Exact Traveling and Nano-Solitons Wave Solitons of the Ionic Waves Propagating along Microtubules in Living Cells

Abdel-Haleem Abdel-Aty,Hichem Eleuch,Raghda A. M. Attia,Mostafa M. A. Khater

doi:10.3390/math8050697

Abstract

In this paper, the weakly nonlinear shallow-water wave model is mathematically investigated by applying the modified Riccati-expansion method and Adomian decomposition method. This model is used to describe the propagation of waves in weakly nonlinear and dispersive media. We obtain exact and solitary wave solutions of this model by using the modified Riccati-expansion method then using these solutions to determine the boundary and initial conditions. These conditions are employed to evaluate the semi-analytical wave solutions and calculate the absolute value of error. The values of absolute error show the accuracy of the obtained solutions. Some solutions are sketched to show the perspective view of the solution of this model. Moreover, the novelty of the obtained solutions is illustrated by showing the similarity and differences between our and previous solutions of the model.

Highlights

The modified Riccati expansion method and Adomian decomposition method were applied to the weakly nonlinear shallow water wave equation for constructing the exact traveling and semi-analytical wave solutions
All our obtained solutions are different from that obtained in References [40,44] where the authors of References [40,44] applied different methods to solve the weakly nonlinear shallow water wave regime
2α, α = 3 δ − 4 σ $, 4 b = 4 σ $ − δ that obtained in this paper where the Khater method was employed to find the exact traveling and solitary wave solutions

Summary

Introduction

The mathematical formulation of this phenomenon depends on real experiments to determine the parameters and empirical functions. These obtained mathematical formulas help us to understand these complex phenomena by explaining their physical and dynamical behavior. Many mathematical and physical researchers have paid attention to solve these nonlinear evolution equations. These works have been deriving many analytical and numerical mathematical schemes to construct analytical, semi-analytical, approximate solutions. These schemes are based on three mainly channels, namely

Objectives

Results

Conclusion