Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 52. no. 3. P. 53 – 62. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-52-3-53-62
Research Article
Full text in English
MSC 35R11

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Nonlocal Problem for the Time-Fractional Hyperbolic-Type Equation with the Prabhakar Fractional Derivative

Kh. N. Turdiev^{\ast}

Fergana state university, 150100, Fergana, Murabbiylar street, 19, Uzbekistan

Abstract. In this paper, we investigate a nonlocal boundary value problem for a time-fractional hyperbolic-type partial differential equation involving fractional derivatives of regularized Prabhakar. The fractional differentiation is defined through the regularized Prabhakar operator, which provides a flexible framework for modeling memory effects with non-singular kernels. The equation is considered on a bounded rectangular domain in the plane with respect to two independent variables. The boundary conditions are nonlocal and are prescribed in the form of partial integral expressions of the unknown solution along each spatial variable, where the corresponding kernels are assumed to be continuous. Building upon previously obtained representation formulas for the solution of the associated Goursat problem in terms of Mittag-Leffler type functions, the original boundary value problem is transformed into a coupled system of Volterra integral equations of the second kind for the traces of the solution on a portion of the boundary. This reduction allows us to apply classical methods of integral equations to analyze the problem. By employing appropriate estimates for the regularized Prabhakar kernels and the properties of the resulting integral operators, we rigorously establish the existence and uniqueness of the solution to the nonlocal boundary value problem. Furthermore, an explicit representation of the solution is derived in terms of the solutions of the obtained system of integral equations. The results demonstrate that the regularized Prabhakar framework provides a robust and analytically tractable approach for treating time-fractional hyperbolic problems with nonlocal boundary interactions.

Key words: Telegraph equation, nonlocal problem, integral equation, Goursat problem, Prabhakar fractional order derivative, Mittag-Leffler type function.

Received: 10.11.2025; Revised: 11.11.2025; Accepted: 12.11.2025; First online: 14.11.2025

For citation. Turdiev Kh.N.  Nonlocal problem for the time-fractional hyperbolic-type equation with
the Prabhakar fractional derivative. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 52: 3, 53-62. EDN: IBVSBJ.
https://doi.org/10.26117/2079-6641-2025-52-3-53-62.

Funding. The study was conducted without the support of foundations.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for the submission of the final version of the article for publication.

^{\ast}Correspondence: E-mail: xurshidjon2801@gmail.com

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Turdiev Kh.N., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

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Information about the author

Turdiev Khurshidjon Nurmahamat ugli – PhD student, Mathematical analysis and differential equations, Fergana state university, Fergana, Uzbekistan, ORCID 0000-0003-4960-8669.

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