Function theories for the Yukawa and Helmholtz equations
type: Item
instance of: scholarly article
title: Function theories for the Yukawa and Helmholtz equations
zbMATH Open document ID: 0844.30032
author: J. W. Dettman
author: L. R. Bragg
published in: Rocky Mountain Journal of Mathematics
publication date: 1996-09-05T00:00:00Z
review text: In preceding papers, the authors have developed analogous function theories for the heat, the wave, and the Laplace equations [Rocky Mt. J. Math. 13, 191-214 (1983; Zbl 0554.35012)] and for the heat, the GASPT, and the Euler-Poisson-Darboux equations [ibid. 14, 383-396 (1984; Zbl 0581.35025)]. The present paper is concerned with function theories for the Yukawa and Helmholtz equations. Here, basic solutions and associated functions for the Yukawa and Helmholtz equations are obtained via transmutations acting on heat polynomials and associated heat functions. In detail, in \S 2, transmutations \((T_1,\dots, T_4)\) are introduced which relate solutions of the heat equation to solutions of the Dirichlet and Neumann problems for the Yukawa equation \((T_1, T_2)\) and the damped wave equation \((T_3, T_4)\). For the case of the Yukawa equation, the transmutation operators \(T_1\) and \(T_2\) are applied in \S 3 in order to obtain basic functions, associated functions, generating functions, recurrence relations and generalized Cauchy-Riemann equations. Similar results are proved in \S 4 for the case of the Helmholtz equation by means of \(T_3\) and \(T_4\). In order to indicate that results for the Laplace equation have their counterparts also in the cases of the Yukawa and Helmholtz equations, solutions with analytic data (\S 5) and with entire data (\S 6) are treated, and series expansions as well as integral type formulas for the solutions are generated. Finally, in \S 7, Fourier transform criteria are obtained for defining expansions of solutions in terms of associated functions.
Mathematics Subject Classification ID: 30G99
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35C10
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35C15
zbMATH DE Number: 878061
zbMATH Keywords: Yukawa equations
zbMATH Keywords: Dirichlet problems
zbMATH Keywords: Helmholtz equations
zbMATH Keywords: transmutations
zbMATH Keywords: heat equation
zbMATH Keywords: Neumann problems
reviewed by: M. Kracht
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072194
OpenAlex ID: W1997299918
cites work: Q5518024
cites work: An outline of the theory of pseudoanalytic functions
cites work: On operator calculus for related partial differential equations
cites work: Expansions of solutions of certain hyperbolic and elliptic problems in terms of Jacobis polynomials
cites work: Analogous function theories for the heat, wave, and Laplace equations
cites work: Analogous function theories for certain singular partial differential equations
cites work: A Class of Related Dirichlet and Initial Value Problems
cites work: Yukawan potential theory
cites work: Hilbert Transforms in Yukawan Potential Theory
cites work: Q4059363
cites work: Q5509892
cites work: Expansions in Terms of Heat Polynomials and Associated Functions
cites work: Some analogies from classical analysis in the theory of heat conduction
cites work: Q5332437
DOI: 10.1216/RMJM/1181072194
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description: scientific article; zbMATH DE number 878061