Since the beginning Fourier himself was interested to find a powerful tool to be used in solving differential equations. Therefore, it is of no surprise that we discuss in this page, the application of Fourier series differential equations. We will only discuss the equations of the form
Note that we will need the complex form of Fourier series of a periodic function. Let us define this object first:
Definition. Let f(x) be -periodic. The complex Fourier series of f(x) is
If you wonder about the existence of a relationship between the real Fourier coefficients and the complex ones, the next theorem answers that worry.
Theoreme. Let f(x) be -periodic. Consider the real Fourier coefficients
and
of f(x), as well as the complex Fourier coefficients
.
We have
The proof is based on Euler's formula for the complex exponential function.
Remark. When f(x) is 2L-periodic, then the complex Fourier series will be defined as before where
Example. Let f(x) = x, for
and
f(x+2) = f(x). Find its complex Fourier coefficients
.
Answer. We have d0 = 0 and
Back to our original problem. In order to apply the Fourier technique to differential equations, we will need to have a result linking the complex coefficients of a function with its derivative. We have:
Theorem. Let f(x) be 2L-periodic. Assume that f(x) is differentiable. If
Example. Find the periodic solutions of the differential equation
Example. Find the periodic solutions of the differential equation
The most important result may be stated as:
Theoreme. Consider the differential equation