Elsayed M. E. Zayed

The (G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics

I the present paper, we construct the traveling wave solutions involving parameters of the combined Korteweg-de Vries–modified Korteweg-de Vries equation, the reaction-diffusion equation, the compound KdV–Burgers equation, and the generalized shallow water wave equation by using a new approach, namely, the (G′/G)-expansion method, where G=G(ξ) satisfies a second order linear ordinary differential equation. When the parameters take special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions.

Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics Using the Modified Simple Equation Method

The modified simple equation method is employed to construct the exact solutions involving parameters of nonlinear evolution equations via the (1+1)-dimensional modified KdV equation, and the (1+1)-dimensional reaction-diffusion equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

A note on the modified simple equation method applied to Sharma–Tasso–Olver equation

Abstract Jawad et al. have applied the modified simple equation method to find the exact solutions of the nonlinear Fitzhugh–Naguma equation and the nonlinear Sharma-Tasso–Olver equation. The analysis of the Sharma–Tasso–Olver equation obtained by Jawad et al. is based on variant of the modified simple equation method. In this paper, we provide its direct application and obtain new 1- soliton solutions.

Applications of an Extended (′/)-Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics

We construct the traveling wave solutions of the (1+1)-dimensional modified Benjamin-Bona-Mahony equation, the (2+1)-dimensional typical breaking soliton equation, the (1+1)-dimensional classical Boussinesq equations, and the (2+1)-dimensional Broer-Kaup-Kuperschmidt equations by using an extended ( 𝐺  / 𝐺 ) -expansion method, where G satisfies the second-order linear ordinary differential equation. By using this method, new exact solutions involving parameters, expressed by three types of functions which are hyperbolic, trigonometric and rational function solutions, are obtained. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions.

DNA Dynamics Studied Using the Homogeneous Balance Method

We employ the homogeneous balance method to construct the traveling waves of the nonlinear vibrational dynamics modeling of DNA. Some new explicit forms of traveling waves are given. It is shown that this method provides us with a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Strengths and weaknesses of the proposed method are discussed.

On the solitary wave solutions for nonlinear Euler equations

In this article, we study the nonlinear Euler equations with respect to the unknown functions . We use an extended hyperbola function method, tanh-function method and sec q  − tanh q method to find some new solitary wave solutions of the nonlinear Euler equations.

On the rational solitary wave solutions for the nonlinear Hirota–Satsuma coupled KdV system

The objective of this article is to investigate an algebraic method for constructing new rational exact wave soliton solutions in terms of hyperbolic and triangular functions for the generalized nonlinear Hirota–Satsuma coupled KdV systems of partial differential equations using symbolic software like Mathematica or Maple. These studies reveal that the generalized nonlinear Hirota–Satsuma coupled KdV system has a rich variety of solutions.

Exact solutions for the nonlinear Schrödinger equation with variable coefficients using the generalized extended tanh-function, the sine–cosine and the exp-function methods

Abstract In this article we find the exact traveling wave solutions of the generalized nonlinear Schrodinger (GNLS) equation with variable coefficients using three methods via the generalized extended tanh-function method, the sine–cosine method and the exp-function method. The main objective of this article is to compare the efficiency of these methods by delivering the exact traveling wave solutions of the proposed nonlinear equation.

Hearing the shape of a general convex domain

Abstract The spectral function Θ ( t ) = ∑ j = 1 ∞ exp(− tλ j ), where { λ j } j = 1 ∞ are the eigenvalues of the Laplace operator Δ = ∑ i = 1 2 ( ∂ ∂x i ) 2 in the x 1 x 2 -plane, is studied for a general convex domain together with an impedance condition on a part of its boundary and another impedance condition on the remaining part of that boundary.

A modified extended method to find a series of exact solutions for a system of complex coupled KdV equations

In this paper an algebraic method is devised to uniformly construct a series of complete new exact solutions for general nonlinear equations. For illustration, we apply the modified proposed method to revisit a complex coupled KdV system and successfully construct a series of new exact solutions including the soliton solutions and elliptic doubly periodic solutions.