The method of potentials for the heat equation with involution
DOI:
https://doi.org/10.70474/wd88jx69Keywords:
nonlocal parabolic equation, involution, potentials of the simple and double layers, Cauchy problem, fundamental solutionAbstract
This work is devoted to the study of the solvability of initial-boundary value problems for parabolic equations with involution using the potential theory method. A nonlocal parabolic equation is introduced using involutive transformations over a spatial variable. The Cauchy problem was solved, a fundamental solution was constructed, using which the heat potentials of the simple and double layers were introduced, and their properties were studied. Further, using these properties, as also applying the methods of the theory of integral equations, the first boundary value problem for a parabolic equation with involution was solved. The solution of this problem is constructed in explicit form, and the uniqueness of the solution is proven using the method of apriori estimates.
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References
Karapetiants N., Samko S., textit{Equations with Involutive Operators}. Boston: Birkhauser, (2001), 427 p.
Litvinchuk G.S., textit{Boundary value problems and singular integral equations with shift}. Moscow: Nauka, (1977), 448 p. Available at: https://booksee.org/book/578151. (Rus.)
Linkov A.V. textit{Substantiation of a method the Fourier for boundary problems with an involute deviation}.Bulletin of Samara State University. Natural Science Series, (1999), 12:2, pp. 60–65. (Rus.)
Karachik V.V., Sarsenbi A., Turmetov B.Kh., textit{On the solvability of the main boundary value problems for a nonlocal Poisson equation}. Turkish Journal of Mathematics, (2019), 43:3, 1604–1625. DOI: http://doi.org/10.3906/mat-1901-71.
Karachik V.V., Turmetov B.Kh., textit{Solvability of one nonlocal Dirichlet problem for the Poisson equation}. Novi Sad Journal of Mathematics, (2020), 50:1, 67–88. DOI: http://doi.org/10.30755/NSJOM.08942.
Nazarova K.Zh., Turmetov B.Kh., Usmanov K.I., textit{On the solvability of some boundary value problems with involution}. Vestnik of Samara University. Natural Science Series. (2020), 26:3, 7–16.
Dildabek G., Ivanova M.B., Sadybekov M.A., textit{On root functions of nonlocal differential second-order operator with boundary conditions of periodic type}. JMMCS. (2021), 4:112.
Sarsenbi A. A., Mussirepova E., textit{Green’s function of a boundary value problem for a second-order differential equation with involution}. Bulletin of the National Engineering Academy of the Republic of Kazakhstan. (2022). 4:86.
Lesev V.N., Bzheumikhova O.I., Jeldasheva A.O., Etuev N.Kh., textit{Boundary value problem for a model differential equation with involution in a rectangular domain}. International Scientific Research Journal, 11:137.
Ahmad B., Alsaedi A., Kirane M., Ramiz G. Tapdigoglu, textit{An inverse problem for space and time fractional evolution equations with an involution perturbation}, Quaestiones Mathematicae, (2017), 40:2, 151-160, DOI:http://doi.org/10.2989/16073606.2017.1283370
Kirane M., Nasser Al-Salti., textit{Inverse problems for a nonlocal wave equation with an involution perturbation}
J. Nonlinear Sci. Appl. 9 (2016), 1243–1251.
Nasser Al-Salti , Kirane M., Torebek B. T., textit{On a class of inverse problems for a heat equation with involution perturbation}. Hacet. J. Math. Stat. (2019), 48:3, 669–681. {https://doi.org/10.15672/HJMS.2017.538}
Kalmenov T.Sh., Tokmagambetov N.E., textit{One nonlocal boundary value problem for a multidimensional heat equation in a non-cylindrical domain}. Siberian Mathematical Journal November - December, (2013), 54:6.
Sobirov Z. A., Rakhimov K. U., textit{The Cauchy problem for the time fractional derivative of the Airy equation on a star graph}. (2019), 5, ISSN-2181-9483. 40-49.
Rakhimov K. U., textit{The method of potentials for the Airy equation of fractional order}. Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences. (2020), 3:2, 222-235.
Tikhonov A.N., Samarskiy A.A., textit{Equations of Mathematical Physics}: Textbook. Manual. -6th ed., revised. And add. - M.: Moscow State University Publishing House, (1999). (Rus.)
Gradshteyn I.S., Ryzhik I.M., textit{Table of Integrals, Series, and Products}, (2007).
Roberts G. E., Kaufman H., textit{Table of Laplace Transforms}, (1966) by W. B. Saunders Company. 367p.
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