Magnetohydrodynamic tangent hyperbolic fluid flow past a stretching sheet

In this article, we investigate the MHD tangent hyperbolic fluid flow along a stretching sheet with suction/injection effect at the boundary. The governing nonlinear partial differential equations are transformed into a set of nonlinear ordinary differential equations using the similarity transformation developed by the Lie group analysis. The transformed non-dimensional ordinary differential equations are solved numerically by a shooting technique. The impacts of the governing parameters on the fluid flow and heat transfer characteristics are investigated and discussed, with the help of graphical representations.     

New Exact Travelling Wave and Periodic Solutions of Discrete Nonlinear Schrödinger Equation

Some new exact travelling wave and period solutions of discrete nonlinear Schrödinger equation are found by using a hyperbolic tangent function approach, which was usually presented to find exact travelling wave solutions of certain nonlinear partial differential models. Now we can further extend the new algorithm to other nonlinear differential-different models.     

A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations

For the Noyes-Fields equations, two-dimensional hyperbolic equations of conversation laws, and theBurgers-KdV equation, a class of traveling wave solutions has been obtained by constructing appropriate functiontransformations. The main idea of solving the equations is that nonlinear partial differential equations are changed intosolving algebraic equations. This method has a wide-rangingpracticability.     

非线性LC电路方程的显式精确行波解

理论上考察了具有耗散的非线性LC电路中的行波. 借助于作者最近发展的精确求解非线性偏微分方程的扩展的双曲函数方法解析地研究了模拟非线性电路中冲击波的四阶耗散非线性波动方程. 一致地获得了丰富的显式精确解析行波解, 包括精确冲击波解和奇异的行波解, 和三角函数有理形式的周期波解. 关键词： LC电路')" href="#">非线性LC电路 非线性耗散波动方程 冲击波 周期波     

Jacobi椭圆函数展开法及其在求解非线性波动方程中的应用

给出了Jacobi椭圆函数展开法,且应用该方法获得了几种非线性波方程的准确周期解.该方法包含了双曲函数展开法,应用该方法得到的周期解包含了冲击波解和孤波解. 关键词： Jacobi椭圆函数 非线性方程 周期解 孤波解     

New explicit and exact solutions for a compound KdV-Burgers equation

In this paper, new explicit and exact solutions for a compound KdV-Burgers equation are obtained using the hyperbolic function method and the Wu elimination method, which include new solitary wave solutions and periodic solutions. Particularly important cases of the equation, such as the compound KdV, mKdV-Burgers and mKdV equations can be solved by this method. The method can also be applied to solve other nonlinear partial differential equation and equations.     

Heat transfer analysis for stretching flow over a curved surface with magnetic field

The analysis of a viscous fluid flow and heat transfer is carried out under the influence of a constant applied magnetic field over a curved stretching sheet. Heat transfer analysis is carried out for two heating processes, namely, prescribed surface temperature (PST) and prescribed heat flux (PHF). The equations governing the flow are modeled in a curvilinear coordinate system (r, s, z). The nonlinear partial differential equations are then transformed to nonlinear ordinary differential equations by using similarity transformations. The obtained system of equations is solved numerically by a shooting method using Runge-Kutta algorithm. The interest lies in determining the influence of dimensionless radius of curvature on the velocity, temperature, skin friction, and rate of heat transfer at the wall prescribed by the Nusselt number. The effects of Hartmann number are also presented for the fluid properties of interest.     

New analytical solutions for nonlinear physical models of the coupled Higgs equation and the Maccari system via rational exp <Emphasis Type="BoldItalic">(−φ(η))</Emphasis>-expansion method

In this article, a variety of solitary wave solutions are found for some nonlinear equations. In mathematical physics, we studied two complex systems, the Maccari system and the coupled Higgs field equation. We construct sufficient exact solutions for nonlinear evolution equations. To study travelling wave solutions, we used a fractional complex transform to convert the particular partial differential equation of fractional order into the corresponding partial differential equation and the rational exp (?φ(η))-expansion method is implemented to find exact solutions of nonlinear equation. We find hyperbolic, trigonometric, rational and exponential function solutions using the above equation. The results of various studies show that the suggested method is very effective and can be used as an alternative for finding exact solutions of nonlinear equations in mathematical physics. A comparative study with the other methods gives validity to the technique and shows that the method provides additional solutions. Graphical representations along with the numerical data reinforce the efficacy of the procedure used. The specified idea is very effective, pragmatic for partial differential equations of fractional order and could be protracted to other physical phenomena.     

New structures for closed-form wave solutions for the dynamical equations model related to the ion sound and Langmuir waves

This treatise analyzes the coupled nonlinear system of the model for the ion sound and Langmuir waves.The modified(G'/G)-expansion procedure is utilized to raise new closed-form wave solutions.Those solutions are investigated through hyperbolic,trigonometric and rational function.The graphical design makes the dynamics of the equations noticeable.It provides the mathematical foundation in diverse sectors of underwater acoustics,electromagnetic wave propagation,design of specific optoelectronic devices and physics quantum mechanics.Herein,we concluded that the studied approach is advanced,meaningful and significant in implementing many solutions of several nonlinear partial differential equations occurring in applied sciences.     

Exact Solutions of (2+1)-Dimensional Boiti-Leon-Pempinelle Equation with (G'/G)-Expansion Method

In this paper, we construct exact solutions for the (2+1)-dimensional Boiti-Leon-Pempinelle equation by using the (G'/G)-expansion method, and with the help of Maple. As a result, non-travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. This method can beapplied to other higher-dimensional nonlinear partial differential equations.     

Deriving the New Traveling Wave Solutions for the Nonlinear Dispersive Equation,KdV-ZK Equation and Complex Coupled KdV System Using Extended Simplest Equation Method

In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov-Kuznetsov (KdV-ZK) equation and complex coupled KdV system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations (NLEEs) in mathematical physics.     

Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach

In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.     

Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach

In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.     

Numerical Simulation of MHD Peristaltic Flow with Variable Electrical Conductivity and Joule Dissipation Using Generalized Differential Quadrature Method

In this paper, the MHD peristaltic flow inside wavy walls of an asymmetric channel is investigated, where the walls of the channel are moving with peristaltic wave velocity along the channel length. During this investigation,the electrical conductivity both in Lorentz force and Joule heating is taken to be temperature dependent. Also, the long wavelength and low Reynolds number assumptions are utilized to reduce the governing partial differential equations into a set of coupled nonlinear ordinary differential equations. The new set of obtained equations is then numerically solved using the generalized differential quadrature method(GDQM). This is the first attempt to solve the nonlinear equations arising in the peristaltic flows using this method in combination with the Newton-Raphson technique. Moreover, in order to check the accuracy of the proposed numerical method, our results are compared with the results of built-in Mathematica command NDSolve. Taking Joule heating and viscous dissipation into account, the effects of various parameters appearing in the problem are used to discuss the fluid flow characteristics and heat transfer in the electrically conducting fluids graphically. In presence of variable electrical conductivity, velocity and temperature profiles are highly decreasing in nature when the intensity of the electrical conductivity parameter is strengthened.     

Hyperbolic reduction

The notions of weak Darboux integrability and hyperbolic reduction are introduced, and their potential is gauged as a means of extending the range of application of geometric methods for solving hyperbolic partial differential equations. For directness, our work is expressed in local coordinates and formulated for semilinear hyperbolic systems in two independent variables. The theory is applied to the study of 1+1-wave maps into surfaces of revolution. It is shown that the differential system for any such wave map may be viewed as an integrable extension of a certain scalar, semilinear, hyperbolic partial differential equation which is explicitly constructed. Using this we discover a new integrable wave map system for which hyperbolic reduction leads to a large family of explicit wave maps.     

New applications of the homogeneous balance principle

The homogeneous balance principle has been widely applied to the exploration of nonlinear transformation, exact solutions (especially solitary wave solution), dromion and similarity reduction to the nonlinear partial differential equations in mathematical physics. In this paper, we use the homogeneous balance principle to derive B?cklund transformations for nonlinear partial differential equations that have more nonlinear terms and more highest-order partial derivative terms. With the aid of the B?cklund transformations derived here, we could obtain exact solutions to the nonlinear partial differential equations. The Davey－Stewartson equation and the Nizhnik－Novikov－Veselov equation are considered as the examples.     

A transformed rational function method for (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation

A direct method, called the transformed rational function method, is used to construct more types of exact solutions of nonlinear partial differential equations by introducing new and more general rational functions. To illustrate the validity and advantages of the introduced general rational functions, the (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama (YTSF) equation is considered and new travelling wave solutions are obtained in a uniform way. Some of the obtained solutions, namely exponential function solutions, hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions and rational solutions, contain an explicit linear function of the independent variables involved in the potential YTSF equation. It is shown that the transformed rational function method provides more powerful mathematical tool for solving nonlinear partial differential equations.     

The Application of Extended Tanh-Function Approach in Toda Lattice Equations

In this paper, we generalize the extended tanh-function approach, which used to find new exact travelling wave solutions of nonlinear partial differential equations (NPDES) or coupled nonlinear partial differential equations, to nonlinear differential-difference equations (NDDES). As illustration, we discuss some Toda lattice equations, and solitary wave and periodic wave solutions of these Toda lattice equations are obtained by means of the extended tanh-function approach. PACS numbers: 05.45.Yv, 02.30.Jr, 02.30.Ik.     

New Solutions of Three Nonlinear Space- and Time-Fractional Partial Differential Equations in Mathematical Physics

Motivated by the widely used ansätz method and starting from the modified Riemann-Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.     

A Generalized Method and Exact Solutions in Bose-Einstein Condensates in an Expulsive Parabolic Potential

In the paper, a generalized sub-equation method is presented to construct some exact analytical solutions of nonlinear partial differential equations. Making use of the method, we present rich exact analytical solutions of the onedimensional nonlinear Schrfdinger equation which describes the dynamics of solitons in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential. The solutions obtained include not only non-traveling wave and coefficient function＇s soliton solutions, but also Jacobi elliptic function solutions and Weierstra.ss elliptic function solutions. Some plots are given to demonstrate the properties of some exact solutions under the Feshbachmanaged nonlinear coefficient and the hyperbolic secant function coefficient.