Nasser AL-SALTİ, Mokhtar KİRANE, Berikbol T. TOREBEK

On a class of inverse problems for a heat equationwith involution perturbation

A class of inverse problems for a heat equation with involution perturbation is consideredusing four different boundary conditions, namely, Dirichlet, Neumann, periodic and anti-periodic boundary conditions. Proved theorems on existence and uniqueness of solutionsto these problems are presented. Solutions are obtained in the form of series expansionusing a set of appropriate orthogonal basis for each problem. Convergence of the obtainedsolutions is also discussed.

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[1]A.R. Aftabizadeh, Y.K. Huang and J. Wiener,Bounded solutions for differentialequations with reflection of the argument, J. Math. Anal. Appl.135(1), 31–37, 1988.
[2]B.D. Aliev and R.M. Aliev,Properties of the solutions of elliptic equations with de-viating arguments(Russian), in: Special Problems of Functional Analysis and theirApplications to the Theory of Differential Equations and the Theory of Functions(Russian), 15–25 Izdat. Akad. Nauk Azerbaijan. SSR, Baku, 1968.
[3]A.A. Andreev,On the well-posedness of boundary value problems for a partial dif-ferential equation with deviating argument(Russian), in: Analytical methods in the theory of differential and integral equations (Russian), 3–6, Kuibyshev. Gos. Univ.,Kuybyshev, 1987.
[4]A.A. Andreev,Analogs of Classical Boundary Value Problems for a Second-OrderDifferential Equation with Deviating Argument, Differ. Equ.40(8), 1192–1194, 2004
[5]A. Ashyralyev and A.M. Sarsenbi,Well-posedness of an elliptic equation with involu-tion, Electron. J. Differ. Equ.2015(284), 1–8, 2015.
[6]A. Ashyralyev, B. Karabaeva and A.M. Sarsenbi,Stable difference scheme for thesolution of an elliptic equation with involution.In: A. Ashyralyev and A. Lukashov,editors. International Conference on Analysis and Applied Mathematics - ICAAM2016 AIP Conference Proceedings; 7–10 September 2016; AIP Publishing, 2016, 1759:020111, 8 pages.
[7]C. Babbage,An essay towards the calculus of calculus of functions, Philos. Trans. R.Soc. Lond.106, 179–256, 1816.
[8]M.Sh. Burlutskayaa and A.P. Khromov,Fourier Method in an Initial-Boundary ValueProblem for a First-Order Partial Differential Equation with Involution, Comput.Math. Math. Phys.51(12), 2102–2114, 2011.
[9]A. Cabada and A.F. Tojo,Equations with involutions, Atlantis Press, 2015.
[10]A. Cabada and A.F. Tojo,Equations with involutions, 2014 [cited 2017, April 14th].Available from: http://users.math.cas.cz/ sremr/wde2014/prezentace/cabada.pdf
[11]T. Carleman,Sur la theorie des équations intégrales et ses applications, Verhandl. desinternat. Mathem. Kongr. I., Zurich, 138–151, 1932.
[12]K.M. Furati, O.S. Iyiola and M. Kirane,An inverse problem for a generalized frac-tional diffusion, Appl. Math. Comput.249, 24–31, 2014.
[13]C.P. Gupta,Boundary value problems for differential equations in Hilbert spaces in-volving reflection of the argument, J. Math. Anal. Appl.128(2), 375–388, 1987.
[14]C.P. Gupta,Existence and uniqueness theorems for boundary value problems involvingreflection of the argument, Nonlinear Anal.11(9), 1075–1083, 1987.
[15]C.P. Gupta,Two-point boundary value problems involving reflection of the argument,Int. J. Math. Sci.10(2), 361–371, 1987.
[16]I.A. Kaliev and M.M. Sabitova,Problems of determining the temperature and densityof heat sources from the initial and final temperatures, ğJ. Appl. Ind. Math.4(3),332–339, 2010.
[17]I.A. Kaliev, M.F. Mugafarov and O.V. Fattahova,Inverse problem for forward-backward parabolic equation with generalized conjugation conditions, Ufa Math. J.3(2), 33–41, 2011.
[18]M. Kirane and N. Al-Salti,Inverse problems for a nonlocal wave equation with aninvolution perturbation, J. Nonlinear Sci. Appl.9, 1243–1251, 2016.
[19]M. Kirane and S.A. Malik,Determination of an unknown source term and the temper-ature distribution for the linear heat equation involving fractional derivative in time,Appl. Math. Comput.218(1), 163–170, 2011.
[20]M. Kirane, B.Kh. Turmetov and B.T. Torebek,A nonlocal fractional Helmholtz equa-tion, Fractional Differential Calculus,7(2), 225–234, 2017.
[21]K. Knopp,Theory of Functions Parts I and II, Two Volumes Bound as One, Part I.Dover, New York, 1996.
[22]A. Kopzhassarova and A. Sarsenbi,Basis Properties of Eigenfunctions of Second-Order Differential Operators with Involution, Abstr. Appl. Anal.2012Article ID:576843, 1–6, 2012.
[23]E.I. Moiseev,On the basis property of systems of sines and cosines, Doklady ANSSSR275(4), 794–798, 1984.
[24]M.A. Naimark,Linear Differential Operators Part II, Ungar, New York, 1968.
[25]I. Orazov and M.A. Sadybekov,One nonlocal problem of determination of the tem-perature and density of heat sources, Russian Math.56(2), 60–64, 2012.
[26]I. Orazov and M.A. Sadybekov,On a class of problems of determining the temperatureand density of heat sources given initial and final temperature, Sib. Math. J.53(1),146–151, 2012.
[27]D. Przeworska-Rolewicz,Sur les équations involutives et leurs applications, StudiaMath.20, 95–117, 1961.
[28]D. Przeworska-Rolewicz,On equations with different involutions of different ordersand their applications to partial differential-difference equations, Studia Mathe.32,101–111, 1969.
[29]D. Przeworska-Rolewicz,On equations with reflection, Studia Math.33, 197–206,1969.
[30]D. Przeworska-Rolewicz,On equations with rotations, Studia Math.35, 51–68, 1970.
[31]D. Przeworska-Rolewicz,Right invertible operators and functional-differential equa-tions with involutions, Demonstration Math.5, 165–177, 1973.
[32]D. Przeworska-Rolewicz,Equations with Transformed Argument. An Algebraic Ap-proach, Modern Analytic and Computational Methods in Science and Mathematics,Elsevier Scientific Publishing and PWN-Polish Scientific Publishers, Amsterdam andWarsaw, 1973.
[33]D. Przeworska-Rolewicz,On linear differential equations with transformed argumentsolvable by means of right invertible operators, Ann. Polon. Math.29, 141–148, 1974.
[34]I.A. Rus,Maximum principles for some nonlinear differential equations with deviatingarguments, Studia Univ. Babes-Bolyai Math.32(2), 53–57, 1987.
[35]M.A. Sadybekov and A.M. Sarsenbi,On the notion of regularity of boundary valueproblems for differential equation of second order with dump argument(Russian),Math. J.7(1), 2007.
[36]A.M. Sarsenbi,Unconditional bases related to a nonclassical second-order differentialoperator, Differ. Equ.46(4), 506–511, 2010.
[37]A.M. Sarsenbi and A.A. Tengaeva,On the basis properties of root functions of twogeneralized eigenvalue problems, Differ. Equ.48(2), 1–3, 2012.
[38]A.L. Skubachevskii,Elliptic functional differential equations and applications,Birkhauser, Basel-Boston-Berlin, 1997.
[39]W. Watkins,Modified Wiener equations, IJMMS,27(6), 347–356, 2001.
[40]J. Wiener,Differential Equation with Involutions, Differ. Equ.5, 1131–1137, 1969.
[41]J. Wiener,Differential Equation in Partial Derivatives with Involutions, Differ. Equ.6, 1320–1322, 1970.
[42]J. Wiener,Generalized Solutions of Functional-Differential Equations, World Scien-tific Publishing, New Jersey, 1993.
[43]J. Wiener and A.R. Aftabizadeh,Boundary value problems for differential equationswith reflection of the argument, Int. J. Math. Sci.8(1), 151–163, 1985.
[44]J. Wu,Theory and Applications of Partial Functional Differential Equations,Springer-Verlag, New York, 1996.