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INITIAL AND BOUNDARY CONDITIONS
THE HEAT EQUATION ON [0, L]
CHAPTER 17 SOLUTIONS IN AN INFINITE MEDIUM
LAPLACE TRANSFORM TECHNIQUES HEAT
The Heat

Equation

17.1 Initial and Boundary Conditions

In Section 12.8, we used Gauss’s divergence theorem to derive a partial differential equation
modeling heat distribution, or diffusion. In the absence of sources or sinks within the medium,
the one-dimensional heat equation is
∂u ∂ u
2
= k (17.1)
∂t ∂x 2
in which k is a constant depending on the medium.
Equation (17.1) can be solved subject to a variety of boundary and initial conditions. For
example,
2
∂u ∂ u
= k for 0 < x < L,t > 0,
∂t ∂x 2
u(0,t) = T 1 ,u(L,t) = T 2 for t ≥ 0,
u(x,0) = f (x) for 0 ≤ x ≤ L
models the temperature distribution in a thin homogeneous bar of length L whose left end is
kept at temperature T 1 and right end at temperature T 2 , and having initial temperature f (x) in the
cross section at x.
The initial-boundary value problem
∂u ∂ u
2
= k for 0 < x < L,t > 0,
∂t ∂x 2
∂u ∂u
(0,t) = (L,t) = 0for t ≥ 0,
∂x ∂x
and
u(x,0) = f (x) for 0 ≤ x ≤ L

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October 14, 2010 15:25 THM/NEIL Page-611 27410_17_ch17_p611-640