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The present study introduces and explores the novel application of the Fractional Residual Power Series Method as a highly efficient and reliable approach for solving the challenging non-linear, time-fractional Kawahara and modified Kawahara equations. These equations, which are fifth-order, non-linear partial differential equations, are of significant importance in modeling shallow water waves. In contrast to existing methods such as the Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), and Adomian Decomposition Method (ADM), our research uniquely demon- strates the superior performance of the Fractional Residual Power Series Method. We emphasize that our approach not only outperforms these conventional techniques but also offers ease of implementation. This novel methodology promises to advance the field of mathematical modeling, providing a powerful tool for solving complex problems in the field of science and engineering. [ABSTRACT FROM AUTHOR]

In this article, we present a novel hybrid approach, by combining the Sawi transform with the homotopy perturbation method, to achieve the approximate and analytic solutions of nonlinear fractional differential equations (ODE as well as PDE) using the time-fractional Caputo derivative. The proposed algorithm is faster and simple compared to other iterative methods. The Sawi transform is used along with the homotopy perturbation method to accelerate the convergence of the series solution. The results discussed using calculations, graphs and tables are compatible for comparison with other known methods like the residual power series method and the exact solution which are discussed in the literature. [ABSTRACT FROM AUTHOR]