An Analytical Approach to Solving Fractional Diffusion and Korteweg–de Vries Equations Using the Laplace Homotopy Perturbation Method

In this study, we introduce the Laplace homotopy perturbation method (LHPM), based on Caputo fractional derivatives (CFDs), and demonstrate its application to approximate solutions of the fractional diffusion equation and the Korteweg–de Vries fractional-order equation. This method’s benefit is that it can produce solutions without requiring limiting assumptions or discretisation. The usefulness and strength of the method are illustrated with four examples. The exact and approximate solutions were compared using four tables and 2D and 3D graphs. The results’ robust convergence attests to the suggested method’s dependability and effectiveness in solving fractional partial differential equations (FPDEs). Fractional diffusion equations are used to simulate anomalous transport in biological tissues, groundwater flow, porous media and finance. Nonlinear wave propagation in shallow water, plasmas, optical fibres and fluid dynamics is described by the fractional Korteweg–de Vries equation. There are many applications of the LHPM in solving complex linear and nonlinear problems in science and engineering. It works especially well with FPDEs, which are frequently found in fluid dynamics, heat transfer and viscoelastic materials. Furthermore, LHPM is used in control theory to model dynamic systems, in electrical engineering to analyse circuits with fractional elements and in biological models to describe population dynamics and the spread of disease. It is an effective tool for researching real-world problems where exact solutions are hard to find, as it combines the Laplace transform with perturbation techniques to obtain accurate analytical approximations with rapid convergence.