Exploration of a New Approach Related to Atangana-Baleanu Derivatives for Solving Fractional Partial Differential Equations

This paper explores the application of fractional calculus to solve fractional partial differential equations (FPDEs) using the Sawi transform in combination with the Atangana-Baleanu fractional derivative. The Atangana-Baleanu derivative, formulated in both Caputo and Riemann-Liouville senses, offers a powerful tool for modeling memory and hereditary properties in complex physical systems. We extend the Sawi transform’s operational framework to efficiently handle FPDEs by deriving new properties and convolution theorems relevant to the fractional derivatives. The combination of the Sawi transform with the homotopy perturbation method yields a novel approach, termed the Sawi-Transform-Homotopy Perturbation Method, which facilitates the analytical solution of nonlinear FPDEs. The proposed method was validated using fractional Kolmogorov and Rosenau-Hyman equations, achieving exact solutions in some cases and series solutions with rapid convergence in others. Numerical results demonstrated a reduction in computational complexity by approximately 30% compared to traditional methods, highlighting its efficiency and accuracy. This work underscores the utility of fractional calculus in solving real-world problems and advances analytical techniques for solving FPDEs using modern fractional operators.