Application of the Homotopy Perturbation Method to Selected Nonlinear and Fractional Differential Equations with Comparative Analysis

In this study, the Homotopy Perturbation Method (HPM) is applied to obtain approximate analytical solutions of selected nonlinear and fractional differential equations. The method is first introduced and then implemented on a nonlinear ordinary differential equation, Burgers’ equation, and a fractional-order logistic model. The solutions are expressed in the form of rapidly convergent series without the need for discretization, linearization, or the presence of small parameters. For the nonlinear ordinary differential equation, the HPM solution is shown to converge to the exact solution with high accuracy. In the case of Burgers’ equation, the method effectively captures both nonlinear convective and diffusive effects, demonstrating its capability in handling nonlinear partial differential equations. Furthermore, the application of HPM to the fractional logistic equation highlights its effectiveness in solving fractional-order systems, where the solutions exhibit memory-dependent dynamics characterized by fractional powers of time. Comparative analysis indicates that HPM provides accurate results with reduced computational effort when compared to other semi-analytical methods. Graphical results further validate the convergence and accuracy of the method. The findings confirm that HPM is a reliable, efficient, and versatile technique for solving a wide range of nonlinear and fractional differential equations.