Heat Equation - Solving The Heat Equation Using Fourier Series

Solving The Heat Equation Using Fourier Series

The following solution technique for the heat equation was proposed by Joseph Fourier in his treatise Théorie analytique de la chaleur, published in 1822. Let us consider the heat equation for one space variable. This could be used to model heat conduction in a rod. The equation is

(1)

where u = u(x, t) is a function of two variables x and t. Here

• x is the space variable, so x ∈, where L is the length of the rod.
• t is the time variable, so t ≥ 0.

We assume the initial condition

(2)

where the function f is given, and the boundary conditions

.

(3)

Let us attempt to find a solution of (1) which is not identically zero satisfying the boundary conditions (3) but with the following property: u is a product in which the dependence of u on x, t is separated, that is:

(4)

This solution technique is called separation of variables. Substituting u back into equation (1),

Since the right hand side depends only on x and the left hand side only on t, both sides are equal to some constant value −λ. Thus:

(5)

and

(6)

We will now show that nontrivial solutions for (6) for values of λ ≤ 0 cannot occur:

1. Suppose that λ < 0. Then there exist real numbers B, C such that
   From (3) we get
   and therefore B = 0 = C which implies u is identically 0.
2. Suppose that λ = 0. Then there exist real numbers B, C such that
   From equation (3) we conclude in the same manner as in 1 that u is identically 0.
3. Therefore, it must be the case that λ > 0. Then there exist real numbers A, B, C such that
   and
   From (3) we get C = 0 and that for some positive integer n,

This solves the heat equation in the special case that the dependence of u has the special form (4).

In general, the sum of solutions to (1) which satisfy the boundary conditions (3) also satisfies (1) and (3). We can show that the solution to (1), (2) and (3) is given by

where