Andriy Beshley, Dmytro Afanasiev


In this paper we investigate a method of approximating the interior Dirichlet problem for the generalized Laplace equation into the boundary value problem for simpler elliptic equations, proposed by Rangogni and Occhi in [10], together with the boundary integral equations approach. Based on some assumptions the considered problem can be substituted by the Dirichlet problem for the Laplace, Klein-Gordon or Helmholtz equations. Afterward, having fundamental solutions for each of these equations, we use the boundary integral equations method representing the solution as a single- or double-layer potential in conjunction with the quadrature method to obtain a fully discrete system of linear equations with unknown approximate values of the density. Сalculating the approximate solution of the problem for a constant-coefficient equation, the approximate solution for the generalized Laplace equation is obtained as well. Finally, several numerical examples with different discretization parameters are provided in order to show the effectiveness of this approach, especially in the case of exact reduction to a constant-coefficient equation.

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[1] Abramowitz M. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables / M. Abramowitz, I. Stegun. New York: Dover Publications, 1972.

[2] Beshley A. An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients
/ A. Beshley, R. Chapko, B.T. Johansson // Journal of Engineering Mathematics. 2018. Vol.112. P.63-73.

[3] Chapko R. An alternating boundary integral based method for a Cauchy problem for Klein-Gordon equation / R. Chapko, D. Laba // Journal of Computational & Applied Mathematics. 2014. Vol.2(116). P. 30-42.

[4] Chapko R. On a quadrature method for a logarithmic integral equation of the first kind / R. Chapko, R. Kress // World Scientific Series in Applicable Analysis. Contributions in Numerical Mathematics. 1993. Vol.2. P. 127-140.

[5] Colton D. Integral Equation Methods in Scattering Theory / D. Colton, R. Kress. New York: Wiley-Interscience, 1983.

[6] Dufera T. T. Analysis of Boundary–Domain Integral Equations for Variable-Coefficient Dirichlet BVP in 2D / T. T. Dufera, S. E. Mikhailov // In: Constanda C., Kirsch A. (eds) Integral Methods in Science and Engineering: Theoretical and Computational Advances. Boston: Springer (Birkhöuser). 2015. P. 163-175.

[7] Fresneda-Portillo C. On the existence of solution of the boundary-domain integral equation system derived from the 2D Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix / C. Fresneda-Portillo, Z. W. Woldemicheal // Complex Variables and Elliptic Equations.– 2019.– P. 1-15.

[8] Kirsch A. An Introduction to the Mathematical Theory of Inverse Problems /
A. Kirsch.– New-York: Springer, 2010.

[9] Kress R. Linear Integral Equations. Third edition / R. Kress.– New-York: Springer-Verlag, 2014.

[10] Rangogni R. Numerical solution of the generalized Laplace equation by the boundary element method / R. Rangogni, R.Occhi // Appl. Math. Modelling.– 1987.– Vol. 11.– P. 393-396.

[11] Rangogni R. Numerical solution of the generalized Laplace equation by coupling the boundary element method and the perturbation method
/ R. Rangogni // Appl. Math. Modelling.– 1986.– Vol. 10.– P. 266-270.

[12] Sveshnikov A.G. Lektsii po matematicheskoj fizike. In Russian [Lectures on mathematical physics] / A.G. Sveshnikov, A.N. Bogoliubov, V.V. Kravtsov.– Moscow: MSU Publ., 1993.



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