Page 145 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

SOLUTIONS WITH LAPLACE TRANSFORMS 135
8.3 DUHAMEL’S THEOREM
We are now prepared to solve the more general problem

∂u
2
∇ u + g(r, t) = (8.1)
∂t
where r may be considered a vector, that is, the problem is in three dimensions. The general
boundary conditions are
∂u
+ h i u = f i (r, t) on the boundary S i (8.2)
∂n i
and

u(r, 0) = F(r) (8.3)

initially. Here ∂u represents the normal derivative of u at the surface. We present Duhamel’s
∂n i
theorem without proof.
Consider the auxiliary problem
∂ P
2
∇ P + g(r,λ) = (8.4)
∂t
where λ is a timelike constant with boundary conditions

∂ P
+ h i P = f i (r,λ) on the boundary S i (8.5)
∂n i
and initial condition

P(r, 0) = F(r) (8.6)

The solution of Eqs. (8.1), (8.2), and (8.3) is as follows:

t t
∂ ∂
u(x, y, z, t) = P(x, y, z,λ, t − λ)dλ = F(x, y, z) + P(x, y, z,λ, t − λ)dλ
∂t ∂t
λ=0 λ=0
(8.7)
This is Duhamel’s theorem. For a proof, refer to the book by Arpaci.

Example 8.8. Consider now the following problem with a time-dependent heat source:

u t = u xx + xe −t
u(0, t) = u(1, t) = 0

u(x, 0) = 0