Fourier Method

a method of solution of partial differential equations by means of separation of variables. Proposed by J. Fourier as a tool for the solution of heat conduction problems, it was formulated in full generality by M. V. Ostrogradskii in 1828.

In the Fourier method a solution of an equation satisfying initial homogeneous and boundary conditions is sought in the form of a sum of solutions satisfying the boundary conditions, each of these solutions being a product of a function of the space variables by a function of time. To find such solutions we must first find the eigenvalues and eigenfunctions of certain differential operators and then obtain the corresponding eigenfunction expansions of the functions involved in the initial conditions.

In particular, the representation of functions in terms of Fourier series and Fourier integrals is made use of in the study of vibrations of a string and in the study of heat conduction in a rod. For example, the study of small-amplitude vibrations of a string of length I and with fixed endpoints reduces to the solution of the equation

with boundary conditions u(0, t) = u(l, t) = 0 and initial conditions . The solutions of the equation that have the form X(x)T(t) and satisfy the boundary conditions are given by the formula

For a suitable choice of the coefficients A_n and B_n the function

is a solution of the problem.

V. A. Steklov solved many important problems involving the use of the Fourier method.