Separation of variables method

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Introduction

We often consider partial differential equations such as

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

which is recognisable as the wave equation in three dimensions, with $\nabla^2$ 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

$YZT \frac{d^2X}{dx^2} + XZT \frac{d^2Y}{dy^2} + XYT \frac{d^2Z}{dz^2} = \frac{1}{c^2} XZY \frac{d^2T}{dt^2}$

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

$\frac{1}{X} \frac{d^2X}{dx^2} + \frac{1}{Y} \frac{d^2Y}{dy^2} + \frac{1}{Z} \frac{d^2Z}{dz^2} = \frac{1}{c^2} \frac{1}{T} \frac{d^2T}{dt^2}$

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

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

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.

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