On a class of inverse problems for a heat equation with involution perturbation

Year 2019, Volume: 48 Issue: 3, 669 - 681, 15.06.2019

Nasser Al-salti Mokhtar Kirane , Berikbol T. Torebek

Abstract

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

Keywords

Inverse problems, heat equation, involution perturbation

References

• A.R. Aftabizadeh, Y.K. Huang and J. Wiener, Bounded solutions for differential equations with reflection of the argument, J. Math. Anal. Appl. 135 (1), 31–37, 1988.
• 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 their Applications to the Theory of Differential Equations and the Theory of Functions (Russian), 15–25 Izdat. Akad. Nauk Azerbaijan. SSR, Baku, 1968.
• 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.
• A.A. Andreev, Analogs of Classical Boundary Value Problems for a Second-Order Differential Equation with Deviating Argument, Differ. Equ. 40 (8), 1192–1194, 2004.
• A. Ashyralyev and A.M. Sarsenbi, Well-posedness of an elliptic equation with involu- tion, Electron. J. Differ. Equ. 2015 (284), 1–8, 2015.
• A. Ashyralyev, B. Karabaeva and A.M. Sarsenbi, Stable difference scheme for the solution of an elliptic equation with involution. In: A. Ashyralyev and A. Lukashov, editors. International Conference on Analysis and Applied Mathematics - ICAAM 2016 AIP Conference Proceedings; 7–10 September 2016; AIP Publishing, 2016, 1759: 020111, 8 pages.
• C. Babbage, An essay towards the calculus of calculus of functions, Philos. Trans. R. Soc. Lond. 106, 179–256, 1816.
• M.Sh. Burlutskayaa and A.P. Khromov, Fourier Method in an Initial-Boundary Value Problem for a First-Order Partial Differential Equation with Involution, Comput. Math. Math. Phys. 51 (12), 2102–2114, 2011.
• A. Cabada and A.F. Tojo, Equations with involutions, Atlantis Press, 2015.
• 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
• T. Carleman, Sur la theorie des équations intégrales et ses applications, Verhandl. des internat. Mathem. Kongr. I., Zurich, 138–151, 1932.
• 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.
• 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.
• C.P. Gupta, Existence and uniqueness theorems for boundary value problems involving reflection of the argument, Nonlinear Anal. 11 (9), 1075–1083, 1987.
• C.P. Gupta, Two-point boundary value problems involving reflection of the argument, Int. J. Math. Sci. 10 (2), 361–371, 1987.
• I.A. Kaliev and M.M. Sabitova, Problems of determining the temperature and density of heat sources from the initial and final temperatures, ğJ. Appl. Ind. Math. 4 (3), 332–339, 2010.
• 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.
• M. Kirane and N. Al-Salti, Inverse problems for a nonlocal wave equation with an involution perturbation, J. Nonlinear Sci. Appl. 9, 1243–1251, 2016.
• 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.
• M. Kirane, B.Kh. Turmetov and B.T. Torebek, A nonlocal fractional Helmholtz equa- tion, Fractional Differential Calculus, 7 (2), 225–234, 2017.
• K. Knopp, Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. Dover, New York, 1996.
• A. Kopzhassarova and A. Sarsenbi, Basis Properties of Eigenfunctions of Second- Order Differential Operators with Involution, Abstr. Appl. Anal. 2012 Article ID: 576843, 1–6, 2012.
• E.I. Moiseev, On the basis property of systems of sines and cosines, Doklady AN SSSR 275 (4), 794–798, 1984.
• M.A. Naimark, Linear Differential Operators Part II, Ungar, New York, 1968.
• 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.
• I. Orazov and M.A. Sadybekov, On a class of problems of determining the temperature and density of heat sources given initial and final temperature, Sib. Math. J. 53 (1), 146–151, 2012.
• D. Przeworska-Rolewicz, Sur les équations involutives et leurs applications, Studia Math. 20, 95–117, 1961.
• D. Przeworska-Rolewicz, On equations with different involutions of different orders and their applications to partial differential-difference equations, Studia Mathe. 32, 101–111, 1969.
• D. Przeworska-Rolewicz, On equations with reflection, Studia Math. 33, 197–206, 1969.
• D. Przeworska-Rolewicz, On equations with rotations, Studia Math. 35, 51–68, 1970.
• D. Przeworska-Rolewicz, Right invertible operators and functional-differential equa- tions with involutions, Demonstration Math. 5, 165–177, 1973.
• 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 and Warsaw, 1973.
• D. Przeworska-Rolewicz, On linear differential equations with transformed argument solvable by means of right invertible operators, Ann. Polon. Math. 29, 141–148, 1974.
• I.A. Rus, Maximum principles for some nonlinear differential equations with deviating arguments, Studia Univ. Babes-Bolyai Math. 32 (2), 53–57, 1987.
• M.A. Sadybekov and A.M. Sarsenbi, On the notion of regularity of boundary value problems for differential equation of second order with dump argument (Russian), Math. J. 7 (1), 2007.
• A.M. Sarsenbi, Unconditional bases related to a nonclassical second-order differential operator, Differ. Equ. 46 (4), 506–511, 2010.
• A.M. Sarsenbi and A.A. Tengaeva, On the basis properties of root functions of two generalized eigenvalue problems, Differ. Equ. 48 (2), 1–3, 2012.
• A.L. Skubachevskii, Elliptic functional differential equations and applications, Birkhauser, Basel-Boston-Berlin, 1997.
• W. Watkins, Modified Wiener equations, IJMMS, 27 (6), 347–356, 2001.
• J. Wiener, Differential Equation with Involutions, Differ. Equ. 5, 1131–1137, 1969.
• J. Wiener, Differential Equation in Partial Derivatives with Involutions, Differ. Equ. 6, 1320–1322, 1970.
• J. Wiener, Generalized Solutions of Functional-Differential Equations, World Scien- tific Publishing, New Jersey, 1993.
• J. Wiener and A.R. Aftabizadeh, Boundary value problems for differential equations with reflection of the argument, Int. J. Math. Sci. 8 (1), 151–163, 1985.
• J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.