Calculation of Green’s Function for Poisson’s Equation in Plane Polar Coordinates using Eigenfunction Expansion in the Angular Variable

Autores

  • Roberto Toscano Couto

DOI:

https://doi.org/10.5540/03.2023.010.01.0029

Palavras-chave:

Green’s function, Poisson, plane polar coordinates, closed form, Dirichlet, Neumann

Resumo

A new calculation of Green’s function for the problem with Poisson’s equation in plane polar coordinates is presented. The method consists in calculating the solution of a problem that is simpler but that has the same Green’s function – the problem that results from the homogenization of the boundary conditions – and then inferring Green’s function by comparing this calculated solution with Green’s formula for the solution. To describe the method, it is applied to the particular case of a disc sector under mixed Dirichlet-Neumann boundary conditions. The solution of the simplified problem is obtained as an eigenfunction expansion in the angular variable. Green’s function arises from the calculations as an infinite series but is finally presented in closed form because it is possible to compute the sum of this series.

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Biografia do Autor

Roberto Toscano Couto

UFF, Niterói, RJ

Referências

W. E. Boyce and R. C. DiPrima. Elementary Differential Equations and Boundary Value Problems. 10th ed. Hoboken, NJ: John Wiley & Sons, 2012. isbn: 978-0-470-45831-0.

R. T. Couto. “A equação de Laplace num semidisco sob a condição de fronteira mista Dirichlet-Neumann”. In: Proceeding Series of the Brazilian Society of Computational and Applied Mathematics. 2021, pp. 010341-1–7. doi: 10.5540/03.2021.008.01.0341.

L. C. Evans. Partial Differential Equations. 2nd ed. Providence, RI: American Mathematical Society, 2010. isbn: 978-0-8218-4974-3.

F. B. Hildebrand. Advanced Calculus for Applications. 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1976. isbn: 0-13-011189-9.

J. D Jackson. Classical Electrodynamics. 3rd ed. New York: John Wiley & Sons, 1999. isbn: 0-471-30932-X.

D. Shirokoff and R. R. Rosales. “An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary”. In: Journal of Computational Physics 23 (2011), pp. 8619–8646. doi: 10.1016/j. jcp.2011.08.011.

E. C. Zachmanoglou and W. T. Thoe. Introduction to Partial Differential Equations with Applications. New York: Dover Publications, 1986. isbn: 0-486-65251-3.

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Publicado

2023-12-18

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