Efficient coupling technique for approximate and exact solutions of nonlinear Caputo fractional differential equations

Abstract

Complex phenomena such as anomalous diffusion, fractal structures, viscoelasticity, and chaotic dynamics are effectively described using fractional-order nonlinear models. Solving these nonlinear fractional models provides deeper insight into the mechanisms and behaviors underlying such processes. In this work, we develop a hybrid analytical technique for solving nonlinear fractional partial differential equations by coupling the residual function with the Laplace transform, referred to as the Laplace residual power series method (LRPSM). This method is constructed upon a modified fractional-order power series formulated within the framework of the Caputo derivative. The accuracy of the proposed approach is confirmed through absolute, relative, and residual error analyses. LRPSM efficiently determines the coefficients of series solutions using a simple limit principle at infinity, in contrast to well-established methods such as the variational iteration method, Adomian decomposition method, and homotopy perturbation method, which typically require complex integrations. Likewise, the classical residual power series method (RPSM) depends on higher-order derivatives, which become increasingly challenging in fractional settings. Since LRPSM avoids the use of Adomian's or He's polynomials for handling nonlinearities, it proves to be a more straightforward and computationally efficient alternative for obtaining accurate series solutions to nonlinear fractional problems.