Heat Conduction Equation

A partial differential diffusion equation of the form

(1)

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

The one-dimensional heat conduction equation is

(2)

This can be solved by separation of variables using

(3)

Then

(4)

Dividing both sides by  gives

(5)

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

(6)

and the  solution is

(7)

The general solution is then

			(8)
			(9)
			(10)

If we are given the boundary conditions

(11)

and

(12)

then applying (11) to (10) gives

(13)

and applying (12) to (10) gives

(14)

so (10) becomes

(15)

Since the general solution can have any ,

(16)

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

(17)

Multiplying both sides by  and integrating from 0 to  gives

(18)

Using the orthogonality of  and ,

			(19)
			(20)
			(21)

so

(22)

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

(23)

(24)

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

(25)

where

(26)