Dirichlet and Poincare in the multidimensional field for a class of singular hyperbolic equations
133 43
Keywords:
solvability problems, multidimensional domain, singular equations, system equationsAbstract
It has been shown in a plane that one of fundamental problems of Math Physics, i.e. studying the behavior of a hesitating string, is not correct when boundary conditions are given on the whole boundary of the domain. As it is shown below, Dirichlet problem is incorrect not just for a wave equation but for general hyperbolic equations. In the works of the author studied the Dirichlet and Poincar? problem for linear multidimensional hyperbolic equations, which shows the correctness of these tasks, depending essentially on the height of the considered cylindrical domain.In this paper we find multi-dimensional area in which the Dirichlet and the Poincare problem solved for a class of singular hyperbolic equations.
References
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[2] D.G.Bourgin and R.Duffin "The Dirichlet problem the vibrating string eguation"//Bulletin of the American Mathematical Society. -1939, -Vol.45, -P. 851-858.
[3] Shabat B. V Examples of solutions of the Dirichlet problem for a mixed-type equation // DAN SSSR. - 1957. - Vol. 112, No 3. - P. 386-389.
[4] Bitsadze A.B. The incorrectness of the Dirichlet problem for mixed-type equations in mixed areas // DAN USSR. - 1958. - Vol. 122, No 2. - P. 167-170.
[5] D.W.Fox and C.Pucci "The Dirichlet problem the wave eguation"// Annali di Mathematica Pura ed Applicata. -1958. - Vol. 46, -P. 155-182.
[6] Dunninger D.R., Zachmanoglou E.C. The condition for uniqueness of the Diriclet problem for hyperbolic equations in cilindrical domains // J.Math.Mech.-1969. Vol.18,-P.8.
[7] Nakhushev A. M. Displacement problems for partial differential equation. - М.: Nauka, 2006. - 287 p.
[8] Aldashev S.A. Aldashev S.A. Oh Dirichlet correctness task mnogomernix volnovogo equation and the equation Lavrenteva- Bicadze // Ukrainian . Matt . Zh .- 1996. - Vol. 4(48). No 5. - P.701-705.
[9] Aldashev S.A. The well - posedness of the Dirichlet problem in the cylindric domain for the multidimensional wave equation// Mathematical problems Engineering, volume 2010, Article ID 653215, 7 pages.
[10] Aldashev S.A. The well - posedness of the Poincare problem in a cylindrical domain for the higher-dimensional wave equation// Journal of Mathematical Science. -2011. - Vol. 173, No 2. -P. 150-154.
[11] Aldashev S.A. A uniqueness criterion for solutions of the Dirichlet problem and Poincare in a cylindrical domain for the multidimensional Euler-Poisson-Darboux // Far Mat. zhurn. - 2012. - Vol. 12, No 1. - P. 3-12.
[12] Aldashev S.A. Dirichlet and Poincare for a class of multidimensional singular hyperbolic equations // Scientific statements BSU, Ser. "Mathematics, Physics".- Belgorod, 2016. Vol. 43. No 13 (234), - P. 18-23.
[13] Mikhlin S.G. Multidimensijnal singular integrals and integral equations.-M.:Physmathgiz,1962.-254 p.
[14] Aldashev S.A. Boundary value problems for multidimensional hyperbolic and mixed equations. -Almaty: Gylym, 1994. -170 p.
[15] Copson E.T. On the Riemann-Green function // J.Rath.Mech and Anal. - 1958, No 1. - P. 324-348.
[16] Weinstein A. // The Fifth simposium in applied Math. MCGraw-Hill. New York. - 1954. - P.137-147.
[17] Nahushev A.M. Elements of fractional calculus and their application. -Nalchik: KBSC RAS, 2000,-298 p.
[18] Tersenov S.A. Introduction to the theory of equations degenerating on the boundary.- Novosibirsk: NSU, 1973, - 144 p.
[19] Aldashev S.A. Some boundary value problems for a class of singular partial differential equations derivatives // Differents.uravneniya. -1976. -Vol.12, No6.- P. 3-14.
[20] Tersenov S.A. Introduction to the theory of parabolic equations with changing time direction. -Novosibirsk: Siberian Branch of the USSR MI, 1982, -167 p.
[21] Aldashev S.A. Degenerate multidimensional hyperbolic equations. -Oral: ZKATU, 2007, -139 p.
[22] Aldashev S.A. Correctness of Dirichlet and Poincare problems in a multidimensional area for wave equalization // Ukrainian math journal. -2014. - Vol. 66, No 10. - P. 1414-1419.
[23] Kolmogorov A. N., Fomin S. V. Elements of function theory and functional analysis. –M.: Nauka, 1976. - 543 p.
[24] G.Beitmen, A.Erdeyee Higher Transcendental Functions. -M.: Nauka, 1973. - Vol. 1. - 294 p.
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Aldashev, S. A. (2018). Dirichlet and Poincare in the multidimensional field for a class of singular hyperbolic equations. Journal of Mathematics, Mechanics and Computer Science, 92(4), 20–31. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/450
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