DIRECT PROBLEM FOR THE SPACE-TIME SINGULAR-NONLOCAL DIFFUSION EQUATION
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Abstract
This article investigates the direct problem for the space-time singular-nonlocal diffusion equation. The study focuses on solutions of spectral fractional-linear equations associated with the Legendre differential equation. First, the existence and uniqueness of the solution are established by introducing certain regularity conditions. The method of separation of variables is applied to construct the solutions, which are represented as series in terms of orthogonal Legendre polynomials. Furthermore, the convergence of these series is thoroughly analyzed, along with the smoothness and functional properties of the obtained solutions.
The paper discusses Caputo fractional derivatives, Riemann–Liouville integrals, and operators of the involution type. Based on several theorems and lemmas, eigenvalues and eigenfunctions of the spectral problems are derived. The uniform and absolute convergence of the solutions is proven, and their consistency with the given initial-boundary conditions is demonstrated.
Overall, this work contributes to the theory of fractional-order differential equations by proposing new approaches to solving direct problems and laying the foundation for further applications in inverse problem theory. The results can be applied to complex problems in mathematical physics and serve as a basis for use in applied models. Moreover, the proposed methods provide opportunities for improving the numerical solutions of specific problems.
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