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

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

1. 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.
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 their Applications 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-Order Differential 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 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.
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 Value Problem 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. des internat. 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 involving reflection 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 density of 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 an involution 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. 2012 Article ID: 576843, 1–6, 2012.
23. E.I. Moiseev, On the basis property of systems of sines and cosines, Doklady AN SSSR 275 (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 temperature and 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, Studia Math. 20, 95–117, 1961.
28. 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.
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 and Warsaw, 1973.
33. D. Przeworska-Rolewicz, On linear differential equations with transformed argument solvable 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 deviating arguments, 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 value problems 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 differential operator, Differ. Equ. 46 (4), 506–511, 2010.
37. 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.
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 equations with 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.