Non-integrable variants of Boussinesq equation with two solitons

Three variants of the Boussinesq equation, namely, the (2 + 1)-dimensional Boussinesq equation, the (3 + 1)-dimensional Boussinesq equation, and the sixth-order Boussinesq equation are studied. The Hirota bilinear method is used to construct two soliton solutions for each equation. The study highlights the fact that these equations are non-integrable and do not admit N-soliton solutions although these equations can be put in bilinear forms.     

A study on KdV and Gardner equations with time-dependent coefficients and forcing terms

In this work we study the KdV equation and the Gardner equation with time-dependent coefficients and forcing term for each equation. A generalized wave transformation is used to convert each equation to a homogeneous equation. The soliton ansatz will be applied to the homogeneous equations to obtain soliton solutions.     

Multiple-front solutions for the Burgers equation and the coupled Burgers equations

In this work, multiple-front solutions for the Burgers equation and the coupled Burgers equations are examined. The tanh–coth method and the Cole–Hopf transformation are used. The work highlights the power of the proposed schemes and the structures of the obtained multiple-front solutions.     

The decomposition method applied to systems of partial differential equations and to the reaction–diffusion Brusselator model

In this work, systems of linear and nonlinear partial differential equations and the reaction–diffusion Brusselator model are handled by applying the decomposition method. The advantage of this work is twofold. Firstly, the decomposition method reduces the computational work. Secondly, in comparison with existing techniques, the decomposition method is an improvement with regard to its accuracy and rapid convergence. The decomposition method has the advantage of being more concise for analytical and numerical purposes.     

A study on two extensions of the Bogoyavlenskii-Schieff equation

In this work, two extensions of the Bogoyavlenskii-Schieff equation are examined. N-soliton solutions are formally determined for each extended equation. We show that the extension terms do not kill the integrability of the typical Bogoyavlenskii-Schieff equation. The simplified Hirota’s method established by Hereman and Nuseir is applied to achieve this goal.     

Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method

In this paper, we consider the coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. First, we utilize systems of Volterra integral forms of the Lane–Emden equations and derive the modified recursion scheme for the components of the decomposition series solutions. The numerical results display that the Adomian decomposition method gives reliable algorithm for analytic approximate solutions of these systems. The error analysis of the sequence of the analytic approximate solutions can be performed by using the error remainder functions and the maximal error remainder parameters, which demonstrate an approximate exponential rate of convergence.     

A variational approach for a class of nonlocal elliptic boundary value problems

The aim of this paper is to apply the variational iteration method to a class of nonlinear, nonlocal, elliptic boundary value problems. The uniform convergence of the scheme is presented and the work is illustrated by considering a number of test examples that confirm the accuracy and efficacy of the iterative process. The computational results show that the scheme is reliable, converges fast and compares very well with the existing analytic solutions.     

Exact solutions for variable coefficients fourth-order parabolic partial differential equations in higher-dimensional spaces

In this paper analytic solution of variable coefficient fourth-order parabolic partial differential equation in two and three space variables are developed. The calculations are accelerated by using the noise terms phenomenon for nonhomogeneous problems. Numerical examples are investigated to illustrate the pertinent features of the proposed algorithm.     

Perturbation,symmetry analysis,Bäcklund and reciprocal transformation for the extended Boussinesq equation in fluid mechanics

In this work, we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics, scientific fields, and ocean engineering. This equation will be reduced to the Korteweg–de Vries equation via using the perturbation analysis. We derive the corresponding vectors, symmetry reduction and explicit solutions for this equation. We readily obtain B?cklund transformation associated with truncated Painlevéexpansion. We also examine the related conservation laws of this equation via using the multiplier method. Moreover, we investigate the reciprocal B?cklund transformations of the derived conservation laws for the first time.