Approximating the Dirac distribution for Fourier analysis

Abstract

For some boundary or initial value problems, the presence of a Dirac distribution on the boundary or in the field results in finite solutions at some points in the domain. However, its presence leads to difficulties if the problem is solved analytically using a Fourier decomposition, since computation and presentation of the solution usually necessitate some sort of truncation. To circumvent this problem, the Dirac distribution is often approximated by a Gaussian distribution, which results in a very simple Fourier transform on an infinite domain. On a finite domain the transform is not as simple, but may still be computed. However, the derivative of the Gaussian is discontinuous on the finite domain, since the smooth function has been truncated. Thus a different approximation, the [beta][pi]-ditribution is proposed. This function satisfies the same criteria which make the Gaussian applicable as an approximation of the Dirac distribution on the infinite domain, but its derivative is continuous everywhere on the finite domain. This article presents a procedure for computing the Fourier coefficients of the [beta][pi]-distribution. Since a large value of the order of the distribution is chosen to approximate the singular behavior, the integral for the Fourier coefficients must be evaluated using a Fourier-Bessel decomposition, which allows the computation to be carried out over large values of the Fourier index. The technique is illustrated with application to a simple two-dimensional boundary value problem containing a singularity in the boundary condition. Convergence is significantly improved if the proposed distribution is used. Values of some Fourier coefficients of the [beta][pi]-distribution are provided in an appendix for several values of its order.