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Mixed partial differential equation: Forward problem linked with the wave-diffusion process

  • Erkinjon Karimov EMAIL logo , Niyaz Tokmagambetov and Muzaffar Toshpulatov
Published/Copyright: October 11, 2025
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Abstract

The main target of the present investigation is a mixed partial differential equation involving the Prabhakar integral-differential operator in the time variable. We begin by briefly describing the several physical processes in which mixed partial differential equations play a key role. Then we formulate an initial-boundary problem for the considered equation linked with the wave-diffusion process. Our primary objective is to demonstrate the unique solvability of the formulated problem under specific conditions for the given data. First, we present the explicit solution of the Cauchy problem for an ordinary differential equation with the regularized Prabhakar fractional derivative. We also present important statements on the bivariate Mittag-Leffler function, namely, Euler-type integral representations and certain estimations for the bi-variate Mittag-Leffler-type function E 2 ( x , y ) . Since the key tool of investigation is the method of separation of variables, most of our evaluations are linked with the proof of uniform convergence of infinite series. We impose certain conditions on given functions to provide uniform convergence of infinite series corresponding to the solution of the formulated problem.

MSC 2020: 35M10; 35R11

Funding statement: This research was funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP23487589). Erkinjon Karimov is partially supported by the Methusalem programme of the Ghent University Special Research Fund (BOF), Grant/Award Number: 01M01021.

References

[1] O. K. Abdullaev, A nonlocal problem with an integral matching condition for a loaded parabolic-hyperbolic equation with a fractional Caputo derivative, Differ. Equ. 59 (2023), no. 3, 351–358. 10.1134/S0012266123030059Search in Google Scholar

[2] R. R. Ashurov and R. T. Zunnunov, An analog of the Tricomi problem for a mixed-type equation with fractional derivative. Inverse problems, Lobachevskii J. Math. 44 (2023), no. 8, 3225–3240. 10.1134/S1995080223080073Search in Google Scholar

[3] A. S. Berdyshev, A. Cabada and E. T. Karimov, On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann–Liouville fractional differential operator, Nonlinear Anal. 75 (2012), no. 6, 3268–3273. 10.1016/j.na.2011.12.033Search in Google Scholar

[4] A. S. Berdyshev, B. E. Eshmatov and B. J. Kadirkulov, Boundary value problems for fourth-order mixed type equation with fractional derivative, Electron. J. Differential Equations 2016 (2016), Paper No. 36. Search in Google Scholar

[5] A. Bokhari, D. Baleanu and R. Belgacem, Regularized Prabhakar derivative for partial differential equations, Comput. Methods Differ. Equ. 10 (2022), no. 3, 726–737. Search in Google Scholar

[6] A. Consiglio and F. Mainardi, On the evolution of fractional diffusive waves, Ric. Mat. 70 (2021), no. 1, 21–33. 10.1007/s11587-019-00476-6Search in Google Scholar

[7] M. D’Ovidio and F. Polito, Fractional diffusion-telegraph equations and their associated stochastic solutions, Teor. Veroyatn. Primen. 62 (2017), no. 4, 692–718; translation in Theory Probab. Appl. 62 (2018), no. 4, 552–574. Search in Google Scholar

[8] H. Elhadedy, M. S. Abdel Latif, H. M. Nour and A. H. Abdel Kader, Exact solution for heat conduction inside a sphere with heat absorption using the regularized Hilfer–Prabhakar derivative, J. Appl. Math. Comput. Mech. 21 (2022), no. 2, 27–37. 10.17512/jamcm.2022.2.03Search in Google Scholar

[9] P. Feng and E. T. Karimov, Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations, J. Inverse Ill-Posed Probl. 23 (2015), no. 4, 339–353. 10.1515/jiip-2014-0022Search in Google Scholar

[10] A. Fernandez, N. Rani and Ž. Tomovski, An operational calculus approach to Hilfer–Prabhakar fractional derivatives, Banach J. Math. Anal. 17 (2023), no. 2, Paper No. 33. 10.1007/s43037-023-00258-1Search in Google Scholar

[11] M. Garg, P. Manohar and S. L. Kalla, A Mittag-Leffler-type function of two variables, Integral Transforms Spec. Funct. 24 (2013), no. 11, 934–944. 10.1080/10652469.2013.789872Search in Google Scholar

[12] R. Garra, R. Gorenflo, F. Polito and Ž. Tomovski, Hilfer–Prabhakar derivatives and some applications, Appl. Math. Comput. 242 (2014), 576–589. 10.1016/j.amc.2014.05.129Search in Google Scholar

[13] B. Y. Irgashev, A nonlocal problem for a mixed equation of high even order with a fractional Caputo derivative, J. Elliptic Parabol. Equ. 9 (2023), no. 1, 389–399. 10.1007/s41808-023-00205-zSearch in Google Scholar

[14] B. J. Kadirkulov and M. A. Jalilov, On a boundary value problem for a third-order equation of parabolic-hyperbolic type with a fractional order operator, Lobachevskii J. Math. 44 (2023), no. 7, 2725–2737. 10.1134/S1995080223070223Search in Google Scholar

[15] E. T. Karimov and A. Hasanov, On a boundary-value problem in a bounded domain for a time-fractional diffusion equation with the Prabhakar fractional derivative, Bull. Karaganda Univ. Math. Ser. (2023), no. 3(111), 39–46. 10.31489/2023m3/39-46Search in Google Scholar

[16] E. T. Karimov, N. Tokmagambetov and M. Toshpulatov, On a mixed equation involving prabhakar fractional order integral-differential operators, Extended Abstracts 2021/2022., Trends Math. 2, Birkhäuser, Cham (2024), 221–230. 10.1007/978-3-031-42539-4_25Search in Google Scholar

[17] S. Kerbal, E. Karimov and N. Rakhmatullaeva, Non-local boundary problem with integral form transmitting condition for fractional mixed type equation in a composite domain, Math. Model. Nat. Phenom. 12 (2017), no. 3, 95–104. 10.1051/mmnp/201712309Search in Google Scholar

[18] J. Khujakulov, Initial-boundary value problem for a time fractional differential equation with the Prabhakar derivative on a star graph, Bull. Inst. Math. 6 (2023), no. 2, 20–30. Search in Google Scholar

[19] A. A. Kilbas and O. A. Repin, An analogue of the Bitsadze–Samarskiĭ problem for an equation of mixed type with a fractional derivative, Differ. Uravn. 39 (2003), no. 5, 638–644, 719. 10.1023/A:1026194020442Search in Google Scholar

[20] A. V. Luikov, Application of the methods of thermodynamics of irreversible processes to the investigation of heat and mass transfer, J. Eng. Phys. 9 (1965), no. 3, 189–202. 10.1007/BF00828333Search in Google Scholar

[21] Y. Povstenko and J. Klekot, Fractional heat conduction with heat absorption in a sphere under Dirichlet boundary condition, Comput. Appl. Math. 37 (2018), no. 4, 4475–4483. 10.1007/s40314-018-0585-7Search in Google Scholar

[22] T. R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7–15. Search in Google Scholar

[23] A. V. Pskhu, Partial Differential Equations of Fractional Order (in Russian), “Nauka”, Moscow, 2005. Search in Google Scholar

[24] N. Rani and A. Fernandez, Solving Prabhakar differential equations using Mikusiński’s operational calculus, Comput. Appl. Math. 41 (2022), no. 3, Paper No. 107. 10.1007/s40314-022-01794-6Search in Google Scholar

[25] K. Sadarangani and O. K. Abdullaev, A non-local problem with discontinuous matching condition for loaded mixed type equation involving the Caputo fractional derivative, Adv. Difference Equ. 2016 (2016), Paper No. 241. 10.1186/s13662-016-0969-1Search in Google Scholar

[26] V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin, 2011. 10.1007/978-3-642-14003-7Search in Google Scholar

[27] V. E. Tarasov, Non-linear fractional field equations: weak non-linearity at power-law non-locality, Nonlinear Dynam. 80 (2015), no. 4, 1665–1672. 10.1007/s11071-014-1342-0Search in Google Scholar

[28] V. E. Tarasov, Fractional dynamics with depreciation and obsolescence: Equations with Prabhakar fractional derivatives, Mathematics 10 (2022), no. 9, Article ID 1540. 10.3390/math10091540Search in Google Scholar

[29] V. E. Tarasov and S. S. Tarasova, Fractional derivatives and integrals: What are they needed for?, Mathematics 8 (2020), no. 2, Paper No. 164. 10.3390/math8020164Search in Google Scholar

[30] V. V. Tarasova and V. E. Tarasov, Economic Dynamics with Memory: Fractional Calculus Approach, De Gruyter, Berlin, 2021. 10.1515/9783110627459Search in Google Scholar

[31] M. Toshpulatov, Initial-boundary problem for time-fractional mixed equation with space-variable coefficients, Bull. Inst. Math. 7 (2024), no. 3, 77–91. Search in Google Scholar

[32] K. Turdiev, Boundary problem for a diffusion-wave equation involving regularized Prabhakar fractional derivative, Bull. Inst. Math. 6 (2023), no. 6, 39–45. Search in Google Scholar

[33] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Volume II, Nonlinear Phys. Sci., Springer, Heidelberg, 2013. 10.1007/978-3-642-33911-0Search in Google Scholar

[34] T. K. Yuldashev and B. J. Kadirkulov, Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator, Ural Math. J. 6 (2020), no. 1, 153–167. 10.15826/umj.2020.1.013Search in Google Scholar

[35] L. A. Zolina, Boundary value problem for the model equation of the hyperbolic-parabolic type, Ž. Vyčisl. Mat i Mat. Fiz. 6 (1966), 991–1001. 10.1016/0041-5553(66)90162-5Search in Google Scholar

Received: 2025-01-03
Revised: 2025-06-11
Accepted: 2025-07-18
Published Online: 2025-10-11

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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