On the Existence, Uniqueness, and Numerical Solutions of a Time-Fractional Pseudo-Parabolic Equation Under Periodic Boundary Conditions

The present work focuses on both the analytical solvability and numerical treatment of a nonlinear time-fractional pseudo-parabolic equation subject to periodic boundary conditions. Rather than imposing the standard global Lipschitz requirement on the nonlinear source term, which is prevalent in the existing literature, the existence and uniqueness results are established by assuming a novel convexity hypothesis. The periodic structure of the spatial domain is incorporated through a generalized Fourier representation, which transforms the original problem into a system of time-fractional equations for the corresponding Fourier coefficients, permitting the analysis to be carried out in an appropriate Banach space. For the numerical investigation, the Caputo derivative with respect to time is discretized by means of the L1 scheme, while a finite difference approximation is employed for the spatial derivatives. Computational experiments demonstrate that the maximum absolute errors are successfully reduced to the order of 10−6, and the calculated numerical convergence rates closely match the theoretical expectation of O(τ2−α) across different fractional orders. These robust numerical results confirm that the proposed scheme faithfully reflects the qualitative properties derived from the theoretical analysis, validating the effectiveness of the convexity-driven analytical approach.