Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: GENERAL AND PARTICULAR SOLUTIONS

will be satisfactory solutions of the equation and that the general solution will be

u(x, t)=f(x−ct)+g(x+ct), (20.23)

wherefandgare arbitrary functions. This solution is discussed further in section 20.4.

The method used to obtain the general solution of the wave equation may also

be applied straightforwardly to Laplace’s equation.

Find the general solution of thetwo-dimensional Laplace equation

∂^2 u

∂x^2

+

∂^2 u

∂y^2

=0. (20.24)

Following the established procedure, we look for a solution that is a functionf(p)of
p=x+λy, where from (20.24)λsatisfies

1+λ^2 =0.

This requires thatλ=±i, and satisfactory variablesparep=x±iy. The general solution
required is therefore, in terms of arbitrary functionsfandg,

u(x, y)=f(x+iy)+g(x−iy).

It will be apparent from the last two examples that the nature of the appropriate

linear combination ofxandydepends upon whetherB^2 > 4 ACorB^2 < 4 AC.

This is exactly the same criterion as determines whether the PDE is hyperbolic

or elliptic. Hence as a general result, hyperbolic and elliptic equations of the

form (20.20), given the restriction that the constantsA,BandCare real, have as

solutions functions whose arguments have the formx+αyandx+iβyrespectively,

whereαandβthemselves are real.

The one case not covered by this result is that in whichB^2 =4AC,i.e.a

parabolic equation. In this caseλ 1 andλ 2 are not different and only one suitable

combination ofxandyresults, namely

u(x, y)=f(x−(B/ 2 C)y).

To find the second part of the general solution we try, in analogy with the

corresponding situation for ordinary differential equations, a solution of the form

u(x, y)=h(x, y)g(x−(B/ 2 C)y).

Substituting this into (20.20) and usingA=B^2 / 4 Cresults in
(
A

∂^2 h

∂x^2

+B

∂^2 h

∂x∂y

+C

∂^2 h

∂y^2

)

g=0.

Therefore we requireh(x, y) to be any solution of the original PDE. There are

several simple solutions of this equation, but as only one is required we take the

simplest non-trivial one,h(x, y)=x, to give the general solution of the parabolic

equation

u(x, y)=f(x−(B/ 2 C)y)+xg(x−(B/ 2 C)y). (20.25)