Fast Discrete Finite Hankel Transform for Equations in a Thin Annulus
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II. NUMERICAL METHODS FAST DISCRETE FINITE HANKEL TRANSFORM FOR EQUATIONS IN A THIN ANNULUS S. S. Budzinskiy1 and T. E. Romanenko2
UDC 519.63
An algorithm is proposed for a fast discrete finite Hankel transform of a function in a thin annulus. The transform arises in a natural way in the Neumann boundary-value problem for the Poisson equation in an annulus when spectral methods are applied for its numerical solution. The proposed algorithm uses the limiting properties of eigenvalues and eigenfunctions of the Laplace operator as the annulus thickness goes to zero. Keywords: Hankel transform, thin annulus, eigenvalues of the Laplace operator.
Introduction Spectral methods constitute one of the approaches to numerical solution of boundary-value problems for elliptical equations and mixed initial-boundary-value problems for parabolic partial-differential equations. Let us briefly recall what spectral methods involve: a system of basis functions is chosen, and then the right-hand side and the sought solution are expanded in the basis functions. By the Hilbert–Schmidt theorem, the basis functions may often be chosen as the eigenfunctions of the linearized operator (Laplace operator), and then it remains to learn to efficiently calculate the expansion coefficients, or conversely evaluate the function from the expansion coefficients. The geometry of the region alongside with the boundary conditions obviously determine the form of the eigenfunctions of the Laplace operator. But the properties of the region also can be utilized to construct fast algorithms for the computation of the expansion coefficient in eigenfunctions. In this article, we consider a fast algorithm for the computation of coefficients (or, in other words, the computation of the discrete finite Hankel transform) for the Laplace operator in a thin annulus. Quasi-linear diffusion equations in such regions arise, for instance, in connection with the modeling of nonlinear optical system [1]. We present a geometrical substantiation of the algorithm and the results of its application on a prototype problem. Our problem is close in spirit to [2], where the Bessel functions are replaced with their asymptotic expressions in terms of sines and cosines. Eigenvalues and Eigenfunctions of the Laplace Operator on an Annulus Consider the eigenvalue problem for the Laplace operator on the annulus K = {1 < x2 + y 2 < 2 } with the Neumann boundary conditions: − ∆u = λu(x, y), @u = 0, @n
(x, y) 2 K,
λ 2 C,
(x, y) 2 @K.
1
Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia; e-mail: [email protected]. 2 Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia; e-mail: romanenko@ cs.msu.ru. Translated from Prikladnaya Matematika i Informatika, No. 63, 2020, pp. 57–62. 364
1046-283X/20/3103–0364
© 2020
Springer Science+Business Media, LLC
FAST D ISCRETE F INITE H ANKEL T RANSFORM FOR E QUATIONS IN A T HIN A NNULUS
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Change to the polar system p x2
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