A Novel Homotopy Perturbation Sumudu Transform Method for Nonlinear Fractional PDEs: Applications and Comparative Analysis

TL;DR

This paper presents a new hybrid method combining Sumudu transform and homotopy perturbation to efficiently solve nonlinear fractional PDEs, demonstrating high accuracy and faster convergence compared to existing methods.

Contribution

The introduction of the Homotopy Perturbation Sumudu Transform Method (HPSTM), a novel hybrid approach for solving nonlinear fractional PDEs with improved convergence and accuracy.

Findings

Achieves low absolute errors (~3.12e-3) for fractional order 0.9.

Faster computational times (~0.5 seconds per example).

Validated against multiple established numerical methods.

Abstract

This study introduces the Homotopy Perturbation Sumudu Transform Method (HPSTM), a novel hybrid approach combining the Sumudu transform with homotopy perturbation to solve nonlinear fractional partial differential equations (FPDEs), including fractional porous medium, heat transfer, and Fisher equations, using the Caputo fractional derivative. HPSTM leverages the linearity-preserving properties of the Sumudu transform and the flexibility of homotopy perturbation, achieving faster convergence than Laplace-HPM or Elzaki-HPM for strongly nonlinear FPDEs. Series solutions yield absolute errors as low as $3.12 \times 10^{-3}$ for $\alpha = 0.9$, with computational times averaging 0.5 seconds per example using 5 series terms on standard hardware. Solutions are validated against exact solutions, Adomian Decomposition Method (ADM), radial basis function (RBF) meshless method, Variational Iteration Method (VIM), Finite Difference Method (FDM), and a spectral method. Numerical examples, sensitivity analysis, and graphical representations for $\alpha = 1.0, 0.9, 0.8, 0.7$ confirm HPSTM's accuracy, efficiency, and robustness. Limitations include challenges with high-order nonlinearities and multi-dimensional domains. HPSTM shows promise for applications in modeling fluid flow in porous media, heat conduction in complex materials, and biological population dynamics.