Truncated series solutions to the(2+1)-dimensional perturbed Boussinesq equation by using the approximate symmetry method

In this paper, the（2＋1）-dimensional perturbed Boussinesq equation is transformed into a series of two-dimensional（2 D） similarity reduction equations by using the approximate symmetry method. A step-by-step procedure is used to acquire Jacobi elliptic function solutions to these similarity equations, which generate the truncated series solutions to the original perturbed Boussinesq equation. Aside from some singular area, the series solutions are convergent when the perturbation parameter is diminished.

Infinite series symmetry reduction solutions to the modified KdV-Burgers equation

From the point of view of approximate symmetry, the modified Korteweg-de Vries-Burgers （mKdV-Burgers） equation with weak dissipation is investigated. The symmetry of a system of the corresponding partial differential equations which approximate the perturbed mKdV-Burgers equation is constructed and the corresponding general approximate symmetry reduction is derived; thereby infinite series solutions and general formulae can be obtained. The obtained result shows that the zero-order similarity solution to the mKdV-Burgers equation satisfies the Painleve II equation. Also, at the level of travelling wave reduction, the general solution formulae are given for any travelling wave solution of an unperturbed mKdV equation. As an illustrative example, when the zero-order tanh profile solution is chosen as an initial approximate solution, physically approximate similarity solutions are obtained recursively under the appropriate choice of parameters occurring during computation.

Approximate Symmetries and Infinite Series Symmetry Reduction Solutions to Perturbed Kuramoto-Sivashinsky Equation

Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto-Sivashinsky （KS） equation. The approximate symmetry reduction and the infinite series symmetry reduction solutions of the perturbed KS equation are constructed. Specially, if selecting the tanh-type travelling wave solution as initial approximate, we not only obtain the general formula of the physical approximate similarity solutions, but also obtain several new explicit solutions of the given equation, which are first reported here.

GERBER TECHNOLOGY LAUNCHES FIRST OF NEW SERIES OF CUTTING SOLUTIONS

Precision, Productivity and Performance def ine the GERBERcutter? Z7Tolland, Conn., USA – Gerber Technology, a business unit of Gerber Scientific, Inc. (NYSE: GRB), and the world leader in providing innovative integrated software and hardware automation systems to

Approximate Homotopy Symmetry Reduction Method:Infinite Series Reductions to Kawahara Equation

The Kawahara equation is studied through the approximate homotopy symmetry method. Under this method we get the similarity reduction solutions of the Kawahara equation, leading to the corresponding homotopy series solutions. Furthermore, the similarity solutions of the corresponding reduced linear ordinary differential equations are also considered.

A series solution for surface motion amplification due to underground twin tunnels:incident SV waves

A series solution of displacement response of the ground surface in the presence of underground twin tunnels subjected to excitation of incident plane SV waves is derived by using Fourier-Bessel series expansion method.The numerical parametric study shows that underground twin tunnels significantly amplify the nearby surface ground motion.It is suggested that the effect of subways on ground motion should be considered when the subways are planned and designed.

AN IMPLICIT SERIES PRECISE INTEGRATION ALGORITHM FOR STRUCTURAL NONLINEAR DYNAMIC EQUATIONS

Nonlinear dynamic equations can be solved accurately using a precise integration method. Some algorithms exist, but the inversion of a matrix must be calculated for these al- gorithms. If the inversion of the matrix doesn’t exist or isn’t stable, the precision and stability of the algorithms will be afected. An explicit series solution of the state equation has been pre- sented. The solution avoids calculating the inversion of a matrix and its precision can be easily controlled. In this paper, an implicit series solution of nonlinear dynamic equations is presented. The algorithm is more precise and stable than the explicit series solution and isn’t sensitive to the time-step. Finally, a numerical example is presented to demonstrate the efectiveness of the algorithm.

A series solution for surface motion amplification due to underground group cavities:Incident P waves

A series solution for surface motion amplification due to underground group cavities for incident plane P waves is derived by Fourier-Bessel series expansion method. It is shown that underground group cavities significantly am-plify the surface ground motion nearby. It is suggested that the effect of subways on ground motion should be con-sidered when the subways are planned and designed.

Approximate Solutions of Primary Resonance for Forced Duffing Equation by Means of the Homotopy Analysis Method

Nonlinear dynamic equation is a common engineering model.There is not precise analytical solution for most of nonlinear differential equations.These nonlinear differential equations should be solved by using approximate methods.Classical perturbation methods such as LP method,KBM method,multi-scale method and the averaging method on weakly nonlinear vibration system is effective,while the strongly nonlinear system is difficult to apply.Approximate solutions of primary resonance for forced Duffing equation is investigated by means of homotopy analysis method （HAM）.Different from other approximate computational method,the HAM is totally independent of small physical parameters,and thus is suitable for most nonlinear problems.The HAM provides a great freedom to choose base functions of solution series,so that a nonlinear problem may be approximated more effectively.The HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter and the auxiliary function.Therefore,HAM not only may solve the weakly non-linear problems but also may be suitable for the strong non-linear problem.Through the approximate solution of forced Duffing equation with cubic non-linearity,the HAM and fourth order Runge-Kutta method of numerical solution were compared,the results show that the HAM not only can solve the steady state solution,but also can calculate the unsteady state solution,and has the good computational accuracy.

The series solution to the metric of stationary vacuum with axisymmetry

The multipole moment method not only conduces to the understanding of the deformation of the space-time, but also serves as an effective tool to approximately solve the Einstein field equation with. However, the usual multipole moments are recursively determined by a sequence of symmetric and trace-free tensors, which is inconvenient for practical resolution. In this paper, we develop a simplified procedure to generate the series solutions to the metric of the stationary vacuum with axisymmetry, and show its validity. In order to understand the free parameters in the solution, we propose to take the Schwarzschild metric as a standard ruler, and some well- known examples are analysed and compared with the series solutions in detail.

A Series Solution Approach to the Circular Restricted Gravitational Three-Body Dynamical Problem

The present manuscript examines the circular restricted gravitational three-body problem (CRGTBP) by the introduction of a new approach through the power series method. In addition, certain computational algorithms with the aid of Mathematica software are specifically designed for the problem. The algorithms or rather mathematical modules are established to determine the velocity and position of the third body’s motion. In fact, the modules led to accurate results and thus proved the new approach to be efficient.

Series Solution for Localization and Entanglement of an Exciton in a Quantum Dot Molecule by an ac Electric Field

The Schroedinger equation involving the phenomenon of the localization and entanglement for an exciton in a quantum dot molecule by an ac electric field is analytically investigated. New exact series solutions for the Schroedinger equation have been obtained for the first time. The analytical expressions can further describe the dynamical behaviors of an interacting electron-hole pair in a double coupled quantum dot molecule under an ac electric field accurately.

Approximate direct reduction method:infinite series reductions to the perturbed mKdV equation

The approximate direct reduction method is applied to the perturbed mKdV equation with weak fourth order dispersion and weak dissipation. The similarity reduction solutions of different orders conform to formal coherence, accounting for infinite series reduction solutions to the original equation and general formulas of similarity reduction equations. Painleve Ⅱ type equations, hyperbolic secant and Jacobi elliptic function solutions are obtained for zeroorder similarity reduction equations. Higher order similarity reduction equations are linear variable coefficient ordinary differential equations.

Analysis of Formal and Analytic Solutions for Singularities of the Vector Fractional Differential Equations

In this article, we study on the existence of solution for a singularities of a system of nonlinear fractional differential equations （FDE）. We construct a formal power series solution for our considering FDE and prove convergence of formal so- lutions under conditions. -We use the Caputo fractional differential operator and the nonlinearity depends on the fractional derivative of an unknown function.

Adomian Modification Methods for the Solution of Chebyshev’s Differential Equations

The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of the employed methods by acquiring exact analytical solutions for the governing equations in most cases;while minimal noisy error terms have been observed in a particular method modification. Above all, the presented approaches have rightly affirmed the exactitude of the available literature. More to the point, the application of this methodology could be extended to examine various forms of high-order differential equations, as approximate exact solutions are rapidly attained with less computation stress.

Nonlinear oscillations with parametric excitation solved by homotopy analysis method

An analytical technique, namely the homotopy analysis method （HAM）, is used to solve problems of nonlinear oscillations with parametric excitation. Unlike perturbation methods, HAM is not dependent on any small physical parameters at all, and thus valid for both weakly and strongly nonlinear problems. In addition, HAM is different from all other analytic techniques in providing a simple way to adjust and control convergence region of the series solution by means of an auxiliary parameter h. In the present paper, a periodic analytic approximations for nonlinear oscillations with parametric excitation are obtained by using HAM, and the results are validated by numerical simulations.

Approximate homotopy similarity reduction for the generalized Kawahara equation via Lie symmetry method and direct method

This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method. Using both methods it obtains the similarity reduction solutions and similarity reduction equations of different orders, showing that the approximate homotopy direct method yields more general approximate similarity reductions than the approximate homotopy symmetry method. The homotopy series solutions to the generalized Kawahara equation are consequently derived.

Symmetries and conservation laws associated with a hyperbolic mean curvature flow

Under investigation in this paper is a hyperbolic mean curvature flow for convex evolving curves.Firstly,in view of Lie group analysis,infinitesimal generators,symmetry groups and an optimal system of symmetries of the considered hyperbolic mean curvature flow are presented.At the same time,some group invariant solutions are computed through reduced equations.In particular,we construct explicit solutions by applying the power series method.Furthermore,the convergence of the solutions of power series is certificated.Finally,conservation laws of the hyperbolic mean curvature flow are established via Ibragimov's approach.

Newtonian heating in stagnation point flow of Burgers fluid

The Newtonian heating effects in the stagnation point flow of a Burgers fluid are addressed in this paper. The boundary layer flow problems are stated in the spatial domain from zero to infinity. The solution expressions for the velocity and the temperature are obtained and examined for the influential variables. The tabulated values show comparison with the previous results. It is observed that the obtained results are in good agreement with the existing results in limiting sense.

CTL Response Suppression in Chronic Phase of Infection:A mathematical study

Human immunodeficiency virus(HIV)has had an insightful impact about the state of healthiness of human immune system.Due to great improvement in drug therapy,HIV infections have been reduced by 17%over the past eight years.It has been proved that most effective treatment HAART(Highly Active Anti Retroviral Therapy)mainly controls the diseases progression but it does not eradicate the diseases completely.Reverse Transcriptase Inhibitor drugs specially associated with virus specific Cytotoxic T-Lymphocyte(CTL)that declines with disease progression. CTL responses against AIDS pathogenesis could be potential in the dynamics of virus replication, recognition and clearance of infected cells.In this research article a mathematical model has been proposed on the basis of CTL response suppression in the chronic phase of infection due to presence of virus.We also consider the growth of the virus population from the infected CD4^+T cells budding process and from the other infected cells like macrophages and thymocytes.Our analytical and numerical studies are consistent with existing observations from allied areas.