Mathematical Methods for Physicists: A concise introduction - Site Map

PARTIAL DIFFERENTIAL EQUATIONS
Figure 10.4.
…; '† ˆ E 0 cos ' ˆE 0 x for !1;
0 for ˆ l;
where the x-axis has been taken in the direction of the uniform ®eld E 0 .
10.6 Obtain the solution of the heat conduction equation
@ 2 u…x; t†
@x 2 ˆ 1 @u…x; t†
@t
which satis®es the boundary conditions
(1) u…0; t† ˆu…l; t† ˆ0; t 0; (2) u…x; 0† ˆf …x†; 0 x, where f …x† is a
given function and l is a constant.
10.7 If a battery is connected to the plates as shown in Fig. 10.4, and if the charge
density distribution between the two plates is still given by …x†, ®nd the
potential distribution between the plates.
10.8 Find the Green's function that satis®es the equation
d 2 G
dx 2 ˆ …x x 0 †
and the boundary conditions G ˆ 0 when x ˆ 0 and G remains bounded
when x approaches in®nity. (This Green's function is the potential due to a
surface charge " per unit area on a plane of in®nite extent located at x ˆ x 0
in a dielectric medium of permittivity " when a grounded conducting plane
of in®nite extent is located at x ˆ 0.)
10.9 Solve by Laplace transforms the boundary-value problem
@ 2 u
@x 2 ˆ 1 @u
for x > 0; t > 0;
K @t
given that u ˆ u 0 (a constant) on x ˆ 0 for t > 0, and u ˆ 0 for x > 0; t ˆ 0.
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