改进同伦分析方法及推广Kuramoto Sivashinsky方程的近似解

首先介绍了带有两个辅助参数的改进同伦分析方法,然后用该方法得到了推广Kuramoto-Sivashin-sky方程的同伦近似解.所得近似解与精确孤立波解进行比较,发现本文得到的近似解更有效地逼近真实解.因为该解包含了两个辅助参数,所以能够更有效地调节和控制其收敛区域和速度.研究表明带有两个辅助参数的改进同伦分析方法对复杂非线性系统的研究更有它的优点.     

New exact solutions of the (n+1)-dimensional sine-Gordon equation using double elliptic equation method

In this paper, the (n+1)-dimensional sine-Gordon equation is studied using double elliptic equation method. With the aid of Maple, more exact solutions expressed by Jacobi elliptic functions are obtained. When the modulus m of Jacobi elliptic function is driven to the limit 1 and 0, some exact solutions expressed by hyperbolic function solutions and trigonometric functions can also be obtained, respectively.     

Application of homotopy perturbation method and homotopy analysis method to fractional vibration equation

In this paper, the approximate analytical solutions of the mathematical model of vibration equation with fractional-order time derivative β (1<β≤2) for very large membranes are obtained with the help of powerful mathematical tools like homotopy perturbation method and homotopy analysis method. Both the methods perform extremely well in terms of efficiency and simplicity. The validity and applicability of the techniques are shown for obtaining approximate numerical solutions for different particular cases which are presented through figures and tables.     

New analytical method for solving Burgers' and nonlinear heat transfer equations and comparison with HAM

In this study, we present a numerical comparison between the differential transform method (DTM) and the homotopy analysis method (HAM) for solving Burgers' and nonlinear heat transfer problems. The first differential equation is the Burgers' equation serves as a useful model for many interesting problems in applied mathematics. The second one is the modeling equation of a straight fin with a temperature dependent thermal conductivity. In order to show the effectiveness of the DTM, the results obtained from the DTM is compared with available solutions obtained using the HAM [M.M. Rashidi, G. Domairry, S. Dinarvand, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 708-717; G. Domairry, M. Fazeli, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 489-499] and whit exact solutions. The method can easily be applied to many linear and nonlinear problems. It illustrates the validity and the great potential of the differential transform method in solving nonlinear partial differential equations. The obtained results reveal that the technique introduced here is very effective and convenient for solving nonlinear partial differential equations and nonlinear ordinary differential equations that we are found to be in good agreement with the exact solutions.     

An analytic algorithm of Lane-Emden type equations arising in astrophysics using modified Homotopy analysis method

Lane-Emden type equation models many phenomena in mathematical physics and astrophysics. It is a nonlinear differential equation which describes the equilibrium density distribution in self-gravitating sphere of polytropic isothermal gas, has a singularity at the origin, and is of fundamental importance in the field of stellar structure, radiative cooling, modeling of clusters of galaxies. An efficient analytic algorithm is provided for Lane-Emden type equations using modified homotopy analysis method, which is different from other analytic techniques as it itself provides us with a convenient way to adjust convergence regions even without Pade technique. Some examples are given to show its validity.     

Homotopy analysis method for higher-order fractional integro-differential equations

In this paper, we present the homotopy analysis method (shortly HAM) for obtaining the numerical solutions of higher-order fractional integro-differential equations with boundary conditions. The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary conditions. The comparison of the results obtained by the HAM with the exact solutions is made, the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of series solution.     

Spectral regularization method for the time fractional inverse advection-dispersion equation

In this paper, we consider the time fractional inverse advection-dispersion problem (TFIADP) in a quarter plane. The solute concentration and dispersion flux are sought from a measured concentration history at a fixed location inside the body. Such problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α(0 < α < 1). We show that the TFIADP is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

Propagation of non-linear waves in a non-ideal relaxing gas

We have derived an evolution equation governing the far-field behaviour of small amplitude waves in a non-ideal relaxing gas for planar and converging flow. Asymptotic expansions of the flow variables for small amplitude waves have been used to derive the evolution equation. This equation turns out to be a generalized Burger's equation. The numerical solution of this equation is obtained by using the homotopy analysis method (HAM) proposed by Liao with two different initial conditions. Using the HAM, we have studied the effect of relaxation and nonlinearity. The convergence control parameter enables us to find a good approximate solution for such a complex flow problem. This method also confirms the capabilities and usefulness of convergence control parameter and HAM for complex and highly non-linear problems.     

Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science

In this paper, we suggest a fractional functional for the variational iteration method to solve the linear and nonlinear fractional order partial differential equations with fractional order initial and boundary conditions by using the modified Riemann-Liouville fractional derivative proposed by G. Jumarie. Fractional order Lagrange multiplier has been considered. Solution has been plotted for different values of α.     

Richardson extrapolation applied to boundary element method results in a Dirichlet problem for the Laplace equation

Richardson extrapolation is used to improve the accuracy of the numerical solutions for the normal boundary flux and for the interior potential resulting from the boundary element method. The boundary integral equations arise from a direct boundary integral formulation for solving a Dirichlet problem for the Laplace equation. The Richardson extrapolation is used in two different applications: (i) to improve the accuracy of the collocation solution for the normal boundary flux and, separately, (ii) to improve the solution for the potential in the domain interior. The main innovative aspects of this work are that the orders of dominant error terms are estimated numerically, and that these estimates are then used to develop an a posteriori technique that predicts if the Richardson extrapolation results for applications (i) and (ii) are reliable. Numerical results from test problems are presented to demonstrate the technique.     

A regularization method for a Cauchy problem of Laplace's equation in an annular domain

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace's equation in an annular domain. Based on a conditional stability for the Cauchy problem of Laplace's equation, we obtain a convergence estimate under the suitable choice of a regularization parameter and an a-priori bound assumption on the solution. A numerical example is provided to show the effectiveness of the proposed method from both accuracy and stability.     

Construction of solitary wave solutions for variants of the K(n,n) equation

In this paper, variational iteration method and Adomian decomposition are implemented to construct solitary solutions for variants of K(n, n) equation. In these schemes the solution takes the form of a convergent series with easily computable components. The chosen initial solution or trial function plays a major role in changing the physical structure of the solution. Comparison between the two methods is made and many models are approached. The obtained results reveal that the two methods are very effective and convenient for constructing solitary solutions.     

分数阶非线性方程精确解的新方法

将分数阶复变换方法和[(G/G)]方法相结合得到了一种辅助方程方法,用来求解分数阶非线性微分方程。利用该方法并借助于软件Mathematica的符号计算功能求解了分数阶Calogero KDV方程,得到了该方程新的精确解。     

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashion as finite difference methods. Numerical results are obtained for various cases involving line and ring solitons.     

Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes

The meshless local boundary integral equation (LBIE) method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular and circular sections with non-conducting walls. Computations have been carried out for different Hartmann numbers and at various time levels. The method is based on the local boundary integral equation with moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain, are utilized to approximate the interior and boundary variables. A time stepping method is employed to deal with the time derivative. Finally, numerical results are presented to show the behaviour of velocity and induced magnetic field.     

The numerical study on the unsteady flow of gas in a semi-infinite porous medium using an RBF collocation method

In this paper, we study a nonlinear two-point boundary value problem on semi-infinite interval that describes the unsteady gas equation. The solution of the mentioned ordinary differential equation (ODE) is investigated by means of the radial basis function (RBF) collocation method. The RBF reduces the solution of the above-mentioned problem to the solution of a system of algebraic equations and finds its numerical solution. To examine the accuracy and stability of the approach, we transform the mentioned problem into another nonlinear ODE which simplifies the original problem. The comparisons are made between the results of the present work and the numerical method by shooting method combined with the Runge–Kutta technique. It is found that our results agree well with those by the numerical method, which verifies the validity of the present work.     

数字测量系统中传感器温度补偿的一种新方法

提出一种对传感器温度系数的校正新方法——采用二次方程校正法和温度补偿算法相结合的温度校正方法。该算法简单可靠、易实现。将A/D转换器的数字输出量和环境温度T都作为微控制器(MCU)的输入,经MCU处理运算后,获得准确的传感器测量值。     

Fast solution of volume–surface integral equation for scattering from composite conducting‐dielectric targets using multilevel fast dipole method

This article presents a fast solution to the volume–surface integral equation for electromagnetic scattering from three‐dimensional (3D) targets comprising both conductors and dielectric materials by using the multilevel fast dipole method (MLFDM). This scheme is based on the concept of equivalent dipole‐moment method (EDM) that views the Rao–Wilton–Glisson and the Schaubert–Wilton–Glisson basis functions as dipole models with equivalent dipole moments. In the MLFDM, a simple Taylor's series expansion of the terms R^α (α = 1, ?1, ?2, ?3) and R? R? in the formulation of the EDM transforms the interaction between two equivalent dipoles into an aggregation–translation–disaggregation form naturally. Furthermore, benefiting from the multilevel grouping scheme, the matrix‐vector product can be accelerated and the memory cost is reduced remarkably. Simulation results are presented to demonstrate the efficiency and accuracy of this method. © 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2012.     

Hybrid method of FG‐FFT and PO for radiation calculation from antennas around an electrically large conducting platform

An efficient hybrid scheme combining fitting Green's function (FG) fast Fourier transform (FFT) and physical optics (PO) is presented to investigate radiation from antennas around an electrically large conducting platform. The whole region is divided into full wave region and PO one. Similar to hybrid method of integral equation FFT (IE‐FFT) and PO, this hybrid method features acceleration by FFT, fewer unknowns, less computing time than traditional IE‐FFT. Differently, realization of FG‐FFT is established by fitting the Green's function onto the nodes of a uniform Cartesian grid, not by Lagrange interpolation. Several examples are given to prove the hybrid method of FG‐FFT and PO in this letter featuring higher accuracy and being not sensitive to both grid spacing and the expansion order compared to hybrid method of IE‐FFT and PO.     

An integrated approach for rating engineering characteristics’ final importance in product-service system development

Product-service system (PSS) approach has emerged as a competitive strategy to impel manufacturers to offer a set of product and services as a whole. A three-domain PSS conceptual design framework based on quality function deployment (QFD) is proposed in this research. QFD is a widely used design tool considering customer requirements (CRs). Since both product and services influence satisfaction of customer, they should be designed simultaneously. Identification of the critical parameters in these domains plays an important role. Engineering characteristics (ECs) in the functional domain include product-related ECs (P-ECs) and service-related ECs (S-ECs). ECs are identified by translating customer requirements (CRs) in the customer domain. Rating ECs’ importance has a great impact on achieving an optimal PSS planning. The rating problem should consider not only the requirements of customer, but also the requirements of manufacturer. From the requirements of customer, the analytic network process (ANP) approach is integrated in QFD to determine the initial importance weights of ECs considering the complex dependency relationships between and within CRs, P-ECs and S-ECs. In order to deal with the vagueness, uncertainty and diversity in decision-making, the fuzzy set theory and group decision-making technique are used in the super-matrix approach of ANP. From the requirements of manufacturer, the data envelopment analysis (DEA) approach is applied to adjust the initial weights of ECs taking into account business competition and implementation difficulty. A case study is carried out to demonstrate the effectiveness of the developed integrated approach for prioritizing ECs in PSS conceptual design.