Mathematical Methods for Physicists: A concise introduction - Site Map

10
Partial di€erential equations
We have met some partial di€erential equations in previous chapters. In this
chapter we will study some elementary methods of solving partial di€erential
equations which occur frequently in physics and in engineering. In general, the
solution of partial di€erential equations presents a much more dicult problem
than the solution of ordinary di€erential equations. A complete discussion of the
general theory of partial di€erential equations is well beyond the scope of this
book. We therefore limit ourselves to a few solvable partial di€erential equations
that are of physical interest.
Any equation that contains an unknown function of two or more variables and
its partial derivatives with respect to these variables is called a partial di€erential
equation, the order of the equation being equal to the order of the highest partial
derivatives present. For example, the equations
3y 2 @u
@x ‡ @u
@y ˆ 2u;
@ 2 u
@x@y ˆ 2x y
are typical partial di€erential equations of the ®rst and second orders, respectively,
x and y being independent variables and u…x; y† the function to be found.
These two equations are linear, because both u and its derivatives occur only to
the ®rst order and products of u and its derivatives are absent. We shall not
consider non-linear partial di€erential equations.
We have seen that the general solution of an ordinary di€erential equation
contains arbitrary constants equal in number to the order of the equation. But
the general solution of a partial di€erential equation contains arbitrary functions
(equal in number to the order of the equation). After the particular choice of the
arbitrary functions is made, the general solution becomes a particular solution.
The problem of ®nding the solution of a given di€erential equation subject to
given initial conditions is called a boundary-value problem or an initial-value
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