﻿ Similarity Method in Constructing Fundamental Solution of the Dirichlet Problem for Equation of Keldysh Type in Half-Space | Herald of the Bauman Moscow State Technical University. Natural Sciences
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# Similarity Method in Constructing Fundamental Solution of the Dirichlet Problem for Equation of Keldysh Type in Half-Space

 Authors: Algazin O.D. Published: 26.01.2018 Published in issue: #1(76)/2018 DOI: 10.18698/1812-3368-2018-1-4-15 Category: Mathematics | Chapter: Differential Equations, Dynamic Systems, and Optimal Control Keywords: equation of Keldysh type, Dirichlet problem, similarity method, self-similar solution, approximate identity, generalized functions

The purpose of this research was to use the similarity method for the equation of Keldysh type, which is elliptic in half-space and degenerating on the boundary. As a result, we found a self-similar solution, which is the approximate identity in the class of integrable functions. It is a fundamental solution of the Dirichlet problem, i. e. the solution of the Dirichlet problem with the Dirac δ-function in the boundary condition. The solution of the Dirichlet problem with an arbitrary function in the boundary condition can be written as the convolution of the function with the fundamental solution of the Dirichlet problem if the convolution exists. For a bounded and piecewise continuous boundary function the convolution exists and is written in the form of an integral, which gives the classical solution of the Dirichlet problem, and is a generalization of the Poisson integral for the Laplace equation. If the boundary function is a generalized function, the convolution is a generalized solution of the Dirichlet problem

## References

[1] Keldysh M.V. On some instances of elliptical equation degeneration at the area boundary. DAN SSSR, 1951, vol. 77, no. 2, pp. 81–83 (in Russ.).

[2] Bers L. Mathematical aspects of subsonic and transonic gas dynamics. New York, Wiley, 1958. 164 p.

[3] Otway T.H. Dirichlet problem for elliptic-hyperbolic equations of Keldysh type. Berlin, Heidelberg, Springer-Verlag, 2012. 214 p.

[4] Ibragimov N.Kh. Group analysis of ordinary differential equations and the invariance principle in mathematical physics (for the 150th anniversary of Sophus Lie). Russian Mathematical Surveys, 1992, vol. 47, no. 4, pp. 89–156. DOI: 10.1070/RM1992v047n04ABEH000916

[5] Bluman G.W., Cole J.D. Similarity methods for differential equations. New York, Heidelberg, Berlin, Springer-Verlag, 1974. 333 p.

[6] Algazin O.D. Exact solution to the Dirichlet problem for degenerating on the boundary elliptic equation of Tricomi — Keldysh type in the half-space. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2016, no. 5, pp. 4–17 (in Russ.). DOI: 10.18698/1812-3368-2016-5-4-17

[7] Parasyuk L.S., Parasyuk I.L. Properties of elementary solutions of the main boundary problems for the mixed types second-order differential equations. Naukovi zapiski [Scientific Papers], 1999, no. 1, pp. 126–129.

[8] Barros-Neto J., Gelfand I.M. Fundamental solutions for the Tricomi operator. Duke Math. J., 1999, vol. 98, no. 3, pp. 465–483. DOI: 10.1215/S0012-7094-99-09814-9

[9] Barros-Neto J., Gelfand I.M. Fundamental solutions for the Tricomi operator, II. Duke Math. J., 2002, vol. 111, no. 3, pp. 561–584. DOI: 10.1215/S0012-7094-02-11137-5

[10] Barros-Neto J., Gelfand I.M. Fundamental solutions for the Tricomi operator, III. Duke Math. J., 2005, vol. 128, no. 1, pp. 119–140. DOI: 10.1215/S0012-7094-04-12815-5

[11] Chen Sh. The fundamental solution of the Keldysh type operator. Science in China Series A: Mathematics, 2009, vol. 52, iss. 9, pp. 1829–1843. DOI: 10.1007/s11425-009-0069-8

[12] Vladimirov V.S. Obobshchennye funktsii v matematicheskoy fizike [Generalised functions in mathematical physics]. Moscow, Nauka Publ., 1979. 320 p.

[13] Gelfand I.M., Shilov G.E. Obobshchennye funktsii i deystviya nad nimi [Generalised functions and operations with them]. Moscow, Fizmatgiz Publ., 1959. 470 p.

[14] Kamke E. Spravochnik po obyknovennym differentsialnym uravneniyam [Handbook on the ordinary differential equations]. Moscow, Nauka Publ., 1971. 576 p.

[15] Gradshteyn I.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products]. Moscow, Nauka Publ., 1971. 1108 p.