21

Partial differential equations:

separation of variables and other

methods

In the previous chapter we demonstrated the methods by which general solutions

of some partial differential equations (PDEs) may be obtained in terms of

arbitrary functions. In particular, solutions containing the independent variables

in definite combinations were sought, thus reducing the effective number of them.

In the present chapter we begin by taking the opposite approach, namely that

of trying to keep the independent variables as separate as possible, using the

method of separation of variables. We then consider integral transform methods

by which one of the independent variables may be eliminated, at least from

differential coefficients. Finally, we discuss the use of Green’s functions in solving

inhomogeneous problems.

21.1 Separation of variables: the general method

Suppose we seek a solutionu(x, y, z, t) to some PDE (expressed in Cartesian

coordinates). Let us attempt to obtain one that has the product form§

u(x, y, z, t)=X(x)Y(y)Z(z)T(t). (21.1)

A solution that has this form is said to beseparableinx,y,zandt, and seeking

solutions of this form is called the method ofseparation of variables.

As simple examples we may observe that, of the functions

(i)xy z^2 sinbt, (ii)xy+zt, (iii) (x^2 +y^2 )zcosωt,

(i) is completely separable, (ii) is inseparable in that no single variable can be

separated out from it and written as a multiplicative factor, whilst (iii) is separable

inzandtbut not inxandy.

§It should be noted that the conventional use here of upper-case (capital) letters to denote the

functions of the corresponding lower-case variable is intended to enable an easy correspondence

between a function and its argument to be made.