Unveiling Approximate Analytical Solutions for Fractional-Order Partial Differential Equations in Physical Processes

This paper aims to tackle the challenge of deriving accurate analytical solutions for three classes of fractional-order partial differential equations (PDEs)—the Sharma–Tasso–Olver (STO), cubic nonlinear Schrödinger (Sch), and Fokker–Planck (FP) equations, which represent intricate phenomena in fluid dynamics, quantum mechanics, and statistical physics. While these equations are fundamental for comprehending stochastic processes and nonlinear wave propagation, they remain analytically intractable using standard techniques. We use two analytical techniques to bridge this gap: a modified generalized Mittag-Leffler function method (MGMLFM) and the Laplace Adomian decomposition method (LADM). These two methods are applied to the proposed problems, andsolutions are presented in a straightforward manner. Our results demonstrate exceptional agreement with known exact solutions (when α=1), with graphical and tabular comparisons revealing how fractional orders govern solution behavior and wave dispersion patterns. Also, the proposed methods reduce computational complexity compared to existing techniques, as the absolute error resulting from our calculations is very small. The LADM and MGMLFM can be easily employed in many linear and nonlinear problems due to their simplicity, low effort in computations, and proven efficiency from the obtained results.