耦合KdV方程组的对称,精确解和守恒律

通过利用修正的CK直接方法建立了耦合KdV方程组的对称群理论.利用对称群理论和耦合KdV方程组的旧解得到了它们的新的精确解.基于上述理论和耦合KdV方程组的共轭方程组的理论,得到了耦合KdV方程组的守恒律.     

基于改进区间数密度集结算子指标群赋权方法

针对指标数据已知,而权重数据未知的群组赋权问题,给出了一种基于改进的区间数密度集结算子来进行指标群组赋权的决策方法。首先给出了区间数和区间数密度集结算子(IDM)的定义及性质,改进了以前区间数聚类的方法,应用直接法对一维区间数据组进行聚类,并定义了模糊统计量,以确定最为合理的一种聚类方式。然后基于改进的区间数密度集结算子这种数学模型,来解决指标值数据已知,而权重未知的群组赋权问题。最后举例说明该方法的可行性和实用性。     

Exact solutions and conservation laws of (2 + 1)-dimensional Boiti-Leon-Pempinelli equation

In this paper, we obtain the symmetry group theorem by using the modified CK’s direct method, and some new exact solutions of (2 + 1)-dimensional BLP equation. Also we derive the corresponding Lie algebra and the conservation laws of BLP equation.     

Symmetry classification of the Gardner equation with time-dependent coefficients arising in stratified fluids

We perform symmetry classification of a variable-coefficient combined KdV-mKdV equation. That is, the equation combining the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations, or also known as the Gardner equation. The direct method of group classification is utilized to specify the forms of these time-dependent coefficients.     

Projective synchronization between different fractional‐order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique

In the present article, the authors have proposed a modified projective adaptive synchronization technique for fractional‐order chaotic systems. The adaptive projective synchronization controller and identification parameters law are developed on the basis of Lyapunov direct stability theory. The proposed method is successfully applied for the projective synchronization between fractional‐order hyperchaotic Lü system as drive system and fractional‐order hyperchaotic Lorenz chaotic system as response system. A comparison between the effects on synchronization time due to the presence of fractional‐order time derivatives for modified projective synchronization method and proposed modified adaptive projective synchronization technique is the key feature of the present article. Numerical simulation results, which are carried out using Adams–Boshforth–Moulton method show that the proposed technique is effective, convenient and also faster for projective synchronization of fractional‐order nonlinear dynamical systems. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Tchebyschev method of solution of ordinary differential equations

Clenshaw's method of computation of the solution of the Cauchy problem is modified for increased speed and efficiency. The complexity of the original method is considerably reduced by replacing coupled sets of simultaneous equations for the linearization coefficients by uncoupled systems. This modification has shown that the method of direct substitution of Tchebyschev series into the differential equation is exactly equivalent to the integrated system and is slightly superior to the mixed system. In addition, the constraints described by Fox and Parker which inhibit the use of the direct method no longer exist. Apart from making calculations manually convenient and more tractable, there is a saving in computation time and labor by several factors.     

On the Acceleration of Kaczmarz Projection Algorithm

In the paper [1], a direct version of the classical Kaczmarz algorithm was proposed, which gives us in only one iteration a solution of an arbitrary consistent system of linear equations. Unfortunately, as any direct method applied to large sparse matrices, this algorithm is based on some modifications of the system matrix sparsity structure such that a big fill-in appears. In order to overcome this difficulty, in the present paper we propose a modified version of this direct Kaczmarz algorithm in which the transformations applied to the system matrix try to conserve the initial sparsity structure. This transformations are done via clustering using Jaccard and Hamming distances. The modified Kaczmarz algorithm is no more a direct method, but we obtain an acceleration of convergence with respect to the classical Kaczmarz algorithm. Numerical experiments which ilustrate the efficiency of our algorithm are also presented. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive use of iterative methods in predictor–corrector interior point methods for linear programming

In this work we devise efficient algorithms for finding the search directions for interior point methods applied to linear programming problems. There are two innovations. The first is the use of updating of preconditioners computed for previous barrier parameters. The second is an adaptive automated procedure for determining whether to use a direct or iterative solver, whether to reinitialize or update the preconditioner, and how many updates to apply. These decisions are based on predictions of the cost of using the different solvers to determine the next search direction, given costs in determining earlier directions. We summarize earlier results using a modified version of the OB1-R code of Lustig, Marsten, and Shanno, and we present results from a predictor–corrector code PCx modified to use adaptive iteration. If a direct method is appropriate for the problem, then our procedure chooses it, but when an iterative procedure is helpful, substantial gains in efficiency can be obtained.     

Modified auxiliary boundary conditions method for an ill-posed problem for the homogeneous biharmonic equation

In this paper, we are interested in the inverse problem for the biharmonic equation posed on a rectangle, which is of great importance in many areas of industry and engineering. We show that the problem under consideration is ill-posed; therefore, to solve it, we opted for a regularization method via modified auxiliary boundary conditions. The numerical implementation is based on the application of the semidiscrete finite difference method for a sequence of well-posed direct problems depending on a small parameter of regularization. Numerical results are performed for a rectangle domain showing the effectiveness of the proposed method.     

New exact travelling wave solutions to the complex coupled KdV equations and modified KdV equation

By using some exact solutions of an auxiliary ordinary differential equation, a new direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the complex coupled KdV equations and modified KdV equation. New exact complex solutions are obtained.     

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over‐specified boundary in the case of the alternating iterative algorithm of Kozlov et al. (USSR Comput Math Math Phys 31 (1991), 45–52) applied to the Cauchy problem for the two‐dimensional modified Helmholtz equation. The two mixed, well‐posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is selected according to the generalized cross‐validation criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the modified Helmholtz equation in two‐dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

一种有效的求解无约束优化问题的共轭方向法

§1.前言 有些实践中的优化问题可以按无约束来处理,而且大量非常有效的约束优化算法都涉及无约束优化方法,因此,无约束优化方法在实用上是很重要的。 考虑下面的二次目标函数F(X)的无约束优化问题:     

Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

     

Traveling wave solutions of the nonlinear Drinfel’d–Sokolov–Wilson equation and modified Benjamin–Bona–Mahony equations

In this paper, the modified simple equation (MSE) method is implemented to find the exact solutions for the nonlinear Drinfel’d–Sokolov–Wilson (DSW) equation and the modified Benjamin–Bona–Mahony (mBBM) equations. The efficiency of this method for constructing these exact solutions has been demonstrated. It is shown that the MSE method is direct, effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this technique reduces the large volume of calculations.     

Symmetry groups of differential-difference equations and their compatibility

It is shown that the intrinsic determining equations of a given differential-difference equation (DDE) can be derived by the compatibility between the original equation and the intrinsic invariant surface condition. The (2+1)-dimensional Toda lattice, the special Toda lattice and the DD-KP equation serving as examples are used to illustrate this approach. Then, Bäcklund transformations of the (2+1)-dimensional DDEs including the special Toda lattice, the modified Toda lattice and the DD-KZ equation are presented by using the non-intrinsic direct method. In addition, the Clarkson-Kruskal direct method is developed to find similarity reductions of the DDEs.     

Solving differential equations using modified Picard iteration

Many classes of differential equation are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and engineering and include non-linear as well as linear differential equations. Examples involving partial as well as ordinary differential equations are presented. The method is easy to implement on a computer and the solutions so obtained are essentially power series. With its conceptual clarity (differential equations are integrated directly), its uniform methodology (the overall approach is the same in all cases) and its straightforward computer implementation (the integration and iteration procedures require only standard commercial software), the modified Picard methods offer obvious benefits for the teaching of differential equations as well as presenting a basic but flexible tool-kit for the solution process itself.     

New exact travelling wave solutions for the Kawahara and modified Kawahara equations

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact travelling wave solutions for nonlinear evolution equations. By this method the Kawahara and the modified Kawahara equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.     

Exact solutions to the combined KdV and mKdV equation

This paper presents two different methods for the construction of exact solutions to the combined KdV and mKdV equation. The first method is a direct one based on a general form of solution to both the KdV and the modified KdV (mKdV) equations. The second method is a leading order analysis method. The method was devised by Jeffrey and Xu. Each of these methods is capable of solving the combined KdV and mKdV equation exactly.     

A note on the modified simple equation method applied to Sharma-Tasso-Olver equation

Jawad et al. have applied the modified simple equation method to find the exact solutions of the nonlinear Fitzhugh-Naguma equation and the nonlinear Sharma-Tasso-Olver equation. The analysis of the Sharma-Tasso-Olver equation obtained by Jawad et al. is based on variant of the modified simple equation method. In this paper, we provide its direct application and obtain new 1- soliton solutions.