Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
amount of heat entering S per unit time is
ZZ
… KrT† ^ndS;
S
S
where ^n is an outward unit vector normal to element surface area dS. Using the
divergence theorem, this can be written as
ZZ
ZZZ
… KrT† ^ndS ˆ r… KrT†dV: …4:38†
Now the heat contained in V is given by
ZZZ
cTdV;
V
where c and are respectively the speci®c heat capacity and density of the solid.
Then the time rate of increase of heat is
ZZZ
ZZZ
@
cTdV ˆ c @T
@t
@t dV:
…4:39†
V
Equating the right hand sides of Eqs. (4.38) and (4.39) yields
ZZZ
c @T
r KrT
@t … † dV ˆ 0:
V
Since V is arbitrary, the integrand (assumed continuous) must be identically zero:
or if K, c, are constants
c @T
@t
V
V
ˆr … KrT †
@T
@t ˆ krrT ˆ kr2 T;
…4:40†
where k ˆ K=c. This is the required equation for heat conduction and was ®rst
developed by Fourier in 1822. For the semiin®nite thin bar, the boundary conditions
are
T…x; 0† ˆf …x†; T…0; t† ˆ0; jT…x; t†j < M; …4:41†
where the last condition means that the temperature must be bounded for physical
reasons.
A solution of Eq. (4.40) can be obtained by separation of variables, that is by
letting
Then
T ˆ X…x†H…t†:
XH 0 ˆ kX 00 H or X 00 =X ˆ H 0 =kH:
180