Numerical Investigation of the Time-Fractional Whitham–Broer–Kaup Equation Involving without Singular Kernel Operators

Kamsing Nonlaopon,Muhammad Naeem,Ahmed Alsanad,Abdu Gumaei,Rasool Shah,Ahmed M Zidan,Muhammad Imran Asjad

doi:10.1155/2021/7979365

Abstract

This paper aims to implement an analytical method, known as the Laplace homotopy perturbation transform technique, for the result of fractional-order Whitham–Broer–Kaup equations. The technique is a mixture of the Laplace transformation and homotopy perturbation technique. Fractional derivatives with Mittag-Leffler and exponential laws in sense of Caputo are considered. Moreover, this paper aims to show the Whitham–Broer–Kaup equations with both derivatives to see their difference in a real-world problem. The efficiency of both operators is confirmed by the outcome of the actual results of the Whitham–Broer–Kaup equations. Some problems have been presented to compare the solutions achieved with both fractional-order derivatives.

Highlights

In engineering and applied sciences and technology, fractional partial differential equations (FPDEs) containing nonlinearities define many phenomena, ranging from gravitation to dynamics. e nonlinear FPDEs are significant tools analyzed to model nonlinear dynamical behaviour in many areas such as plasma physics, mathematical biology, fluid dynamics, and solid-state physics. e widely held dynamical schemes can be denoted by an appropriate set of FPDEs
Whitham–Broer–Kaup equations are used by other scholars who implemented several numerical techniques, such as residual power series technique [4], reduced differential transformation technique [21], Adomian decomposition technique [7], homotopy perturbation technique [22, 23], Lie symmetry analysis [24, 25], exp-function technique [26], G′/G-expansion technique [27], and homotopy analysis technique [28]
The LHPTM was considered to achieve an analytical result for the fractional-order Whitham– Broer–Kaup equations considering the Caputo–Fabrizio and Atangana–Baleanu operators

Summary

Introduction

In engineering and applied sciences and technology, fractional partial differential equations (FPDEs) containing nonlinearities define many phenomena, ranging from gravitation to dynamics. e nonlinear FPDEs are significant tools analyzed to model nonlinear dynamical behaviour in many areas such as plasma physics, mathematical biology, fluid dynamics, and solid-state physics. e widely held dynamical schemes can be denoted by an appropriate set of FPDEs. Amjad et al [19] applied the solution of standard order coupled with fractional-order Whitham–Broer–Kaup equation by Laplace decomposition technique. Noor et al [20] used the homotopy perturbation technique to investigate the results of much classical order of PDEs. Whitham–Broer–Kaup equations are used by other scholars who implemented several numerical techniques, such as residual power series technique [4], reduced differential transformation technique [21], Adomian decomposition technique [7], homotopy perturbation technique [22, 23], Lie symmetry analysis [24, 25], exp-function technique [26], G′/G-expansion technique [27], and homotopy analysis technique [28]. We solved the fractional-order Whitham–Broer–Kaup equations in the Atangana–Baleanu and Caputo–Fabrizio senses using the LHPM

Implementation of Caputo–Fabrizio Operator

Conclusions