Poisson and Kirchhoff formulas deduced by control volumes
DOI:
https://doi.org/10.35819/remat2022v8i1id5516Keywords:
Differential Equations, Integral Equations, Integral TransformsAbstract
This paper presents an alternative deduction for the Poisson and Kirchhoff formulas, which solve the initial value problem defined by the wave equation in two and three dimensions, respectively. This deduction was inspired by a known approach for the one-dimensional case, associated with D'Alembert's formula and which is developed through an integration in a control volume defined in the domain of space and time. It is, therefore, an extension of that approach to two- and three-dimensional cases.
Downloads
References
BORTHWICK, D. Introduction to Partial Differential Equations. Universitext. [S. l.]: Springer, 2016. DOI: https://doi.org/10.1007/978-3-319-48936-0.
COUTO, R. T. Sobre a Dedução da Equação da Onda e da Solução Segundo a Fórmula de Kirchooff. TEMA: Tendências em Matemática Aplicada e Computacional, São Carlos, v. 11, n. 1, p. 49-58, 2010. Disponível em: https://tema.sbmac.org.br/tema/article/view/112. Acesso em: 21 abr. 2022.
DOETSCH, G. Introduction to the Theory and Application of the Laplace Transform. Berlin: Springer, 1974. DOI: https://doi.org/10.1007/978-3-642-65690-3.
DYKE, P. P. G. An Introduction to Laplace Transforms and Fourier Series. Springer Undergraduate Mathematics Series. Berlin: Springer, 1999.
EVANS, L. C. Partial differential equations. Graduate Studies in Mathematics. v. 19. 2. ed. Providence, Rhode Island: American Mathematical Society, 2010.
FOLLAND, G. B. Introduction to Partial Differential Equations. Mathematical Notes. New Jersey: Princeton University Press, 1995.
FRITZ, J. Partial Differential Equations. Applied Mathematical Sciences. New York: Springer, 1982.
HOUNIE, J. Teoria Elementar das Distribuições. Rio de Janeiro: IMPA, 1979.
JOST, J. Partial Differential Equations. Graduate Texts in Mathematics. 3. ed. New York: Springer, 2012.
MELO, A. R.; GRAMANI, L. M.; KAVISKI, E. Esquema Explícito Semi-Analítico para a Solução da Equação da Onda Unidimensional com Condições de Contorno Naturais. TEMA: Tendências em Matemática Aplicada e Computacional, São Carlos, v. 20, n. 1, p. 77-93, 2019. DOI: https://doi.org/10.5540/tema.2019.020.01.77.
OLVER, P. J. Introduction to Partial Differential Equations. Undergraduate Texts in Mathematics. New York: Springer, 2014. DOI: https://doi.org/10.1007/978-3-319-02099-0.
SCHIFF, J. L. The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics. Berlin: Springer, 1999.
SPIEGEL, M. R. Theory and Problems of Laplace Transforms. Schaum's Outline. New York: McGraw-Hill, 1965.
STRAUSS, W. A. Partial Differential Equations: An Introduction. United States of America: John Whiley & Sons, 1992.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 REMAT: Revista Eletrônica da Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
REMAT retains the copyright of published articles, having the right to first publication of the work, mention of first publication in the journal in other published media and distribution of parts or of the work as a whole in order to promote the magazine.
This is an open access journal, which means that all content is available free of charge, at no cost to the user or his institution. Users are permitted to read, download, copy, distribute, print, search or link the full texts of the articles, or use them for any other legal purpose, without requesting prior permission from the magazine or the author. This statement is in accordance with the BOAI definition of open access.
