arXiv subjects

Damir Shamuratov

Publications and source records attributed to Damir Shamuratov.

3 recordsLinked to original sources

arXiv2026

Integral representations and asymptotic behaviors of the Multivariate Mittag-Leffler function

In this paper, a multivariate Mittag--Leffler-type function arising in the theory of fractional differential equations with several fractional parameters is investigated. New Hankel contour integral representations are derived for the three-variable Mittag--Leffler function, and complete asymptotic expansions are established in different sectors of the complex plane. The proposed approach is based on the classical Hankel integral representation of the reciprocal Gamma function together with suitable contour transformations. Furthermore, the corresponding integral representations and asymptotic expansions are established for the multivariate Mittag-Leffler function with an arbitrary number of variables. The obtained formulas extend several known results for one- and two-variable Mittag--Leffler functions and provide useful analytical tools for the qualitative analysis of fractional differential equations involving multiple fractional derivatives.

math.AP↗

arXiv2026

The backward problem for a multi-term time-fractional diffusion equation

This paper is devoted to the investigation of the backward problem for a multi-term time-fractional diffusion equation. Backward problems for fractional diffusion equations are typically studied using regularization methods due to their ill-posedness in the sense of Hadamard; that is, a small change in u(T) may lead to large changes in the initial data. Nevertheless, we show that if sufficiently smooth current data are considered, then the solution exists, is unique, and is stable. A principal difficulty in the analysis of the backward problem stems from the structure of the solution, in which the multinomial Mittag-Leffler function appears in the denominator. Accordingly, a precise characterization of the asymptotic behavior of this function is required. Such asymptotic properties are nontrivial and have been rigorously established in the authors' recent work, which serves as a fundamental basis for the present study. In addition, we investigate the conditional stability of the backward problem. It is shown that, although the problem is ill-posed in general, stability can be restored under an appropriate a priori bound imposed on the initial data. The main novelty of the paper lies in proving the best smoothing property of the solution, showing that it belongs to the domain of the operator A for any positive time.

math.AP↗

arXiv2026

Inverse problem for a multi-term time-fractional diffusion equation with the Caputo derivatives

This paper investigates an inverse source problem for a multi-term time-fractional diffusion equation with Caputo derivatives. The source term is separable as \(f(x)g(t)\), with the unknown spatial component \(f(x)\) reconstructed from an overdetermination condition at interior time \(t_0 \in (0, T]\). The elliptic part is governed by a self-adjoint positive differential operator \(A(x, D)\) of order \(m \ge 2\). The solution features a spectral representation using the multinomial Mittag-Leffler function, for which we derive novel precise asymptotic expansions. These asymptotics provide a uniform lower bound for the solution's characteristic denominator, enabling sufficient conditions for the existence of a classical solution. Uniqueness of the reconstructed source holds under natural assumptions on the data and \(g(t)\). Despite the problem's ill-posedness, high-regularity classical solutions are achievable under suitable structural conditions.

math.AP↗