Journal of Advanced Analysis and Applications
Volume 1, Issue 1 (2026), Pages 26–53

A Nonlocal Problem for the Time-Fractional Diffusion-Wave Equation
Authors: D.K. Durdiev1,2 and A.A. Rahmonov1,2,*

1V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences
2Bukhara State University, Bukhara, Uzbekistan
*Correspondence: Email: araxmonov@mail.ru

Abstract. In this paper, we study a nonlocal initial boundary value problem for an abstract time-fractional
diffusion-wave equation involving an unbounded, positive, self-adjoint operator on a Hilbert space. The
existence of a mild solution is investigated using the eigenfunction decomposition method. By expanding
the solution in terms of the operators eigenfunctions, the problem is reduced to a system of ordinary
fractional differential equations with nonlocal conditions. The solutions of these equations are expressed
in terms of the Mittag-Leffler function. By examining the zeros of the denominator, we identify the appropriate
interval of definition, which excludes the right endpoint, ensuring the correctness of the solution. The solution
to the original problem is expressed as a series expansion in terms of the eigenfunctions of an abstract operator.
By applying estimation techniques in the corresponding Hilbert spaces, we establish the existence, uniqueness,
and regularity of the solution.

Keywords: fractional diffusion-wave equation; Mittag-Leffler function; H\”{o}lder continuity; existence;
uniqueness; regularity

MSC (2020): 35A09, 35R11, 35R30, 45B05

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Submitted: December 27, 2025 / Accepted: April 9, 2026 / Published: April 17, 2026

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How to cite: D.K. Durdiev and A.A. Rahmonov, A nonlocal problem for the time-fractional diffusion-wave equation,
Journal of Advanced Analysis and Applications, 1(1) (2026), 26–53.
DOI: https://doi.org/10.66745/jaaa.v1i1.003

This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). https://creativecommons.org/licenses/by/4.0/

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