Radial basis functions method for numerical solution of the modified equal width equation

The numerical solution of the modified equal width equation is investigated by using meshless method based on collocation with the well-known radial basis functions. Single solitary wave motion, two solitary waves interaction and three solitary waves interaction are studied. Results of the meshless methods with different radial basis functions are presented.     

A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines

The numerical solution of the one-dimensional modified equal width wave (MEW) equation is obtained by using a lumped Galerkin method based on quadratic B-spline finite elements. The motion of a single solitary wave and the interaction of two solitary waves are studied. The numerical results obtained show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis of the scheme is also investigated.     

A non-periodic Fourier method for solution of the classical wave equation

A Fourier method solution for the classical wave equation which overcomes the difficulty of the periodicity of the trigonometric functions is presented.     

A split-step Fourier method for the complex modified Korteweg-de Vries equation

In this study, the complex modified Korteweg-de Vries (CMKdV) equation is solved numerically by three different split-step Fourier schemes. The main difference among the three schemes is in the order of the splitting approximation used to factorize the exponential operator. The space variable is discretized by means of a Fourier method for both linear and nonlinear subproblems. A fourth-order Runge-Kutta scheme is used for the time integration of the nonlinear subproblem. Classical problems concerning the motion of a single solitary wave with a constant polarization angle are used to compare the schemes in terms of the accuracy and the computational cost. Furthermore, the interaction of two solitary waves with orthogonal polarizations is investigated and particular attention is paid to the conserved quantities as an indicator of the accuracy. Numerical tests show that the split-step Fourier method provides highly accurate solutions for the CMKdV equation.     

A parallel DSDADI method for solution of the steady state diffusion equation

A parallel diagonally scaled dynamic alternating-direction-implicit (DSDADI) method is shown to be an effective algorithm for solving the 2D and 3D steady-state diffusion equation on large uniform Cartesian grids. Empirical evidence from the parallel solution of large gridsize problems suggests that the computational work done by DSDADI to converge over an N^d grid with continuous diffusivity is of lower order than O(N^d+α) for any fixed α > 0. This is in contrast to the method of diagonally scaled conjugate gradients (DSCG), for which the computational work necessary for convergence is O(N^d+1). Furthermore, the combination of diagonal scaling, spatial domain decomposition (SDD), and distributed tridiagonal system solution gives the DSDADI algorithm reasonable scalability on distributed-memory multiprocessors such as the CRAY T3D. Finally, an approximate parallel tridiagonal system solver with diminished interprocessor communication exhibits additional utility for DSDADI.     

Parallel solution of large-scale eigenvalue problem for master equation in protein folding dynamics

It is known that a master equation characterizes time evolution of trajectories and transition of states in protein folding dynamics. Solution of the master equation may require calculating eigenvalues for the corresponding eigenvalue problem. In this paper, we numerically study the folding rate for a dynamic problem of protein folding by solving a large-scale eigenvalue problem. Three methods, the implicitly restarted Arnoldi, Jacobi–Davidson, and QR methods are employed in solving the corresponding large-scale eigenvalue problem for the transition matrix of master equation. Comparison shows that the QR method demands tremendous computing resource when the length of sequence L>10$L>10$ due to extremely large size of matrix and CPU time limitation. The Jacobi–Davidson method may encounter convergence issue, for cases of L>9$L>9$. The implicitly restarted Arnoldi method is suitable for solving problems among them. Parallelization of the implicitly restarted Arnoldi method is successfully implemented on a PC-based Linux cluster. The parallelization scheme mainly partitions the operation of matrix. For the Arnoldi factorization, we replicate the upper Hessenberg matrix _Hm${\mathbf{H}}_{\mathbf{m}}$ for each processor, and distribute the set of Arnoldi vectors _Vm${\mathbf{V}}_{\mathbf{m}}$ among processors. Each processor performs its own operation. The algorithm is implemented on a PC-based Linux cluster with message passing interface (MPI) libraries. Numerical experiment performing on our 32-nodes PC-based Linux cluster shows that the maximum difference among processors is within 10%. A 23-times speedup and 72% parallel efficiency are achieved when the matrix size is greater than 2×10⁶$2\times {10}^6$ on the 32-nodes PC-based Linux cluster. This parallel approach enables us to explore large-scale dynamics of protein folding.     

On Newton's method for Riccati equation solution

The purpose of this note is to point out that the assumptions of two theorems of Kleinman concerning Newton's method for the Riccati equation can be weakened.     

A parallel method for the numerical solution of integro-differential equation with positive memory

An efficient parallel numerical method is proposed for an integro-differential equation with positive memory. Instead of solving the equation in classical time-marching methods which require massive storage of solutions of previous time steps in order to advance to a next time step, the Fourier–Laplace transformation in time is applied to obtain a set of complex-valued, elliptic problems parameterized by points on a contour in the complex plane. Using the independence of an elliptic problem corresponding to one contour point is independent of those elliptic problems corresponding to other contour points, all elliptic problems can be solved in parallel essentially without data communications. Then the time domain solution can be obtained by the Fourier–Laplace inversion formula. An error analysis and the numerical implementation of this parallel method is presented.     

The Group Explicit method for the solution of Burger's equation

The aim of this paper is to extend the application of the Group Explicit method [1, 2] to the numerical solution of a non-linear parabolic partial differential equation of second order. The method was tested out on Burger's equation for various initial and boundary conditions. It can be seen that the method is accurate and comparable to existing finite difference methods.     

Jumarie’s修正的R-L分数阶系统的新解法

结合同伦摄动理论和Sumudu变换方法,提出了一种简单有效的摄动方法,并应用该方法求解了Jumarie’s修正的Riemann-Liouville(R-L)分数阶的方程,该方程带有分数阶的初值条件,而以前的文献中很少讨论分数阶的初值条件。结果表明该方法具有较高的精度和有效性。     

A modified NBI and NC method for the solution of N-multiobjective optimization problems

Multiobjective optimization (MO) techniques allow a designer to model a specific problem considering a more realistic behavior, which commonly involves the satisfaction of several targets simultaneously. A fundamental concept, which is adopted in the multicriteria optimization task, is that of Pareto optimality. In this paper we test several well-known procedures to deal with multiobjective optimization problems (MOP) and propose a novel modified procedure that when applied to the existing Normal Boundary Intersection (NBI) method and Normal Constraint (NC) method for more than two objectives overcomes some of their deficiencies. For the three and four objective applications analyzed here, the proposed scheme presents the best performance both in terms of quality and efficiency to obtain a set of proper Pareto points, when compared to the analyzed existing approaches.     

A stabilizing solution to the algebraic Riccati equation the resolvent method

A new approach to the problem of analytic representation of the stabilizing solution to the algebraic Riccati equation is proposed. The quadratic matrix equation is reduced to a linear one using the resolvent (sI _2n -H)^?1 of the Hamilton matrix. The symmetric solution to the obtained linear equation defines a stabilizing solution to the Riccati equation. Matrix coefficients of the linear equation are defined by the integral of resolvent in the complex domain over the closed contour which contains all its right poles. This construction of the solution to the problem gives rise to the development of important parts of the analysis and of the corresponding computing procedures.     

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.     

A generalization of Numerov's method for the numerical solution of the Schrodinger equation in two dimensions

In this paper generalizations of the well known Numerov's method are obtained. The local truncation errors of the new methods are presented and the result of the application of the new methods to a two-dimensional Schrodinger equation in an equal space discretization is presented. Numerical illustrations show the efficiency of the new methods compared with the known five-point formula in two coulombic potentials.     

拟合实验数据的新方程--非整数幂多项式方程

将实验数据拟合成方程在科学研究和工程计算上具有十分重要的作用。目前,拟合实验数据最常用的是整数幂多项式f(x)=c0 c1x c2x^2… cnx^n但是用此多项式拟合各类数据时有时误差较大。本文提出一个新方程,即双系列非整数幂多项式g(x)=c0 c1x^a…… ckx^ka ck 1x^(k 1)b ……cnx^nb式中a,b为参数ci(i=0,1,2,3,……n)为待定系数。在拟合各类实验数据时,新方程总是优于整数幂多项式。     

Practical applications of an integral equation method for the solution of Laplace's equation

An integral equation method for the solution of Laplace's equation, originally proposed for boundary value problems in a single medium, is here extended to problems involving multiple media. The extended method has been used to compute the internal thermal resistance of electric cables and some numerical results are presented.     

A parallel monotone iterative method for the numerical solution of multi-dimensional semiconductor Poisson equation

Various self-consistent semiconductor device simulation approaches require the solution of Poisson equation that describes the potential distribution for a specified doping profile (or charge density). In this paper, we solve the multi-dimensional semiconductor nonlinear Poisson equation numerically with the finite volume method and the monotone iterative method on a Linux-cluster. Based on the nonlinear property of the Poisson equation, the proposed method converges monotonically for arbitrary initial guesses. Compared with the Newton's iterative method, it is easy implementing, relatively robust and fast with much less computation time, and its algorithm is inherently parallel in large-scale computing. The presented method has been successfully implemented; the developed parallel nonlinear Poisson solver tested on a variety of devices shows it has good efficiency and robustness. Benchmarks are also included to demonstrate the excellent parallel performance of the method.