Calculation of Green's Function for Poisson's Equation in Plane Polar Coordinates
DOI:
https://doi.org/10.5540/tcam.2024.025.e01797Keywords:
Green's function, Poisson, plane polar coordinates, disc sector, closed form, Dirichlet, NeumannAbstract
A new calculation of Green's function for Poisson's equation in plane polar coordinates is presented. The method consists in first calculating the solution to the simpler problem, but with the same Green's function, that is obtained with the homogenization of the boundary conditions and then inferring Green's function by comparing this calculated solution with Green's solution formula. Depending on how the solution to the simplified problem is calculated, Green's function may result as an integral or an infinite series, but it is finally presented in a closed form, because it is possible to calculate the integral or the sum of this series.References
L. C. Evans, Partial Differential Equations. Providence, RI: American Mathematical Society, 2nd ed., 2010.
J. D. Jackson, Classical Electrodynamics. New York: John Wiley & Sons, 3rd ed., 1999.
W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems. Hoboken, NJ: John Wiley & Sons, 10th ed., 2012.
F. B. Hildebrand, Advanced Calculus for Applications. Englewood Cliffs, NJ: Prentice-Hall, 2nd ed., 1976.
E. Butkov, Mathematical Physics. Reading, MA: Addison-Wesley, 1973.
J. W. Brown and R. V. Churchill, Fourier Series and Boundary Value Problems. New York, NY: McGraw-Hill, 8th ed., 2012.
G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists. San Diego, CA: Academic Press, 5th ed., 2001.
E. C. Zachmanoglou and D. W. Thoe, Introduction to Partial Differential Equations with Applications. New York, NY: Dover Publications, 1986.
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, vol. 8, pp. 010341–1 to 7, SBMAC, 2021.
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,” Journal of Computational Physics, no. 23, pp. 8619–8646, 2011.
L. Krähenbühl, F. Buret, R. Perrussel, and D. Voyer, “Numerical treatment of rounded and sharp corners in the modeling of 2D electrostatic fields,” Journal of Microwaves, Optoelectronics and Electromagnetic Applications, vol. 10, no. 1, pp. 66–81, 2011.
D. F. Brailsford and A. J. B. Robertson, “Calculation of electric field strengths at a sharp edge,” International Journal of Mass Spectrometry and Ion Physics, vol. 1, no. 1, pp. 75–85, 1968.
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