An integral equation method for the electromagnetic scattering from a scatterer on an absorbing plane

Consider a time-harmonic electromagnetic plane wave incident on a scatterer on a grounded absorbing plane modelized as an infinite impedance plane. In this paper, a new integral representation formula is rigorously derived. Existence and uniqueness of weak solutions for the model problem are also established. The proof of existence is based on an extension of the Hodge decomposition technique to open boundaries. The results reported in this paper form a basis for numerical solutions of the electromagnetic scattering problem from a scatterer on an absorbing plane.     

Lie symmetry analysis,optimal system,and generalized group invariant solutions of the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, Lie group theoretic method is used to carry out the similarity reduction and solitary wave solutions of (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation. The equation describes the propagation of nonlinear dispersive waves in inhomogeneous media. Under the invariance property of Lie groups, the infinitesimal generators for the governing equation have been obtained. Thereafter, commutator table, adjoint table, invariant functions, and one-dimensional optimal system of subalgebras are derived by using Lie point symmetries. The symmetry reductions and some group invariant solutions of the DJKM equation are obtained based on some subalgebras. The obtained solutions are new and more general than the rest while known results reported in the literature. In order to show the physical affirmation of the results, the obtained solutions are supplemented through numerical simulation. Thus, the solitary wave, doubly soliton, multi soliton, and dark soliton profiles of the solutions are traced to make this research physically meaningful.     

Dynamic analysis of plane elasticity with new complex Fourier radial basis functions in the dual reciprocity boundary element method

In this paper, dual reciprocity (DR) boundary element method (BEM) is reformulated using new radial basis function (RBF) to approximate the inhomogeneous term of Navier’s differential equation (i.e., inertia term). This new RBF, which is in the form of exp(iωr), is called complex Fourier RBF hereafter. The present RBF has simultaneously collected the properties of Gaussian and real Fourier RBF reported in literature together. Consequently, this promising feature has provided more robustness and potency of the proposed method. The required kernels for displacement and traction particular solutions are derived by employing the method of variation of parameters. As some terms of these kernels are singular, a new simple smoothing trick is employed to resolve the singularity problem. Moreover, the limiting values of relevant kernels are evaluated. The validity, accuracy, and strength of the present formulation are illustrated throughout several numerical examples. The numerical results show that the proposed complex Fourier RBF represents more accurate solutions, using less degree of freedom compared to other RBFs available in the literature.     

Volterra series approximation for rational nonlinear system

Rational nonlinear systems are widely used to model the phenomena in mechanics, biology, physics and engineering. However, there are no exact analytical solutions for rational nonlinear system. Hence, the approximate analytical solutions are good choices as they can give the estimation of the states for system analysis, controller design and reduction. In this paper, an approximate analytical solution for rational nonlinear system is derived in terms of the solution of a polynomial system by Volterra series theory. The rational nonlinear system is transformed to a singular polynomial system with finite terms by adding some algebraic constraints related to the rational terms. The analytical solution of singular polynomial system is approximated by the summation of the solutions of Volterra singular subsystems. Their analytical solutions are derived by a novel regularization algorithm. The first fourth Volterra subsystems are enough to approximate the analytical solution to guarantee the accuracy. Results of numerical experiments are reported to show the effectiveness of the proposed method.     

Differential Operator Solutions of the Bauer-Peschl Equation

In this paper new solutions of the Bauer-Peschl equation represented by differential operators are derived. A relation to the solutions of Bauer is given. Furthermore, it will be said in which new way Bauer's solutions can be obtained. Also three other ways for obtaining the solutions of the Bauer-Peschl equation are sketched.     

On generalized vector equilibrium problems

In this paper, we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for the weak solutions and strong solutions of generalized vector equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.     

New travelling wave solutions to the Ostrovsky equation

In this work, new travelling wave solutions to the Ostrovsky equation are studied by employing the improved tanh function method. With this method, the Ostrovsky equation is reduced to the nonlinear ordinary differential equation and then the different types of exact solutions are derived based on the solutions of the Riccati equation. We will compare our solutions with those gained by the other methods.     

Maximizing a submodular function by integer programming: Polyhedral results for the quadratic case

We give a new mixed integer programming (MIP) formulation for the quadratic cost partition problem that is derived from a MIP formulation for maximizing a submodular function. Several classes of valid inequalities for the convex hull of the feasible solutions are derived using the valid inequalities for the node packing polyhedron. Facet defining conditions and separation algorithms are discussed and computational results are reported.     

Symbolic analysis and exact travelling wave solutions to a new modified Novikov equation

In this paper, a new modified Novikov equation is introduced. Multiple exact travelling wave solutions are obtained by using the extended-tanh function method. Furthermore, based on the homogeneous balance method, an auto-Bäcklund transformation is derived and some solitary wave solutions to this new equation are established.     

一类四阶奇异半正Sturm-Liouville边值问题的正解

在Sturm-Liouville边界条件下研究较广泛的一类四阶奇异半正微分方程,得到其C2[0,1]正解与C3[0,1]正解存在的新结果,并给出了其正解与该边值问题的格林函数之间的某些联系.     

New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations

In this paper we consider a special fifth-order KdV equation with constant coefficients and we obtain traveling wave solutions for it, using the projective Riccati equation method. By mean of a scaling, exact solutions to general Kaup-Kupershmidt (KK) equation are obtained. As a particular case, exact solutions to standard KK equation can be derived. Using the same method, we obtain exact solutions to standard Ito equation. By mean of scaling, new exact solutions to general Ito equation are formally derived.     

Solitons solutions for some nonlinear evolution equations

In this paper, we establish new solitary wave solutions to the modified Kawahara equation by the sine-cosine method. Moreover, the periodic solutions and bell-shaped solitons solutions to the generalized fifth-order KdV equation are obtained. The tanh method is used to handle the double sine-Gordon equation and the double sinh-Gordon equation. Families of exact travelling wave solutions are formally derived. The rational triangle sine-cosine method is introduced and to be constructed complex solutions to the modified Degasperis-Procesi (DP) equation and the modified Camassa-Holm (CH) equation.     

A new integrable equation that combines the KdV equation with the negative‐order KdV equation

In this work, we develop a new integrable equation by combining the KdV equation and the negative‐order KdV equation. We use concurrently the KdV recursion operator and the inverse KdV recursion operator to construct this new integrable equation. We show that this equation nicely passes the Painlevé test. As a result, multiple soliton solutions and other soliton and periodic solutions are guaranteed and formally derived.     

半无限长圆管内Stokes流的入口流

本文研究了半无限长圆管内Stokes流的入口流问题.我们导出了一种新的级数解,它与文献[1,2]的解有一个明显的区别就是该解中不包含无穷积分,因此有利于计算.本文利用配点法进行了计算,得到了满意的结果.     

两个非线性发展方程精确解析解的研究

对齐次平衡法进行了改进并将其应用于两个非线性发展方程中,通过一些新的假设,获得了若干精确解析解,这些解包含王和张的结论及其它新类型的解析解,如果理分式解和周期解,这种方法也可以应用于求解更多的非线性偏微分方程。     

The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation

We obtain new exact solutions to generalized Sawada-Kotera equation. Using the variational iteration method combined with the improved generalized tanh-coth method, we construct new traveling wave solutions for the standard Sawada-Kotera equation and, by means of scaling, we obtain new solutions to general Sawada-Kotera equation. Periodic and soliton solutions are formally derived for both models.     

A new sine-Gordon equation expansion algorithm to investigate some special nonlinear differential equations

A new transformation method is developed using the general sine-Gordon travelling wave reduction equation and a generalized transformation. With the aid of symbolic computation, this method can be used to seek more types of solutions of nonlinear differential equations, which include not only the known solutions derived by some known methods but new solutions. Here we choose the double sine-Gordon equation, the Magma equation and the generalized Pochhammer–Chree (PC) equation to illustrate the method. As a result, many types of new doubly periodic solutions are obtained. Moreover when using the method to these special nonlinear differential equations, some transformations are firstly needed. The method can be also extended to other nonlinear differential equations.     

利用改进的(G′/G)-展开法求广义的(2+1)维Boussinesq方程的精确解

利用改进的(G′/G)-展开法,求广义的(2+1)维Boussinesq方程的精确解,得到了该方程含有较多任意参数的用双曲函数、三角函数和有理函数表示的精确解,当双曲函数表示的行波解中参数取特殊值时,便得到广义的(2+1)维Boussinesq方程的孤立波解.     

On mass transfer in three‐dimensional flow of a viscoelastic fluid

This attempt theoratically examines the mass transfer analysis of the boundary layer flow caused by a stretching surface. Analytic solutions are derived with the Homptopy analysis method (HAM). Numerical values of skin friction coefficients and the surface mass transfer are reported. The effectiveness of this method is explicitly illustrated by discussing the convergence of the obtained series solutions. Velocity and gradient of mass transfer are shown graphically and discussed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 915–936, 2011     

New periodic and soliton solutions for the Generalized BBM and Burgers-BBM equations

In this paper, we consider the Benjamin Bona Mahony equation (BBM), and we obtain new exact solutions for it by using a generalization of the well-known tanh-coth method. New periodic and soliton solutions for the Generalized BBM and Burgers-BBM equations are formally derived.