Solvability of a Mixed Problem for an Integro-Differential Equation with the Hilfer Type Fractional Analogue of the Barenblatt–Zheltov–Kochina Operator

Authors: Tursun K. Yuldashev, Bakhtiyor J. Kadirkulov, Aygul A. Matchanova

Keywords: integro-differential equation, Hilfer fractional analog of the Barenblatt–Zheltov–Kochina operator, degenerate kernel, regular solvability

Abstract:

In the article a partial integro-differential equation with degenerate kernel and Hilfer fractional operator is considered. The issues of unique classical solvability and construction of a solution to a mixed problem for a homogeneous integro-differential equation containing a Hilfer fractional analogue of the Barenblatt–Zheltov–Kochina operator are studied. The Fourier series method based on the separation of variables was used. By the aid of the Mittag–Leffler function is obtained a countable system. Sufficient coefficient conditions for unique classical solvability of the mixed problem are established. The absolute and uniform convergence of the obtained series is proved.

References:

[1] O. KH. ABDULLAEV, O. SH. SALMANOV, T. K. YULDASHEV, Direct and inverse problems for a parabolic-hyperbolic equation involving Riemann–Liouville derivatives, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. – Mathematics Vol. 43, 1 (2023), 21–33. [2] O. KH. ABDULLAEV, T. K. YULDASHEV, Inverse problems for the loaded parabolichyperbolic equation involves Riemann–Liouville operator, Lobachevskii J. Math. Vol. 44, 3 (2023), 1080–1090. https://doi.org/10.1134/S1995080223030034. [3] E. ALVAREZ, A. GOMEZ, M. PINTO, (w,c)−periodic functions and mild solutions to abstract fractional integro-differential equations, Electron. J. Qual. Theory Differ. Equ. Vol. 16 (2018), 1–8. https://doi.org/10.14232/ejqtde.2018.1.16 [4] B. AHMAD, A. ALSAEDI, M. KIRANE, R. G. TAPDIGOGLU, An inverse problem for space and time fractional evolution equations with an involution perturbation, Quaestiones Mathematicae Vol. 40, 2 (2017), 151–160. https://doi.org/10.2989/16073606.2017.1283370 [5] R. R. ASHUROV, O. T. MUHIDDINOVA, Initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary elliptic differential operator, Lobachevskii J. Math. Vol. 42, 3 (2021), 517–525. [6] R. R. ASHUROV, O. T. MUHIDDINOVA, S. R. UMAROV, A Non-local problem for the fractional-order Rayleigh–Stokes equation, Fractal and Fractional Vol. 7, 6, ID 490 (2023), 1–16. https://doi.org/10.3390/ fractalfract7060490 [7] R. R. ASHUROV, O. T. MUHIDDINOVA, Inverse problem of determining the order of the fractional derivative in the Rayleigh-Stokes equation, Fractional Calculus and Applied Analysis Vol. 26 (2023), 1691–1708. https://doi.org/10.1007/s13540-023-00178-9 [8] R. R. ASHUROV, YU. E. FAYZIEV, Uniqueness and existence for inverse problem of determining an order of time-fractional derivative of subdiffusion equation, Lobachevskii J. Math. Vol. 42, 3 (2021), 508–516. https://doi.org/10.1134/S1995080221030069 [9] R. R. ASHUROV, YU. E. FAYZIEV, N. KH. KHUSHVAKTOV, Some problems for the Barenblatt–Zheltov–Kochina type fractional equations, Bulletin of the Institute of Mathematics Vol. 5, 5 (2022), 97–104. [10] R. R. ASHUROV, YU. E. FAYZIEV, N. KH. KHUSHVAKTOV, Forward and inverse problems for the Barenblatt–Zheltov–Kochina type fractional equations, Lobachevskii J.Math. Vol. 44, 7 (2023), 2563–2572. [11] R. R. ASHUROV, M. D. SHAKAROVA, Time-Dependent source identification problem for fractional Schr¨odinger type equations, Lobachevskii J. Math. Vol. 43, 5 (2022), 1053–1064. [12] R. R. ASHUROV, N. Z. VAISOVA, Backward and non-local problems for the Rayleigh–Stokes equation, Fractal and Fractional Vol. 6, 587 (2022), 1–18. https://doi.org/10.3390/fractalfract6100587 [13] E. ALVAREZ, S. DIAZ, C. LIZAMA, On the existence and uniqueness of (w,c)−periodic solutions to a class of Volterra difference equations, Adv. Differ. Equ. Vol. 2019, 105 (2019), 1–12. https://doi.org/10.1186/s13662-019-2053-0 [14] M. AGAOGLOU, M. FECKAN, A. P. PANAGIOTIDOU, Existence and uniqueness of (w,c)−periodic solutions of semilinear evolution equations, Int. J. Dyn. Sys. Diff. Equ. Vol. 10, 2 (2020), 149–166. https://doi.org/10.1504/IJDSDE.2020.106027 [15] T. AKRAM, M. ABBAS, A. ALI, A numerical study on time fractional Fisher equation using an extended cubic B-spline approximation, J. Math. Comput. Sci. Vol. 22, 1 (2021), 85–96. http://dx.doi.org/10.22436/jmcs.022.01.08 [16] R. ALAHMAD, Q. ALAHMAD, A. ABDELHADI, Solution of fractional autonomous ordinary differential equations, J. Math. Comput. Sci. Vol. 27, 1 (2020), 59–64. http://dx.doi.org/10.22436/jmcs.027.01.05 [17] R. ASHUROV, B. KADIRKULOV, M. JALILOV, On an inverse problem of the Bitsadze–Samarskii type for a parabolic equation of fractional order, Bolet´ın de la Sociedad Matem´atica Mexicana Vol. 29, 3 (2023), 1–21. https://doi.org/10.1007/s40590-023-00542-y [18] A. YU. ANIKIN, S. YU. DOBROKHOTOV, A. A. SHKALIKOV, On expansions in the exact and asymptotic eigenfunctions of the one-dimensional Schr¨odinger operator, Math. Notes Vol. 112, 5 (2022), 623–641. [19] G. I. BARENBLATT, YU. P. ZHELTOV, I. N. KOCHINA, On finitness conditions in the mechanics of continuous media. Static problems of the theory of elasticity, Prikl. Mat. i Mekh. Vol. 24, 5 (1960), 316–322. [20] V. D. BUDAEV, Orthogonal and biorthogonal bases, Proceedings of the Russian State Pedagogical University named by A. I. Herzen Vol. 5 (2005), 7–38. [in Russian] [21] A. T. BULABAEV, M. OTELBAEV, L. A. SHUSTER, Properties of Green’s function of the Sturm–Liouville operator and their applications, Differ. Equations Vol. 25, 7 (1989), 773–779. [22] G. I. BARENBLATT, YU. P. ZHELTOV, Fundamental equations of filtration of homogeneous liquids in fissured rocks, Sov. Phys., Dokl., Vol. 132, 5, (1960) 522–525. [23] A. S. BERDYSHEV, J. B. KADIRKULOV, On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan–Nersesyan operator, Differential Equations Vol. 52, 1 (2016), 122–127. https://doi.org/10.1134/S0012266116010109 [24] S. A. BERDYSHEV, A. CABADA, J. B. KADIRKULOV, The Samarskii–Ionkin type problem for the fourth order parabolic equation with fractional differential operator, Computers and Mathematics with Applications Vol. 62 (2011), 3884–3893. [25] M. KH. BESHTOKOV, To boundary-value problems for degenerating pseudoparabolic equations with Gerasimov–Caputo fractional derivative, Russian Mathematics (Izv. vuz) Vol. 62, 10 (2018), 1–14. [26] G. I. BARENBLATT, V. M. YENTOV, V. M. RYZHIK, Movement of Liquids and Gases in Natural Reservoirs, Nedra, Moscow, 1984. [in Russian] [27] K. S. BASNIEV, I. N. KOCHINA, V. M. MAKSIMOV, Underground Hydromechanics, Nedra, Moscow, 1993. [in Russian] [28] P. N. DUC, H. D. BINH, L. D. LONG, H. T. KIM VAN, Reconstructing the right-hand side of the Rayleigh–Stokes problem with nonlocal in time condition, Advances in Difference Equations Vol. 470 (2021), 1–18. https://doi.org/10.1186/s13662-021-03626-z [29] J. T. EDWARDS, N. J. FORD, A. C. SIMPSON, The numerical solution of linear multi-term fractional differential equations: systems of equations, J. Computational and Appl. Math., Vol. 148, 2 (2002), 401–418. https://doi.org/10.1016/S0377-0427(02)00558-7 [30] V. E. FEDOROV, A. V. NAGUMANOVA, Inverse linear problems for a certain class of degenerate fractional evolution equations, Itogi nauki i tekhniki. Seriya Sovremennaya matematika i ikh prilozheniya. Tematicheskiye obzory, Vol. 167 (2019), 97–111. [in Russian] https://doi.org/10.36535/0233-6723-2019-167-97-111 [31] R. GORENFLO, A. A. KILBAS, F. MAINARDI, S. V. ROGOZIN, Mittag–Leffler Functions, Related Topics and Applications, Springer, Berlin – Heidelberg, Germany, 2014, https://doi.org/10.1007/978-3-662-61550-8. [32] V. A. IL’IN, Necessary and sufficient conditions for being a Riesz basis of root vectors of second-order discontinuous operators, Differ. Uravneniya Vol. 22, 12 (1986), 2059–2071. [in Russian] [33] H. K. JASSIM, M. SHAREEF, On approximate solutions for fractional system of differential equations with Caputo-Fabrizio fractional operator, J. Math. Comput. Sci. Vol. 23, 1 (2021), 58–66. http://dx.doi.org/10.22436/jmcs.023.01.06 [34] P. YA. KOCHINA AND OTHER, Development of Research on the Theory of Filtration in the USSR (1917–1967) Nauka, Moscow, 1969 [in Russian]. 138–144. [35] A. A. KILBAS, H. M. SRIVASTAVA, J. J. TRUJILLO, Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland,Mathematics studies, 2006. [36] D. KUMAR, D. BALEANU, Fractional calculus and its applications in physics, Front. Phys. Vol. 7, 6 (2019). https://doi.org/10.3389/fphy.2019.00081. [37] M-H. KIM, G-CH. RI, H-CH. O, Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives Fract. Calc. Appl. Anal. Vol. 17, 1 (2014), 79–95. https://doi.org/10.2478/s13540-014-0156-6. [38] M. T. KHALLADI, A. RAHMANI, (w,c)−pseudo almost periodic distributions, Nonauton. Dyn. Syst. No. 7 (2020), 237–248. https://doi.org/10.1515/msds-2020-0119 [39] L. V. KRITSKOV, A. M. SARSENBI, Spectral properties of a nonlocal problem for a second-order differential equation with an involution, Differential Equations Vol. 51, 8 (2015), 984–990. https://doi.org/10.1134/S0012266115080029 [40] L. V. KRITSKOV, M. A. SADYBEKOV, A. M. SARSENBI, Properties in Lp of root functions for a nonlocal problem with involution, Turkish J. Math. Vol. 43 (2019), 393–401. [41] S. KERBAL, B. J. KADIRKULOV, M. KIRANE, Direct and inverse problems for a Samarskii–Ionkin type problem for a two dimensional fractional parabolic equation, Progr. Fract. Differ. Appl. Vol. 4, 3 (2018), 1–14. http://dx.doi.org/10.18576/pfda/01010 [42] C. LIZAMA, Abstract Linear Fractional Evolution Equations, Handbook of Fractional Calculus with Applications Vol. 2, J.A.T. Marchado Ed. DeGruyter, Berlin, 2019. p. 465–497. [43] Y. LIU, ZH. LI, M. YAMAMOTO, Inverse problems of determining sources of the fractional partial differential equations, Handbook of Fractional Calculus with Applications Vol. 2, J.A.T. Marchado Ed. DeGruyter, Berlin, 2019. p. 411–429. [44] N. K. OCHILOVA, T. K. YULDASHEV, On a nonlocal boundary value problem for a degenerate parabolic-hyperbolic equation with fractional derivative, Lobachevskii J. Math. Vol. 43, 1 (2022), 229–236. https://doi.org/10.1134/S1995080222040175 [45] M. PAKDAMAN, A. AHMADIAN, S. EFFATI, S. SALAHSHOUR, D. BALEANU, Solving differential equations of fractional order using an optimization technique based on training artificial neural network, Appl. Math. Comput. Vol. 293 (2017), 81–95. https://doi.org/10.1016/j.amc.2016.07.021 [46] A. V. PSKHU, Fractional Partial Differential Equations, Nauka,Moscow, 2005. [in Russian] https://www.elibrary.ru/item.asp? id=19448655 [47] S. PATNAIK, J. P. HOLLKAMP, F. SEMPERLOTTI, Applications of variable-order fractional operators: a review, Proceedings A, Royal Society, A476: 20190498. http://dx.doi.org/10.1098/rspa.2019.0498 [48] P. YA. POLUBARINOVA-KOCHINA, The Theory of Groundwater Movement, Nauka, Moscow, 1977. [in Russian] [49] M. A. SADYBEKOV, G. DILDABEK, A. TENGAYEVA, Constructing a basis from systems of eigenfunctions of one not strengthened regular boundary value problem, Filomat Vol. 31, 4 (2017), 981–987. [50] R. K. SAXENA, R. GARRA, E. ORSINGHER, Analytical solution of space-time fractional telegraph-type equations involving Hilfer and Hadamard derivatives, Integral Transforms and Special Functions No. 6 (2015). https://doi.org/10.1080/10652469.2015.1092142. [51] G. A. SVIRIDYUK, D. E. SHAFRANOV, The Cauchy problem for the Barenblatt–Zheltov–Kochina equation on a smooth manifold, Vest. Chel’yabinskGU Vol. 9 (2003), 171–177. [52] M. A. SAGADEEVA, F. L. HASAN, Bounded solutions of Barenblatt–Zheltov–Kochinamodel in Quasi-Sobolev spaces, Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming and Computer Software Vol. 8, 4 (2015), [53] H. SUN, A. CHANG, Y. ZHANG, W. CHEN, A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications, Fract. Calc. Appl. Anal. Vol. 22 (2019), 27–59. https://doi.org/10.1515/fca-2019-0003 [54] A. M. SAVCHUK, A. A. SHKALIKOV, Recovering a potential of Sturm–Liouville problem from finite sets of spectral data, American Mathematical Society Translations – Series 2, Advances in the Mathematical Sciences Vol. 233 (2014), 211–224. [55] A. A. SHKALIKOV, The basis problem of the eigenfunctions of ordinary differential operators with integral boundary conditions, Moscow Univ. Math. Bull. Vol. 37, 6 (1982), 10–20. [56] A. A. SHKALIKOV, Basis properties of root functions of differential operators with spectral parameter in the boundary conditions, Differen. Equations Vol. 55, 5 (2019), 631–643. [57] J. A. TENREIRO MACHADO Handbook of Fractional Calculus with Applications in eight volumes , Walter de Gruyter GmbH, Berlin – Boston, 2019, p. 47–85. [58] V. E. VLADYKINA, A. A. SHKALIKOV, Spectral properties of ordinary differential operators with involution, Doklady Mathematics Vol. 99, 1 (2019), 5–10. [59] T. K. YULDASHEV, O. KH. ABDULLAEV, Unique solvability of a boundary value problem for a loaded fractional parabolic-hyperbolic equation with nonlinear terms, Lobachevskii J. Math. Vol. 42, 5 (2021), 1113–1123. https://doi.org/10.1134/S1995080221050218 [60] T. K. YULDASHEV, B. J. KADIRKULOV, Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator, Ural Math. J. Vol. 6, 1 (2020), 153–167. [61] T. K. YULDASHEV, T. G. ERGASHEV, T. A. ABDUVAHOBOV, Nonlinear system of impulsive integro-differential equations with Hilfer fractional operator and mixed maxima, Chelyabinsk. Phys. Math. J. Vol. 7, 3 (2022), 312–325. [62] T. K. YULDASHEV, KH. KH. SABUROV, T. A. ABDUVAHOBOV, Nonlocal problem for a nonlinear system of fractional order impulsive integro-differential equations with maxima, Chelyab. Phys.-Math. J. Vol. 7, 1 (2022), 113–122. [63] T. K. YULDASHEV, T. A. ABDUVAHOBOV, Periodic solutions for an impulsive system of fractional order integro-differential equations with maxima, Lobachevskii J. Math. Vol. 44, 10 (2023), 4393–4401 , https://doi.org/10.1134/S1995080223100451 [64] T. K. YULDASHEV, KH. R. MAMEDOV, T. A. ABDUVAHOBOV, On a periodic solution for an impulsive system of differential equations with Gerasimov–Caputo fractional operator and maxima, J. Contemporary Applied Math. Vol. 13, 1 (2023), 111–122. http://journalcam.com [65] T. K. YULDASHEV, KH. R. MAMEDOV, T. A. ABDUVAHOBOV, (w,c)−periodic solution for an impulsive system of differential equations with the quadrate of Gerasimov–Caputo fractional operator and maxima, J. Contemporary Appl. Math., Vol. 13, 2 (2023), 65–79, http://doi.org/10.5281/zenodo.8425548, http://journalcam.com [66] T. K. YULDASHEV, B. J. KADIRKULOV, Boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator, Axioms Vol. 9, 2 (2020), 1–19. [67] T. K. YULDASHEV, B. J. KADIRKULOV, Inverse boundary value problem for a fractional differential equations of mixed type with integral redefinition conditions, Lobachevskii J. Math. Vol. 42, 3 (2021), 649–662. [68] T. K. YULDASHEV, B. J. KADIRKULOV, Inverse problem for a partial differential equation with Gerasimov–Caputo-type operator and degeneration, Fractal Fract Vol. 5, 2, ID 58 (2021), 1–13. https://doi.org/10.3390/fractalfract5020058 [69] T. K. YULDASHEV, B. J. KADIRKULOV, On a boundary value problem for a mixed type fractional differential equations with parameters, Proceedings of the Institute of Mathematics and Mechanics of Azerbaijan National Academy of Sciences Vol. 47, 1 (2021), 112–123. [70] T. K. YULDASHEV, B. J. KADIRKULOV, R. A. BANDALIYEV, On a mixed problem for Hilfer type fractional differential equation with degeneration, Lobachevskii J. Math. Vol. 43, 1 (2022), 263–274. https://doi.org/10.1134/S1995080222040229 [71] T. K. YULDASHEV, E. T. KARIMOV, Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters, Axioms Vol 9, 4, ID 121 (2020), 1–24. https://doi.org/10.3390/axioms9040121 [72] Y. ZHANG, X. XU, Inverse source problem for a fractional differential equations, Inverse Prob. Vol. 27, 3 (2011), 31–42.