Graphical and numerical validation of the least-square Laplace residual power series method for fractional Fokker-Planck equations: a study of anomalous diffusion

The time-fractional Fokker-Planck equations are used to provide basic modeling of anomalous diffusion and non-Markovian processes in transport that occur in physics, biology and complex systems. Fractional formulations have nonlocal operators that reflect memory effects unlike the classical integer-order derivative, which thus give a more realistic description of subdiffusive phenomena. This work is motivated by such applications and examines the time-fractional Fokker-Planck equation that has been developed in the Caputo sense, making it compatible with initial conditions that have a physical interpretation. In order to derive the approximate solutions, we use the least-square Laplace residual power series method. The suggested method integrates Laplace transform framework and least-square residual approximation to build power series solutions. The accuracy and the efficiency of the method is tested on various problems and the findings are compared to those of the new iterative transform method and homotopy perturbation transform method. The graphical analysis highlights the role of memory effects in anomalous diffusion, showing that the fractional-order parameter strongly influences the dynamical behavior of the solutions. In addition, tabulated comparisons indicate that least-square Laplace residual power series method produces much smaller absolute errors than those obtained in competing methods. These results prove that the least-square Laplace residual power series method is a stable, accurate, and computationally efficient approach to the solution of time-fractional Fokker-Planck equations and help to understand the concept of the use of fractional operators in diffusion modeling better.