Mathematical Methods for Physicists: A concise introduction - Site Map

PARTIAL DIFFERENTIAL EQUATIONS
Now think of this as a di€erential equation in terms of x, withp as a parameter.
Then taking the Laplace transform of the given conditions u…0; t† ˆu…3; t† ˆ0,
we have
or
L‰u…0; t†Š ˆ 0; L‰u…3; t†Š ˆ 0
U…0; p† ˆ0; U…3; p† ˆ0:
These are the boundary conditions on U…x; p†. Solving Eq. (10.76) subject to these
conditions we ®nd
sin 2x 3 sin 4x
U…x; p† ˆ5
p ‡ 162 p ‡ 64 2 :
The solution to Eq. (10.74) can now be obtained by taking the inverse Laplace
transform
u…x; t† ˆL 1 ‰U…x; p†Š ˆ 5e 162t sin 2x 3e 642 sin 4x:
The Fourier transform method was used in Chapter 4 for the solution of
ordinary linear ordinary di€erential equations with constant coecients. It can
be extended to solve a variety of partial di€erential equations. However, we shall
not discuss this here. Also, there are other methods for the solution of linear
partial di€erential equations. In general, it is a dicult task to solve partial
di€erential equations analytically, and very often a numerical method is the
best way of obtaining a solution that satis®es given boundary conditions.
Problems
10.1 (a) Show that y…x; t† ˆF…2x ‡ 5t†‡G…2x 5t† is a general solution of
4 @2 y
@t 2 ˆ 25 @2 y
@x 2 :
(b) Find a particular solution satisfying the conditions
y…0; t† ˆy…; t† ˆ0; y…x; 0† ˆsin 2x; y 0 …x; 0† ˆ0:
10.2. State the nature of each of the following equations (that is, whether elliptic,
parabolic, or hyperbolic)
…a† @2 y
@t 2 ‡ @2 y
@x 2 ˆ 0;
…b† x @2 u
@x 2 ‡ y @2 u
@y 2 ‡ 3y2 @u
@x :
10.3 The electromagnetic wave equation: Classical electromagnetic theory was
worked out experimentally in bits and pieces by Coulomb, Oersted, Ampere,
Faraday and many others, but the man who put it all together and built it
into the compact and consistent theory it is today was James Clerk Maxwell.
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