Inverse Problems for the Fractional Diffusion Equation with the Hilfer Operator D. K. Durdiev, H. H. Turdiev, R. R. Rashidov Russian Mathematics, 2026 Abstract The article discusses inverse problems for the fractional diffusion equation with the Hilfer operator in time. The direct problem is the initial-boundary value problem for this equation with Cauchy-type initial data and Dirichlet boundary conditions. The first inverse problem, which involves determining a time-dependent coefficient, is reduced to an equivalent Volterra-type integral equation. The existence and uniqueness of the solution are proven using the contraction mapping principle. The second inverse problem involves determining a function dependent on the spatial variable on the right-hand side of the equation. This problem is studied using the Fourier method and the properties of the Mittag–Leffler function. The solution is constructed in the form of a series based on eigenfunctions.
Direct and Inverse Coefficient Problems for the Fractional Diffusion Wave Equation with the Riemann–Liouville Time Derivative H. H. Turdiev, M. O. Rajabova, S. H. Xoliqov, B. T. Karamatov Russian Mathematics, 2025 Abstract The inverse problem in the fractional wave equation with the Riemann–Louville derivative is considered. In this case, the direct problem is an initial nonlocal boundary value problem for this equation with initial Cauchy type and nonlocal boundary conditions. As a redefinition condition, a nonlocal integral condition with respect to the direct solution of the problem is specified. Using the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag-Leffler function and the generalized singular Grönwall inequality, we obtain an a priori estimate of the solution in terms of the unknown coefficient, which we will need to investigate for the inverse problem. The inverse problem is reduced to the equivalent integral of a Volterra type equation. To solve this equation, the contraction mapping principle is used. Local existence and global uniqueness have been proven.
Problem of Determining a Multidimensional Kernel in a Diffusion-Wave Equation with a Time-Fractional Derivative D. K. Durdiev, Z. A. Subhonova, H. H. Turdiev Russian Mathematics, 2025 For a time-fractional wave equation with an integral term of the convolution type, we study the direct Cauchy problem and the inverse problem of finding a multidimensional kernel of the integral, depending not only on the time variable, but also on the first $$n - 1$$ components of the spatial variable $$x = ({{x}_{1}},{{x}_{2}}, \ldots ,{{x}_{n}}) \in {{\mathbb{R}}^{n}}$$ . In this case, the known parameters of the problems are the Cauchy data specified at time $$t = 0$$ and the overdetermination condition on the hyperplane $${{x}_{n}} = 0.$$ The problems are equivalently reduced to problems that are convenient for further study. Using the fundamental solution to the time-fractional wave operator, which contains the generalized hypergeometric Fox function, the solution to the direct problem is written in the form of an integral equation of Volterra type and its properties are studied. Using the results of the direct problem, the solution to the inverse problem is also represented as a nonlinear integral equation. By applying the contraction mapping principle to this equation, the local solvability of the problem is established.
An Inverse Coefficient Problem for the Fractional Telegraph Equation with the Corresponding Fractional Derivative in Time D. K. Durdiev, T. R. Suyarov, Kh. Kh. Turdiev Russian Mathematics, 2025 Abstract This work investigates an initial-boundary value and an inverse coefficient problem of determining a time dependent coefficient in the fractional wave equation with the conformable fractional derivative and an integral. In the beginning, the initial boundary value problem (direct problem) is considered. By the Fourier method this problem is reduced to equivalent integral equations. Then, using the technique of estimating these functions and the generalized Grönwall inequality, we get the a priori estimate for the solution via the unknown coefficient which is used to study the inverse problem. The inverse problem is reduced to an equivalent integral equation of Volterra type. To show the existence and uniqueness of the solution to this equation, the Banach principle is applied. The local existence and uniqueness results are obtained.
Solvability Cauchy Problem for Time-Space Fractional Diffusion-Wave Equation with Variable Coefficient H. H. Turdiev Lobachevskii Journal of Mathematics, 2024 This work investigates the Cauchy problem for the $$n$$ -dimensional time-space fractional diffusion wave equation with variable coefficient. In order to provide solution to the Cauchy problem, a fundamental solution of this equation is constructed and the properties of this solution are studied. The fundamental solutions of the considered equations, which can be expressed in the H-function, are constructed and checked using the asymptotic expansions of the H-function. By the Fourier method, this problem is reduced to equivalent integral equations, which contain Mittag-Leffler type functions in free terms and kernels. An integral equation equivalent to Cauchy’s problem is obtained. For this equation, we construct their fundamental solution, analyze asymptotic behaviors of solutions, and study gradient estimates and large time behaviors. It was verified that the solution of the Cauchy problem satisfies the equation in the sense of a classical solution. The existence and uniqueness of the solution of the integral equation was proved.
Inverse coefficient problem for a time-fractional wave equation with initial-boundary and integral type overdetermination conditions D. K. Durdiev, H. H. Turdiev Mathematical Methods in the Applied Sciences, 2024 This paper considers the inverse problem of determining the time‐dependent coefficient in the time‐fractional diffusion‐wave equation. In this case, an initial boundary value problem was set for the fractional diffusion‐wave equation, and an additional condition was given for the inverse problem of determining the coefficient from this equation. First of all, it was considered the initial boundary value problem. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag‐Leffler function and the generalized singular Gronwall inequality, we get a priori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved. The stability estimate is also obtained.
THE PROBLEM OF DETERMINING THE MEMORY IN TWO-DIMENSIONAL SYSTEM OF INTEGRO-DIFFERENTIAL MAXWELL’S EQUATIONS Bulletin of the Institute of Mathematics, 2021
