Heat Conduction Equation

A partial differential diffusion equation of the form

Physically, the equation commonly arises in situations where is the thermal diffusivity and the temperature.

The one-dimensional heat conduction equation is

This can be solved by separation of variables using

Then

Dividing both sides by gives

where each side must be equal to a constant. Anticipating the exponential solution in , we have picked a negative separation constant so that the solution remains finite at all times and has units of length. The solution is

and the solution is

The general solution is then

If we are given the boundary conditions

and

then applying (11) to (10) gives

and applying (12) to (10) gives

so (10) becomes

Since the general solution can have any ,

Now, if we are given an initial condition , we have

Multiplying both sides by and integrating from 0 to gives

Using the orthogonality of and ,

so

If the boundary conditions are replaced by the requirement that the derivative of the temperature be zero at the edges, then (◇) and (◇) are replaced by

Following the same procedure as before, a similar answer is found, but with sine replaced by cosine:

where