Regularized long wave equations and diffusion equation using Tantawy least square method

This paper presents a new semi-analytical method, the Tantawy Least Squares Method (TLSM), to solve fractional-order regularised long wave (RLW) equation, diffusion equations, and Rosenau-Hyman (RH) equation. Compared with the existing semi-analytical approaches, TLSM combines the iterational framework of the Tantawy method with the least squares approximation, greatly improving the reliability and accuracy of the wide range of linear and nonlinear fractional partial differential equations (FPDEs). Integrating Caputo fractional derivative enables the method to encompass the memory and hereditary, which are inherent in presenting complex physical phenomena like anomalous diffusion, nonlinear shallow water dynamics and wave propagation. To demonstrate the effectiveness of its generalizability, TLSM is employed to solve four typical problems, including a traditional fractional linear diffusion equation, fourth-order RLW equation, homogeneous RLW equation and RH equation. Comparative error and graphical analyses indicate superior convergence and accuracy of proposed method as compared to other existing methods. The novelty of the TLSM is its systematic framework where the computational complexity in numerical solution is minimized while still preserving analytical tractability, thus offering a versatile and precise tool of fractional differential modeling applications in science and engineering.