Laplace’s and Poisson’s Equations in a Semi-Disc under the Dirichlet-Neumann Mixed Boundary Condition

R. T. Couto

Abstract


In this work, the solution of Poisson's equation in a semi-disc under a Dirichlet boundary condition at the base and a Neumann boundary condition on the circumference is calculated. The solution is determined in terms of Green's function, which is calculated in two ways, by the method of images and by solving its equation. In the particular case of Laplace's equation, it is presented a second way to solve it, which uses separation of variables and a Fourier transform.

Keywords


Laplace; Poisson; semi-disk; Dirichlet; Neumann; Green's; images.

Full Text:

PDF

References


D. G. Zill and M. R. Cullen, Differential Equations with Boundary-Value Prob lems. Belmont, CA: Brooks/Cole, 7 ed., 2009.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. Amsterdan: Academic Press (Elsevier), 2007.

J. D. Jackson, Classical Electrodynamics. New York: John Wiley & Sons, 3 ed., 1999.

D. G. Duffy, Green’s Functions with Applications. Boca Raton: Chapman & Hall/CRC, 2001.

A. N. Tijonov and A. A. Samarsky, Ecuaciones de la Fisica Matematica. Moscow: Editorial Mir (translated from Russian to Spanish), 2 ed., 1980.

E. C. Zachmanoglou and D. W. Thoe, Introduction to Partial Differential Equa tions with Applications. New York: Dover Publications, 1986.

I. Rubinstein and L. Rubinstein, Partial Differential Equations. Cambridge University Press, 1998.

G. Barton, Elements of Green’s Functions and Propagation: Potentials, Diffu sion, and Waves. Oxford University Press, 1989.

E. Butkov, Mathematical Physics. Reading, MA: Addison-Wesley, 1973.

J. Franklin, Classical Electromagnetism. Mineola, NY: John Wiley & Sons, 2 ed., 2017.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists. San Diego: Academic Press, 5 ed., 2001.

R. Reynolds and A. Stauffer, “A method for evaluating definite integrals in terms of special functions with examples,” International Mathematical Forum, vol. 15, no. 5, pp. 235–244, 2020.




DOI: https://doi.org/10.5540/tcam.2023.024.02.00191

Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM

Refbacks

  • There are currently no refbacks.


Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

 

Indexed in:

                       

         

 

Desenvolvido por:

Logomarca da Lepidus Tecnologia