Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: GENERAL AND PARTICULAR SOLUTIONS

describes the quantum mechanical wavefunctionu(r,t) of a non-relativistic particle

of massm;
is Planck’s constant divided by 2π. Like the diffusion equation it is

second order in the three spatial variables and first order in time.

20.2 General form of solution

Before turning to the methods by which we may hope to solve PDEs such as

those listed in the previous section, it is instructive, as for ODEs in chapter 14, to

study how PDEs may be formed from a set of possible solutions. Such a study

can provide an indication of how equations obtained not from possible solutions

but from physical arguments might be solved.

For definiteness let us suppose we have a set of functions involving two

independent variablesxandy. Without further specification this is of course a

very wide set of functions, and we could not expect to find a useful equation that

they all satisfy. However, let us consider a type of functionui(x, y)inwhichxand

yappear in a particular way, such thatuican be written as a function (however

complicated)of a single variablep, itself a simple function ofxandy.

Let us illustrate this by considering the three functions

u 1 (x, y)=x^4 +4(x^2 y+y^2 +1),

u 2 (x, y)=sinx^2 cos 2y+cosx^2 sin 2y,

u 3 (x, y)=

x^2 +2y+2

3 x^2 +6y+5

.

These are all fairly complicated functions ofxandyand a single differential

equation of which each one is a solution is not obvious. However, if we observe

that in fact each can be expressed as a function of the variablep=x^2 +2yalone

(withnootherxoryinvolved) then a great simplification takes place. Written

in terms ofpthe above equations become

u 1 (x, y)=(x^2 +2y)^2 +4=p^2 +4=f 1 (p),

u 2 (x, y)=sin(x^2 +2y)=sinp=f 2 (p),

u 3 (x, y)=

(x^2 +2y)+2

3(x^2 +2y)+5

=

p+2

3 p+5

=f 3 (p).

Let us now form, for eachui, the partial derivatives∂ui/∂xand∂ui/∂y.Ineach

case these are (writing both the form for generalpand the one appropriate to

our particular case,p=x^2 +2y)

∂ui

∂x

=

dfi(p)

dp

∂p

∂x

=2xfi′,

∂ui

∂y

=

dfi(p)

dp

∂p

∂y

=2fi′,

fori= 1, 2, 3. All reference to the form offican be eliminated from these