On the Volterra property of a boundary problem with integral gluing condition for a mixed parabolic-hyperbolic equation
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On the Volterra property of a boundary problem with integral gluing condition for a mixed parabolic-hyperbolic equation Abdumauvlen S Berdyshev1 , Alberto Cabada2 , Erkinjon T Karimov3* and Nazgul S Akhtaeva1 *
Correspondence: [email protected] 3 Institute of Mathematics, National University of Uzbekistan named after Mirzo Ulughbek, Tashkent, Uzbekistan Full list of author information is available at the end of the article
Abstract In the present work, we consider a boundary value problem with gluing conditions of an integral form for the parabolic-hyperbolic type equation. We prove that the considered problem has the Volterra property. The main tools used in the work are related to the method of the integral equations and functional analysis.
Introduction The theory of mixed type equations is one of the principal parts of the general theory of partial differential equations. The interest for these kinds of equations arises intensively because of both theoretical and practical uses of their applications. Many mathematical models of applied problems require investigations of this type of equations. The actuality of the consideration of mixed type equations has been mentioned, for the first time, by S. A. Chaplygin in in his famous work ‘On gas streams’ []. The first fundamental results in this direction was obtained in - by Tricomi [] and Gellerstedt []. The works of Lavrent’ev [], Bitsadze [, ], Frankl [], Protter [, ] and Morawetz [], have had a great impact in this theory, where outstanding theoretical results were obtained and pointed out important practical values of them. Bibliography of the main fundamental results on this direction can be found, among others, in the monographs of Bitsadze [], Berezansky [], Bers [], Salakhitdinov and Urinov [] and Nakhushev []. In most of the works devoted to the study of mixed type equations, the object of study was mixed elliptic-hyperbolic type equations. Comparatively, few results have been obtained on the study of mixed parabolic-hyperbolic type equations. However, this last type of equations have also numerous applications in the real life processes (see [] for an interesting example in mechanics). The reader can find a nice example given, for the first time, by Gelfand in [], and connect with the movement of the gas in a channel surrounded by a porous environment. Inside the channel, the movement of gas was described by the wave equation and outside by the diffusion one. Mathematic models of this kind of problems arise in the study of electromagnetic fields, in a heterogeneous environment, consisting of dielectric and conductive environment for modeling the movement of a little compressible fluid in a channel surrounded by a porous medium []. Here, the wave equation describes the hydrodynamic pressure of the fluid in the channel, and the equation of filtration-pressure fluid in a porous medium. Similar problems arise in the study of the magnetic intensity of the electromagnetic field []. © 2013 Berdyshev et al.;
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