Mathematical Methods for Physicists: A concise introduction - Site Map

PARTIAL DIFFERENTIAL EQUATIONS
problem. We have seen already that such problems often lead to eigenvalue
problems.
Linear second-order partial di€erential equations
Many physical processes can be described to some degree of accuracy by linear
second-order partial di€erential equations. For simplicity, we shall restrict our
discussion to the second-order linear partial di€erential equation in two independent
variables, which has the general form
A @2 u
@x 2 ‡ B @2 u
@x@y ‡ C @2 u
@y 2 ‡ D @u
@x ‡ E @u
@y
‡ Fu ˆ G; …10:1†
where A; B; C; ...; G may be dependent on variables x and y.
If G is a zero function, then Eq. (10.1) is called homogeneous; otherwise it is
said to be non-homogeneous. If u 1 ; u 2 ; ...; u n are solutions of a linear homogeneous
partial di€erential equation, then c 1 u 1 ‡ c 2 u 2 ‡‡c n u n is also a solution,
where c 1 ; c 2 ; ...are constants. This is known as the superposition principle; it does
not apply to non-linear equations. The general solution of a linear non-homogeneous
partial di€erential equation is obtained by adding a particular solution
of the non-homogeneous equation to the general solution of the homogeneous
equation.
The homogeneous form of Eq. (10.1) resembles the equation of a general conic:
ax 2 ‡ bxy ‡ cy 2 ‡ dx ‡ ey ‡ f ˆ 0:
We thus say that Eq. (10.1) is of
9
8
elliptic >=
>< B 2 4AC < 0
hyperbolic
parabolic
>; type when B 2 4AC > 0 :
>:
B 2 4AC ˆ 0
For example, according to this classi®cation the two-dimensional Laplace equation
@ 2 u
@x 2 ‡ @2 u
@y 2 ˆ 0
is of elliptic type (A ˆ C ˆ 1; B ˆ D ˆ E ˆ F ˆ G ˆ 0†, and the equation
@ 2 u
@x 2 @ 2 u
2
@y 2 ˆ 0
is of hyperbolic type. Similarly, the equation
is of parabolic type.
@ 2 u
@x 2 @u
@y ˆ 0
… is a real constant†
… is a real constant†
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