The (G′/G)-expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines

Ayad M Shahoot,Khaled A E Alurrfi,Mohamed O M Elmrid,Ali M Almsiri,Abdullah M H Arwiniya

doi:10.1080/16583655.2018.1528663

Ayad M Shahoot, Khaled A E Alurrfi + Show 3 more

Open Access

https://doi.org/10.1080/16583655.2018.1528663

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Abstract

ABSTRACTIn this paper, we apply the (G′/G)-expansion method based on three auxiliary equations, namely, the generalized Riccati equation , the Jacobi elliptic equation and the second order linear ordinary differential equation (ODE) to find many new exact solutions of a nonlinear partial differential equation (PDE) describing the nonlinear low-pass electrical lines. The given nonlinear PDE has been derived and can be reduced to a nonlinear ODE using a simple transformation. Soliton wave solutions, periodic function solutions, rational function solutions and Jacobi elliptic function solutions are obtained. Comparing our new solutions obtained in this paper with the well-known solutions is given. Furthermore, plotting 2D and 3D graphics of the exact solutions is shown.

Highlights

In the recent years, investigations of exact solutions to nonlinear partial differential equation (PDE) play an important role in the study of nonlinear physical phenomena in such as fluid mechanics, hydrodynamics, optics, plasma physics, solid state physics, biology and so on
Investigations of exact solutions to nonlinear PDEs play an important role in the study of nonlinear physical phenomena in such as fluid mechanics, hydrodynamics, optics, plasma physics, solid state physics, biology and so on
With reference to solving Equation (2.6) [20], we deduce that the Jacobi elliptic functions solutions and other exact solutions of Equation (1.1) as follows: Case 1

Summary

Introduction

Investigations of exact solutions to nonlinear PDEs play an important role in the study of nonlinear physical phenomena in such as fluid mechanics, hydrodynamics, optics, plasma physics, solid state physics, biology and so on. -expansion method of Section 2 to find new exact solutions of Equation (1.1). With reference to solving Equation (2.7) [19], we deduce that the exact solutions of Equation (1.1) as follows: V1(ξ ). 216α2β (2α 2 (p2 −4qr)−9β p2 ) t, 2α2δ2(p2 − 4qr) < 9βp and β > 0 In this result, we deduce that the exact solutions of Equation (1.1) as follows:. ) t, With reference to solving Equation (2.6) [20], we deduce that the Jacobi elliptic functions solutions and other exact solutions of Equation (1.1) as follows: Case 1. Substituting (3.3) along with the second order linear ODE (2.7) into Equation (3.2) and collecting all the coefficients of (i = 0, 1, 2, 3) and setting them to be zero, we have the following algebraic equations: Uβ a31.

Physical explanations of some results

Conclusions