A higher-order L1-2-3 tension spline scheme for nonlinear two-dimensional time-fractional reaction–diffusion equation

Abstract

The paper presents the design and analysis of a high-order numerical scheme for nonlinear two-dimensional time-fractional reaction–diffusion equations. The Caputo time-fractional derivative is approximated using the L1-2-3 scheme, while the spatial derivatives are discretized by means of tension spline difference scheme. The existence, uniqueness, and regularity of the solution are established to support the theoretical analysis. The truncation error, stability, and convergence of the proposed method are rigorously investigated. Two test problems are considered to validate the accuracy and efficiency of the method. For sufficiently smooth solutions, the proposed method is shown to be unconditionally stable and to achieve fourth-order spatial accuracy and ( 4 − α ) -order temporal accuracy. The numerical results confirm the theoretical convergence rates and demonstrates that the proposed method provides higher accuracy, improved convergence behavior, and competitive computational efficiency compared with existing numerical methods. These results indicate that the proposed method is an effective and reliable approach for solving nonlinear two-dimensional time-fractional reaction–diffusion equations with sufficiently smooth solutions arising in models with memory and hereditary effects.