A Linearized Finite Difference Scheme for Two‐Dimensional Time‐Fractional Pseudo‐Parabolic Equation on Variable Time Steps

ABSTRACT This paper presents an efficient approach for solving a two‐dimensional time‐fractional pseudo‐parabolic equation using a linearized finite difference scheme on variable time steps. The equation incorporates fractional derivatives in temporal dimensions to accurately model anomalous diffusion and memory effects in diverse physical phenomena. The study focuses on the challenges associated with solving multidimensional nonlinear time‐fractional pseudo‐parabolic equations, emphasizing the need for advanced numerical techniques to overcome computational complexities. Considering the localized characteristics of the fractional derivative and the memory property of fractional differential equations, the proposed method offers an efficient and accurate solution strategy tailored for time‐fractional pseudo‐parabolic equations. Building on previous research on nonuniform finite difference schemes for time‐fractional diffusion problems, this study highlights the effectiveness of nonuniform time‐stepping methods in achieving higher convergence orders in discretization schemes. The results demonstrate the utility of the proposed approach in accurately capturing the dynamics of complex transport processes in various physical systems. We rigorously prove that the proposed method has a convergence rate of order . In addition, several numerical simulations are presented to verify the efficiency of the proposed method. The numerical results confirm the theoretical analysis.