance, we easily find the transforms

L

(

e−atsin(πx)

)

=

1

s+asin(πx),

L

(

sin(x+t)

)

=

ssin(x)+cos(x)

s^2 + 1.

The transformUnaturally is a function not only ofsbut also of the “untrans-
formed” variablex. We assume that derivatives or integrals with respect to the
untransformed variable pass through the transform

L

(∂u

∂x

)

=

∫∞

0

∂u(x,t)

∂x

e−stdt

= ∂

∂x

∫∞

0

u(x,t)e−stdt= ∂

∂x

(

U(x,s)

)

.

Ifwewishtofocusontheroleofxas a variable and keepsin the background
as a parameter, we might use the symbol for the ordinary derivative:

L

(∂u

∂x

)

=

dU

dx.

The rule for transforming a derivative with respect totcan be found, as
before, with integration by parts:

L

(∂u

∂t

)

=sL

(

u(x,t)

)

−u(x, 0 ).

If the Laplace transform is applied to a boundary value–initial value prob-
lem inxandt, all time derivatives disappear, leaving an ordinary differential
equation inx. We shall illustrate this technique with some trivial examples.
Incidentally,weassumefromhereonthatproblemshavebeenprepared(for
example, by dimensional analysis) so as to eliminate as many parameters as
possible.