Fourier Method

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 Fourier Method. 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 An and Bn the function

is a solution of the problem.

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

References in periodicals archive ?
Finally, we compare the efficacy of the new imaging method with the Direct Fourier Method.
It still employs the traditional planar array but theoretically modifies the measured visibility by adding a correction phase term to get the equivalent far-field visibility, which is also the modified Fourier method [18,19].
Secondly, the proposed methods are proven to be unconditionally stable by using the Fourier method.
The shape of the shell from the two populations of the bivalve were analyzed using the Elliptic Fourier method that allowed various simple transformations so that the results will be invariant to size, location, rotation, and starting point of the digitized outline [9].
We can only present in this latter case the Fourier method results for we do not have, as far as we know, explicit or quasi explicit formula.
Forssen, On the direct Fourier method for computer tomography, IEEE Trans.
Then by using the Fourier method, we can obtain the Dirichlet data exactly:
Note that this result implies that Krylov subspace spectral methods reduce to the Fourier method in the case where L has constant coefficients.
A comparison of Fourier methods for the description of wing shape in mosquitoes (Ritera culicidae).
They and a third associate, Steve Kranz, an engineering research assistant at the university, saw the machine as a visual means to demonstrate Fourier methods to students.