A CCD–ADI method for unsteady convection–diffusion equations

With a combined compact difference scheme for the spatial discretization and the Crank–Nicolson scheme for the temporal discretization, respectively, a high-order alternating direction implicit method (ADI) is proposed for solving unsteady two dimensional convection–diffusion equations. The method is sixth-order accurate in space and second-order accurate in time. The resulting matrix at each ADI computation step corresponds to a triple-tridiagonal system which can be effectively solved with a considerable saving in computing time. In practice, Richardson extrapolation is exploited to increase the temporal accuracy. The unconditional stability is proved by means of Fourier analysis for two dimensional convection–diffusion problems with periodic boundary conditions. Numerical experiments are conducted to demonstrate the efficiency of the proposed method. Moreover, the present method preserves the higher order accuracy for convection-dominated problems.     

Riccati discrete time transfer matrix method for elastic beam undergoing large overall motion

An efficient method for dynamics simulation for elastic beam with large overall spatial motion and nonlinear deformation, namely, the Riccati discrete time transfer matrix method (Riccati-DT-TMM), is proposed in this investigation. With finite segments, continuous deformation field of a beam can be decomposed into many rigid bodies connected by rotational springs. Discrete time transfer matrices of rigid bodies and rotational springs are used to analyze the dynamic characteristic of the beam, and the Riccati transform is used to improve the numerical stability of discrete time transfer matrix method of multibody system dynamics. A predictor-corrector method is used to improve the numerical accuracy of the Riccati-DT-TMM. Using the Riccati-DT-TMM in dynamics analysis, the global dynamics equations of the system are not needed and the computation time required increases linearly with the system’s number of degrees of freedom. Three numerical examples are given to validate the method for the dynamic simulation of a geometric nonlinear beam undergoing large overall motion.     

A modified predictor-corrector scheme for the Klein–Gordon equation

A numerical method based on a three-time level finite-difference scheme has been proposed for the solution of the two forms of the Klein–Gordon equation. The method, which is analysed for local truncation error and stability, leads to the solution of a nonlinear system. To avoid solving it, a predictor-corrector scheme using as predictor a second-order explicit scheme is proposed. The procedure of the corrector is modified by considering, as known, the already evaluated corrected values instead of the predictor ones. This modified scheme is applied to problems possessing periodic, kinks and soliton waves. The accuracy as well as the long-time behaviour of the proposed scheme is discussed and comparisons with the relevant known in the bibliography schemes are given.     

A Fully Discrete Fast Fourier–Galerkin Method Solving a Boundary Integral Equation for the Biharmonic Equation

We develop a fully discrete fast Fourier–Galerkin method for solving a boundary integral equation for the biharmonic equation by introducing a quadrature scheme for computing the integrals of non-smooth functions that appear in the Fourier–Galerkin method. A key step in developing the fully discrete fast Fourier–Galerkin method is the design of a fast quadrature scheme for computing the Fourier coefficients of the non-smooth kernel function involved in the boundary integral equation. We prove that with the proposed quadrature algorithm, the total number of additions and multiplications used in generating the compressed coefficient matrix for the proposed method is quasi-linear (linear with a logarithmic factor), and the resulting numerical solution of the equation preserves the optimal convergence order. Numerical examples are presented to demonstrate the approximation accuracy and computational efficiency of the proposed method.     

Calculation of Fourier transform using symmetrical properties

A matrix method which computes discrete Fourier transforms using a digital computer program is presented in this paper. The proposed technique takes advantage of symmetry of the complex functions about the real and imaginary axes to reduce the number of calculations necessary in a given Fourier transform. Computationally the method described is not as efficient, especially for a large N, as the well-known Cooley-Tukey method. However, it differs from the Cooley-Tukey formulation in two notable respects: first, the present method is not as restrictive in the selection of values of N as the Cooley-Tukey method; and second, the calculations can proceed with the first value of the time function, thus eliminating the need for storing data before beginning with the transform calculations as is the case with the Cooley-Tukey method. In a number of applications, realizing these two conditions is more important than computational efficiency.The logic of a computer program which calculates the Fourier transform using the symmetry properties is described by a flow chart. This paper also includes numerical examples, using the computer program, of a Fourier transform from the time domain to the frequency domain. In addition, the program calculates the inverse Fourier transform, reconstructing the original time function from its frequency contents.     

Numerical solution of a coupled modified Korteweg–de Vries equation by the Galerkin method using quadratic B-splines

In this paper, numerical solutions of a coupled modified Korteweg–de Vries equation have been obtained by the quadratic B-spline Galerkin finite element method. The accuracy of the method has been demonstrated by five test problems. The obtained numerical results are found to be in good agreement with the exact solutions. A Fourier stability analysis of the method is also investigated.     

Lax-Wendroff差分法在高速互连线分析中的应用

将Lax-Wendroff差分法用于互连传输线的分析，充分地利用了电报方程结构特性，给出了能够有效应用于互连线分析的二阶精度的Lax-Wendroff差分格式．通过数值实验将本文方法同特征法及快速傅里叶变换(FFT)法进行了比较．这种直接的时—空离散数值方法可应用于包括非均匀传输线在内的一般的互连传输线的瞬态分析，计算效率近于特征法，但适用范围要广得多．     

A Fast Solver for Boundary Integral Equations of the Modified Helmholtz Equation

The main purpose of this paper is to develop a fast fully discrete Fourier–Galerkin method for solving the boundary integral equations reformulated from the modified Helmholtz equation with boundary conditions. We consider both the nonlinear and the Robin boundary conditions. To tackle the difficulties caused by the two types of boundary conditions, we provide an improved version of the Galerkin method based on the Fourier basis. By employing a matrix compression strategy and efficient numerical quadrature schemes for oscillatory integrals, we obtain fully discrete nonlinear or linear system. Finally, we use the multilevel augmentation method to solve the resulting systems. We point out that the proposed method enjoys an optimal convergence order and a nearly linear computational complexity. The theoretical estimates are confirmed by the performance of this method on several numerical examples.     

The Use of the Ritz Method and Laplace Transform for Solving 2D Fractional‐Order Optimal Control Problems Described by the Roesser Model

In this article a numerical solution is presented for a class of two‐dimensional fractional‐order optimal control problems (2D‐FOOCPs) with one input and two outputs. To implement the numerical method, the Legendre polynomial basis is used with the aid of the Ritz method and the Laplace transform. By taking the Ritz method as a basic scheme into account and applying a new constructed fractional operational matrix to estimate the fractional and integer order derivatives of the basis, the given 2D‐FOOCP is reduced to a system of algebraic equations. One of the advantages of the proposed method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed. Moreover, satisfactory results are obtained in just a small number of polynomials order. The convergence of the method is extensively investigated and finally two illustrative examples are included to show the validity and applicability of the novel proposed technique in the current work.     

Fast spherical Bessel transform via fast Fourier transform and recurrence formula

We propose a new method for the numerical evaluation of the spherical Bessel transform. A formula is derived for the transform by using an integral representation of the spherical Bessel function and by changing the integration variable. The resultant algorithm consists of a set of the Fourier transforms and numerical integrations over a linearly spaced grid of variable k in Fourier space. Because the k-dependence appears in the upper limit of the integration range, the integrations can be performed effectively in a recurrence formula. Several types of atomic orbital functions are transformed with the proposed method to illustrate its accuracy and efficiency, demonstrating its applicability for transforms of general order with high accuracy.     

基于快速Fourier变换法的广义特征值问题重根辨识方法

对具有重根的广义特征值问题,采用基于快速Fourier变换的方法进行求解,实现重根辨识.文章中采用多次单点初始激励的方式,仿真计算测点上的自由振动响应,对响应进行快速Fourier变换后得到频域数据.而后对频域数据分析,得到固有频率和多组测点振型数据.根据单频和重频处的振型特性,引入振型的余弦相似度为判别参数,辨识重根.数值算例表明,该方法可有效实现重根辨识,同时特征值的计算能达到较高精度.     

Selective Regional Correlation for Pattern Recognition

In this paper, a novel correlation-based pattern classifier that relies on the analysis of time-frequency decomposition of a template and signals is proposed. Significant improvements in resolution and accuracy are obtained using this new classifier when compared to a conventional correlation-based one. The short-time Fourier transform, continuous wavelet transform, and S-transform are considered in the time-frequency decomposition process. To evaluate the performance of the proposed scheme, numerical studies are performed on a set of synthetic test signals, and excellent results have been obtained. This paper also presents an illustrative example where two types of heart sounds are classified. The classification error percentage for the heart sounds using the new classifier is only 6.670% as compared to 56.67% when a general correlation-based classifier is used     

Spectral multidomain approximation of elliptic problems with mixed conditions on the interfaces

The multidomain technique for elliptic problems, that allows the fulfillment of the interface conditions by means of a suitable combination of the continuity of the solution and of its normal derivative, is considered. Some choices of this combination are investigated and, in particular, a choice that allows the solution of the multidomain problem through two solutions for each subproblem, is proposed. The scheme has been discretized with a collocation method and some numerical tests are reported. Moreover the method is compared with the more classical Dirichlet/Neumann one as well as with the capacitance matrix method.This research has been supported by a grant from M.P.I. (40%).     

Delay-dependent stability analysis and stabilization for discrete-time fuzzy systems with state delay: a fuzzy Lyapunov-Krasovskii functional approach.

This correspondence studies stability analysis and stabilization for discrete-time Takagi and Sugeno fuzzy systems with state delay. First, a new fuzzy Lyapunov-Krasovskii functional (LKF) is constructed to derive a delay-dependent stability condition for open-loop fuzzy systems. Then, a delay-dependent stabilization approach based on a nonparallel distributed compensation scheme is provided for closed-loop fuzzy systems. Both state feedback and observer-based control cases are considered. The proposed stability and stabilization conditions are represented in terms of linear matrix inequalities (LMIs), which can be solved efficiently by using existing LMI optimization techniques. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.     

带输入延时的连续时间系统的离散化

含多项式插值的Runge-Kutta方法应用于对带输入延时的连续时间系统的离散化中. 与传统的离散化方法相比, 本文提出的方法是有效且精度高阶的. 此方法的精度与Runge-Kutta法及插值多项式的精度紧密相关. 本文讨论了离散化方法的近似精度阶及最大可达的精度阶. 除此之外, 也分析了方法的输入状态稳定性. 为保证相应离散系统的稳定性, 可通过考察RK法的绝对稳定域来选择采样时间. 特别当RK法是A-稳定时, 可以不受稳定性的约束选择采样时间. 最后提供了一个数值例子来证明方法的优越性.     

A new fractional Chebyshev FDM: an application for solving the fractional differential equations generated by optimisation problem

In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method. The algorithm is based on a combination of the useful properties of Chebyshev polynomial approximation and finite difference method. We implement this technique to solve numerically the non-linear programming problem which are governed by fractional differential equations (FDEs). The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the Caputo fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The application of the method to the generated FDEs leads to algebraic systems which can be solved by an appropriate method. Two numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method. A comparison with the fourth-order Runge–Kutta method is given.     

L1 Fourier spectral methods for a class of generalized two-dimensional time fractional nonlinear anomalous diffusion equations

In this paper, L1 Fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two-dimensional time-fractional nonlinear anomalous diffusion equations involving Caputo fractional derivative. Firstly, we establish the L1 Fourier Galerkin full discrete and L1 Fourier collocation schemes with Fourier spectral discretization in spatial direction and L1 difference method in temporal direction. Secondly, stability and convergence for both Galerkin and collocation approximations are proved. It is shown that the proposed methods are convergent with spectral accuracy in space and $(2?\alpha )$ order accuracy in time. For implementation, the equivalence between pseudospectral method and collocation method is discussed. Furthermore, a numerical algorithm of Fourier pseudospectral method is developed based on two-dimensional fast Fourier transform (FFT2) technique. Finally, numerical examples are provided to test the theoretical claims. As is shown in the numerical experiments, Fourier spectral methods are powerful enough with excellent efficiency and accuracy.     

Assessment and improvement of precise time step integration method

In this paper, the numerical stability and accuracy of Precise Time Step Integration Method are discussed in detail. It is shown that the method is conditionally stable and it has inherent algorithmic damping, algorithmic period error and algorithmic amplitude decay. However for discretized structural models, it is relatively easy for this time integration scheme to satisfy the stability conditions and required accuracy. Based on the above results, the optimum values of the truncation order L and bisection order N are presented. The Gauss quadrature method is used to improve the accuracy of the Precise Time Step Integration Method. Finally, two numerical examples are presented to show the feasibility of this improvement method.     

Robust stability criteria for uncertain linear systems with interval time-varying delay

This paper considers the robust delay-dependent stability problem of a class of linear uncertain system with interval time-varying delay and proposes less conservative stability criteria for computing the maximum allowable bound of the delay range. Less conservatism of the proposed stability criteria is attributed to the delay-central point method of stability analysis, wherein the delay interval is partitioned into two subintervals of equal length, and the time derivative of a candidate Lyapunov-Krasovskii functional based on delay decomposition technique is evaluated in each of these delay segments. In deriving the stability conditions in LMI framework, neither model transformations nor bounding techniques using free-weighting matrix variables are employed for dealing the cross-terms that emerge from the time derivative of the Lyapunov-Krasovskii functional; instead, they are dealt using tighter integral inequalities. The proposed analysis subsequently yields a stability condition in convex LMI framework that can be solved using standard numerical packages. For deriving robust stability conditions, two categories of system uncertainties, namely, time-varying structured and polytopic-type uncertainties, are considered. The effectiveness of the proposed stability criteria is validated through standard numerical examples.     

A conservative numerical method for the Cahn–Hilliard equation with Dirichlet boundary conditions in complex domains

In this paper we present a conservative numerical method for the Cahn–Hilliard equation with Dirichlet boundary conditions in complex domains. The method uses an unconditionally gradient stable nonlinear splitting numerical scheme to remove the high-order time-step stability constraints. The continuous problem has the conservation of mass and we prove the conservative property of the proposed discrete scheme in complex domains. We describe the implementation of the proposed numerical scheme in detail. The resulting system of discrete equations is solved by a nonlinear multigrid method. We demonstrate the accuracy and robustness of the proposed Dirichlet boundary formulation using various numerical experiments. We numerically show the total energy decrease and the unconditionally gradient stability. In particular, the numerical results indicate the potential usefulness of the proposed method for accurately calculating biological membrane dynamics in confined domains.