Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.2 SUPERPOSITION OF SEPARATED SOLUTIONS

In order to satisfy the boundary conditionu→0ast→∞,λ^2 κmust be>0. Sinceκ
is real and>0, this implies thatλis a real non-zero number and that the solution is
sinusoidal inxand is not a disguised hyperbolic function; this was our reason for choosing
the separation constant as−λ^2 .

As a final example we consider Laplace’s equation in Cartesian coordinates;

this may be treated in a similar manner.

Use the method of separation of variables toobtain a solution for the two-dimensional

Laplace equation,

∂^2 u

∂x^2

+

∂^2 u

∂y^2

=0. (21.13)

If we assume a solution of the formu(x, y)=X(x)Y(y) then, following the above method,
and taking the separation constant asλ^2 , we find

X′′=λ^2 X, Y′′=−λ^2 Y.

Takingλ^2 as>0, the general solution becomes

u(x, y)=(Acoshλx+Bsinhλx)(Ccosλy+Dsinλy). (21.14)

An alternative form, in which the exponentials are written explicitly, may be useful for
other geometries or boundary conditions:

u(x, y)=[Aexpλx+Bexp(−λx)](Ccosλy+Dsinλy), (21.15)

with different constantsAandB.
Ifλ^2 <0 then the roles ofxandyinterchange. The particular combination of sinusoidal
and hyperbolic functions and the values ofλallowed will be determined by the geometrical
properties of any specific problem, together with any prescribed or necessary boundary
conditions.

We note here that a particular case of the solution (21.14) links up with the

‘combination’ resultu(x, y)=f(x+iy) of the previous chapter (equations (20.24)

and following), namely that ifA=BandD=iCthen the solution is the same

asf(p)=ACexpλpwithp=x+iy.

21.2 Superposition of separated solutions

It will be noticed in the previous two examples that there is considerable freedom

in the values of the separation constantλ, the only essential requirement being

thatλhas thesamevalue in both parts of the solution, i.e. the part depending

onxand the part depending ony(ort). This is a general feature for solutions

in separated form, which, if the original PDE hasnindependent variables, will

containn−1 separation constants. All that is required in general is that we

associate the correct function of one independent variable with the appropriate

functions of the others, the correct function being the one with the same values

of the separation constants.

If the original PDE is linear (as are the Laplace, Schr ̈odinger, diffusion and

wave equations) then mathematically acceptable solutions can be formed by