Partial differential equations/Separation of variables method

Introduction

We often consider partial differential equations such as

${\mathrm{\nabla }}^2\psi =\frac{1}{c^2}\frac{{\partial }^2\psi }{\partial t^2}$ ,

which is recognisable as the wave equation in three dimensions, with ${\mathrm{\nabla }}^2$ being the Laplacian operator, $\psi$ being some function of three spacial dimensions and time, and c being the speed of the wave.

These are often found by considering the physical connotations of a system, but how can we find a form of $\psi$ such that the equation is true?

Finding General Solutions

One way of doing this is to make the assumption that $\psi$ itself is a product of several other functions, each of which is itself a function of only one variable. In the case of the wave equation shown above, we make the assumption that

$\psi (x,y,z,t)=X(x)\times Y(y)\times Z(z)\times T(t)$

(NB Remember that the upper case characters are functions of the variables denoted by their lower case counterparts, not the variables themselves)

By substituting this form of $\psi$ into the original wave equation and using the three dimensional cartesian form of the Laplacian operator, we find that

$YZT\frac{d^{2}X}{d{x}^2}+XZT\frac{d^{2}Y}{d{y}^2}+XYT\frac{d^{2}Z}{d{z}^2}=\frac{1}{c^2}XZY\frac{d^{2}T}{d{t}^2}$

We can then divide this equation through by $\psi$ to produce the following equation:

$\frac{1}{X}\frac{d^{2}X}{d{x}^2}+\frac{1}{Y}\frac{d^{2}Y}{d{y}^2}+\frac{1}{Z}\frac{d^{2}Z}{d{z}^2}=\frac{1}{c^2}\frac{1}{T}\frac{d^{2}T}{d{t}^2}$

Both sides of this equation must be equal for all values of x, y, z and t. This can only be true if both sides are equal to a constant, which can be chosen for convenience, and in this case is -(k²).

The time-dependent part of this equation now becomes an ordinary differential equation of form

$\frac{d^{2}T}{d{t}^2}=-c^2{k}^{2}T$

This is easily soluble, with general solution

$T(t)=A\cos (ckt)+B\sin (ckt)$

with A and B being arbitrary constants, which are defined by the specific boundary conditions of the physical system. Note that the key to finding the time-dependent part of the original function was to find an ODE in terms of time. This general process of finding ODEs from PDEs is the essence of this method.