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612 CHAPTER 17 The Heat Equation

models the distribution in a bar of length L having no heat loss across its ends (insulation
conditions) and initial temperature function f .
Other kinds of boundary conditions also can be specified. If the left end is kept at constant
temperature T and the right end is insulated, then we would have
∂u
u(0,t) = T and (L,t) = 0for t > 0.
∂x
Free radiation or convection occurs when the bar loses energy by radiation from its ends into
the surrounding medium, which is assumed to be maintained at constant temperature T .Now the
boundary conditions have the form
∂u ∂u
(0,t) = A[u(0,t) − T ], (L,t) =−A[u(L,t) − T ] for t ≥ 0,
∂x ∂x
in which A is a positive constant. Notice that, if the bar is kept hotter than the surrounding
medium, then the heat flow as measured by ∂u/∂x must be positive at one end and negative at
the other.
Boundary conditions
∂u
u(0,t) = T 1 , (L,t) =−A[u(L,t) − T 2 ]
∂x
are used if the left end is kept at constant temperature T 1 while the right end radiates heat energy
into a medium of constant temperature T 2 .
As with the wave equation, we also consider the heat equation on the line or half-line, subject
to various conditions.

SECTION 17.1 PROBLEMS

1. Formulate an initial-boundary value problem modeling right end at temperature β(t). The initial temperature
heat conduction in a thin homogeneous bar of length L function in the cross section at x is f (x).
if the left end is kept at temperature zero and the right
3. Formulate an initial-boundary value problem for the
end is insulated. The initial temperature function is f .
temperature distribution in a thin bar of length L if
2. Formulate an initial-boundary value problem modeling the left end is insulated and the right end is kept at
heat conduction in a thin homogeneous bar of length temperature β(t). The initial temperature function is f .
L if the left end is kept at temperature α(t) and the

17.2 The Heat Equation on [0, L]

We will solve several initial-boundary value problems on an interval [0, L].

17.2.1 Ends Kept at Temperature Zero

If the initial temperature in the cross section at x is f (x) and the ends of the bar are kept at
temperature zero, the problem for the temperature distribution function is
2
∂u ∂ u
= k for 0 < x < L,t > 0,
∂t ∂x 2
u(0,t) = u(L,t) = 0for t ≥ 0,
and
u(x,0) = f (x) for 0 ≤ x ≤ L.

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