Application of Analytical Techniques for Solving Fractional Physical Models Arising in Applied Sciences

TLDR

The time‑fractional Kawahara equation models nonlinear water waves and signal transmission in long‑wavelength regimes, and existing methods are straightforward yet effective for solving such fractional‑order problems. The study aims to provide a tool for solving fractional partial differential equations by applying the derived analytical techniques. The authors applied the homotopy perturbation transform method and the Elzaki transform decomposition method, employing Caputo fractional derivatives and Adomian/He’s polynomials to manage nonlinear terms. Numerical results show that the proposed methods converge to exact solutions, accurately capture wave behavior, and significantly reduce computational time.

Abstract

In this paper, we examined the approximations to the time-fractional Kawahara equation and modified Kawahara equation, which model the creation of nonlinear water waves in the long wavelength area and the transmission of signals. We implemented two novel techniques, namely the homotopy perturbation transform method and the Elzaki transform decomposition method. The derivative having fractional-order is taken in Caputo sense. The Adomian and He’s polynomials make it simple to handle the nonlinear terms. To illustrate the adaptability and effectiveness of derivatives with fractional order to represent the water waves in long wavelength regions, numerical data have been given graphically. A key component of the Kawahara equation is the symmetry pattern, and the symmetrical nature of the solution may be observed in the graphs. The importance of our suggested methods is illustrated by the convergence of analytical solutions to the precise solutions. The techniques currently in use are straightforward and effective for solving fractional-order issues. The offered methods reduced computational time is their main advantage. It will be possible to solve fractional partial differential equations using the study’s findings as a tool.