Existence and Uniqueness Analysis of a Nonlinear Time–Fractional Diffusion Equation with Periodic Boundary Conditions

This study addresses the analysis and numerical solution of a nonlinear time fractional diffusion problem with periodic boundary conditions. A generalized Fourier-based approach is applied to obtain a representation of the solution within a suitable Banach space, taking advantage of the periodic nature of the problem. Existence and uniqueness analysis of the solution is performed under the assumption of coordinated convexity instead of the commonly used global Lipschitz condition on the source term. Furthermore, a finite difference scheme based on the L1 approximation is proposed for the numerical solution of the problem. The consistency of the proposed method is verified through numerical experiments. In addition, the effect of fractional order on the solution behavior is investigated, and the results are supported by theoretical findings.