A Novel Efficient Approach for Solving Nonlinear Caputo Fractional Differential Equations

Several scientific areas utilize fractional nonlinear partial differential equations (PDEs) to model various phenomena, yet most of these equations lack exact solutions (Ex-Ss). Consequently, techniques for obtaining approximate solutions (App-S), which sometimes yield Ex-Ss, are essential for solving these equations. This study employs a novel technique by combining the residual function and modified fractional power series (FPS) with the Aboodh transform (A-T) to solve various nonlinear problems within the framework of the Caputo derivative. Studies on absolute error (Abs-E), relative error (Rel-E), residual error (Res-E), and recurrence error (Rec-E) validate the accuracy and effectiveness of our approach. We apply the limit principle at infinity to determine the coefficients of the series solution terms. In contrast, other methods, such as variational iteration, homotopy perturbation, and Elzaki Adomian decomposition, rely on integration, while the residual power series method (RPSM) employs differentiation, both of which face challenges in fractional scenarios. Moreover, the efficiency of our approach in solving nonlinear problems without depending on Adomian and He polynomials makes it more effective than various approximate series solution techniques. Our method yields results that are very similar to those obtained from the differential transform, the homotopy perturbation, the analytical computational, and Adomian decomposition methods (ADMs). This demonstrates that our technique is a suitable alternative tool for solving nonlinear models.