Abstract:
Four-dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time. Moreover, as an application of four-dimensional differential transform, exact solutions of nonlinear system of partial differential equations have been investigated. The results of the present method are compared very well with analytical solution of the system. Differential transform method can easily be applied to linear or nonlinear problems and reduces the size of computational work. With this method, exact solutions may be obtained without any need of cumbersome work, and it is a useful tool for analytical and numerical solutions.

Abstract:
Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSP-RK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies. 1. Introduction In this paper we consider the one-dimensional sine-Gordon equation with initial conditions The Dirichlet boundary conditions are given by The nonlinear sine-Gordon equation arises in many different applications such as propagation of fluxion in Josephson junctions [1], differential geometry, stability of fluid motion, nonlinear physics, and applied sciences [2]. The sine-Gordon equation (1) is a particular case of Klein-Gordon equation, which plays a significant role in many scientific applications such as solid state physics, nonlinear optics and quantum field theory [3], given by where is a nonlinear force and is a constant. In the literature several schemes have been developed for the numerical solution of sine-Gordon equation. Ben-Yu et al. [4] proposed two difference schemes; Bratsos and Twizell [5] used method of lines to transform the initial/boundary value problem associated with (1) into a first order nonlinear initial value problem. Mohebbi and Dehghan [6] presented a combination of a compact finite difference approximation of fourth order and a fourth-order A-stable DIRKN method. Kuang and Lu [7] proposed two classes of finite difference method for generalized sine-Gordon equation; Bratsos and Twizell [8] presented a family of finite difference method, in which time and space derivatives are replaced by finite-difference approximations and then the equation is converted into a linear algebraic system. Wei [9] used the discrete singular convolution algorithm for the integration of (1). A variational iteration method to obtain approximate analytical solution of the sine-Gordon equation without any discretization has been developed by Batiha et al. [10]. Zheng [11] presented a numerical solution of sine-Gordon

Abstract:
We develop a direct method of solution for finding the bright $N$-soliton solution of the Fokas-Lenells derivative nonlinear Schr\"odinger equation. The construction of the solution is performed by means of a purely algebraic procedure using an elementary theory of determinants and does not rely on the inverse scattering transform method. We present two different expressions of the solution both of which are expressed as a ratio of determinants. We then investigate the properties of the solutions and find several new features. Specifically, we derive the formula for the phase shift caused by the collisions of bright solitons.

Abstract:
This study is devoted to studying the (2+1)-dimensional ZK-BBM (Zakharov-Kuznetsov-Benjamin-Bona-Mahony) wave equation in an analytical solution. The analysis is based on the implementation a new method, called Exp-function method. The obtained results from the proposed approximate solution have been verified with those obtained by the extended tanh method. It shows that the obtained results of these methods are the same; while Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear partial differential equations of engineering problems in the terms of accuracy and efficiency.

Abstract:
In a previous study (Matsuno Y, J. Phys. A: Math. Theor. 45(2012)23202), we have developed a systematic method for obtaining the bright soliton solutions of the Fokas-Lenells derivative nonlinear Schr\"odinger equation (FL equation shortly) under vanishing boundary condition. In this paper, we apply the method to the FL equation with nonvanishing boundary condition. In particular, we deal with a more sophisticated problem on the dark soliton solutions with a plane wave boundary condition. We first derive the novel system of bilinear equations which is reduced from the FL equation through a dependent variable transformation and then construct the general dark $N$-soliton solution of the system, where $N$ is an arbitrary positive integer. In the process, a trilinear equation derived from the system of bilinear equations plays an important role. As a byproduct, this equation gives the dark $N$-soliton solution of the derivative nonlinear Schr\"odinger equation on the background of a plane wave. We then investigate the properties of the one-soliton solutions in detail, showing that both the dark and bright solitons appear on the nonzero background which reduce to algebraic solitons in specific limits. Last, we perform the asymptotic analysis of the two- and $N$-soliton solutions for large time and clarify their structure and dynamics.

Abstract:
We present a finite-difference integration algorithm for solution of a system of differential equations containing a diffusion equation with nonlinear terms. The approach is based on Crank-Nicolson method with predictor-corrector algorithm and provides high stability and precision. Using a specific example of short-pulse laser interaction with semiconductors, we give a detailed description of the method and apply it for the solution of the corresponding system of differential equations, one of which is a nonlinear diffusion equation. The calculated dynamics of the energy density and the number density of photoexcited free carriers upon the absorption of laser energy are presented for the irradiated thin silicon film. The energy conservation within 0.2% has been achieved for the time step $10^4$ times larger than that in case of the explicit scheme, for the chosen numerical setup. We also present a few examples of successful application of the method demonstrating its benefits for the theoretical studies of laser-matter interaction problems.

Abstract:
In this paper we study the analytic solutions of Burgers-type nonlinear fractional equations by means of the Invariant Subspace Method. We first study a class of nonlinear equations directly related to the time-fractional Burgers equation. Some generalizations linked to the forced time-fractional Burgers equations and variable-coefficient diffusion are also considered. Finally we study a Burgers-type equation involving both space and time-fractional derivatives.

Abstract:
In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a one-dimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation is proposed. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equation is recovered correctly from the lattice Boltzmann equation. The method is applied on some test examples, and the numerical results have been compared with the analytical solutions or the numerical solutions reported in previous studies. The L_{2}, L_{∞} and Root-Mean-Square (RMS) errors in the solutions show the efficiency of the method computationally.

Abstract:
For the H-nonlinear equation systems produced by stiffnonlinear function f(y): y Rm Rm, the paper presents a new Newton-like iterative solution method:completely-square method, establishes its convergence theory and offersfour simple algorithms for approximate calculation of optimum iterativeparameter in this method. The iterative method do not need tocompute(f)2, and LU-decomposition only need to be done forsome m m matrix. Numerical examples show that if appropriateapproximate optimum iterative parameter is selected on the coefficientsin the hybrid method that products the H-nonlinear equation systems then theiterative solution method in the paper is high efficieney.

Abstract:
Using the perturbation method, a class of nonlinear generalized Landau-Ginzburg-Higgs equations are studied. Firstly, by introducing a varitational iteration, the Lagrange multiplicator is accounted for. Then the iteration of the solution for original equation is constructed and the approximate solution is obtained.