A collocation approach for solving systems of linear Volterra integral equations with variable coefficients

In this paper, a numerical method is introduced to solve a system of linear Volterra integral equations (VIEs). By using the Bessel polynomials and the collocation points, this method transforms the system of linear Volterra integral equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives an analytic solution when the exact solutions are polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with existing results. All of the numerical computations have been performed on computer using a program written in MATLAB v7.6.0 (R2008a).     

A Bessel polynomial approach for solving general linear Fredholm integro-differential–difference equations

In this paper, to find an approximate solution of general linear Fredholm integro-differential–difference equations (FIDDEs) under the initial-boundary conditions in terms of the Bessel polynomials, a practical matrix method is presented. The idea behind the method is that it converts FIDDEs to a matrix equation which corresponds to a system of linear algebraic equations and is based on the matrix forms of the Bessel polynomials and their derivatives by means of collocation points. The solutions are obtained as the truncated Bessel series in terms of the Bessel polynomials J _n (x) of the first kind defined in the interval [0, ∞). The error analysis and the numerical examples are included to demonstrate the validity and applicability of the technique.     

A collocation approach to solve the Riccati-type differential equation systems

In this paper, a collocation method is presented for the solutions of the system of the Riccati-type differential equations with variable coefficients. The proposed approach consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions in terms of the Bessel polynomials with unknown coefficients. The unknown coefficients of the Bessel polynomials are found by using the matrix operations of derivatives together with the collocation method. The proposed method gives the analytic solutions when the exact solutions are polynomials. Also, an error analysis technique based on the residual function is introduced for the suggested method. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. Comparing the methodology with some known techniques shows that the presented approach is relatively easy and highly accurate. All of the numerical calculations have been done by using a program written in Maple.     

Bessel polynomial solutions of high-order linear Volterra integro-differential equations

In this study, a practical matrix method, which is based on collocation points, is presented to find approximate solutions of high-order linear Volterra integro-differential equations (VIDEs) under the mixed conditions in terms of Bessel polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results. The results show the efficiency and accuracy of the present work. All of the numerical computations have been performed on the computer using a program written in MATLAB v7.6.0 (R2008a).     

On the solutions of a system of linear retarded and advanced differential equations by the Bessel collocation approximation

In this study, we present a numerical approximation for the solutions of the system of high-order linear retarded and advanced differential equations with variable coefficients under the mixed conditions. This method is based on taking the truncated Bessel expansion of the functions in the retarded and advanced differential equation system. By the aid of the matrix operations and the collocation points, the problem is transformed into a matrix equation with the unknown Bessel coefficients. By solving this matrix equation, the unknown coefficients of the approximate solutions are computed. In addition, examples that illustrate the pertinent features of the method are presented, and the results of this investigation are discussed.     

Normalization techniques for the SVE of the Green function of Helmholtz operator

Electromagnetic and acoustic scattering problems can be usually formulated by suitable integral equations, where the kernel is given in terms of the fundamental solution of the Helmholtz operator. We can consider a special analytic method for the singular value expansion (SVE) of this integral kernel. Note that this is an important tool for the numerical solution of scattering problems, in fact, from the knowledge of the SVE of the integral kernel, we can easily solve the corresponding integral equation. In this paper, we study the numerical approximation of the SVE of this integral kernel, where we have to consider the asymptotic behavior of the Bessel functions.     

Numerical solutions of systems of linear Fredholm integro-differential equations with Bessel polynomial bases

In this paper, a numerical matrix method based on collocation points is presented for the approximate solution of the systems of high-order linear Fredholm integro-differential equations with variable coefficients under the mixed conditions in terms of the Bessel polynomials. Numerical examples are included to demonstrate the validity and the applicability of the technique and also the results are compared with the different methods. The results show the efficiently and the accuracy of the present work. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a).     

Numerical approach for solving linear Fredholm integro-differential equation with piecewise intervals by Bernoulli polynomials

In this paper a numerical method is given for the solution of linear Fredholm integro-differential equation (FIDE) with piecewise intervals under the mixed conditions using the Bernoulli polynomials. The aim of this article is to present an efficient numerical procedure for solving linear FIDE with piecewise intervals. This method transforms linear FIDE with piecewise intervals and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. Finally, some experiments and their numerical solutions are given. The results reveal that this method is reliable and efficient.     

求解非线性最小二乘问题的实用型方法

１．引言对于非线性最小二乘问题其中,为残差向量且,这里是指通常意义下的范数,即二范数．目标函数的梯度和Ｈｅｓｓｅ矩阵为其中　矩阵, 求解非线性最小二乘问题（１．１）的最基本方法是Ｇａｕｓｓ－Ｎｅｗｔｏｎ法,迭代格式为其中ｄｋ为线性方程组的解,这． 当人为满秩矩阵时,线性方程组（１．５）有唯一解,即并且有如下不等式：其中　是矩阵　的最小特征值．当　人接近奇异时,因此有可能存在着 ｄｋ,使得,即某一步迭代的步长太大,导致 Ｇａｕｓｓ－Ｎｅｗｔｏｎ法迭代失败． 另外,当　为奇异矩阵时,线性方程组（１．５）…     

The Unified Frame of Alternating Direction Method of Multipliers for Three Classes of Matrix Equations Arising in Control Theory

In this paper, the unified frame of alternating direction method of multipliers (ADMM) is proposed for solving three classes of matrix equations arising in control theory including the linear matrix equation, the generalized Sylvester matrix equation and the quadratic matrix equation. The convergence properties of ADMM and numerical results are presented. The numerical results show that ADMM tends to deliver higher quality solutions with less computing time on the tested problems.     

Taylor collocation method for solution of systems of high-order linear Fredholm–Volterra integro-differential equations

A Taylor collocation method is presented for numerically solving the system of high-order linear Fredholm–Volterra integro-differential equations in terms of Taylor polynomials. Using the Taylor collocations points, the method transforms the system of linear integro-differential equations (IDEs) and the given conditions into a matrix equation in the unknown Taylor coefficients. The Taylor coefficients can be found easily, and hence the Taylor polynomial approach can be applied. This method is also valid for the systems of differential and integral equations. Numerical examples are presented to illusturate the accuracy of the method. The symbolic algebra program Maple is used to prove the results.     

基于GPU-CUDA的共轭斜量法实现及性能对比

偏微分方程数值解法(包括有限差分法、有限元法)以及大量的数学物理方程数值解法最终都会演变成求解大型线性方程组。因此,探讨快速、稳定、精确的大型线性方程组解法一直是数值计算领域不断深入研究的课题且具有特别重要的意义。在迭代法中,共轭斜量法(又称共轭梯度法)被公认为最好的方法之一。但是,该方法最大缺点是仅适用于线性方程组系数矩阵为对称正定矩阵的情况,而且常规的CPU算法实现非常耗时。为此,通过将线性方程组系数矩阵作转换成对称矩阵后实施基于GPU-CUDA的快速共轭斜量法来解决一般性大型线性方程组的求解问题。试验结果表明:在求解效率方面,基于GPU-CUDA的共轭斜量法运行效率高,当线性方程组阶数超过3000时,其加速比将超过14;在解的精确性与求解过程的稳定性方面,与高斯列主元消去法相当。基于GPU-CUDA的快速共轭斜量法是求解一般性大型线性方程组快速而非常有效的方法。     

Integration of Dynamic Equation of Spring-Mass-Damper Systems via Chebyshev Polynomials

A numerical scheme based on Chebyshev polynomials for the determination of the response of spring-mass-damper systems is presented. The state vector of the differential equation of the spring-mass-damper system is expanded in terms of Chebyshev polynomials. This expansion reduces the original differential equations to a set of linear algebraic equations where the unknowns are the coefficient of Chebyshev polynomials. A formal procedure to generate the coefficient matrix and the right-hand side vector of this system of algebraic equations is discussed. The numerical efficiency of the proposed method is compared with that of Runge-Kutta method. It is shown that this scheme is accurate and is computationally efficient.     

Numerical Verification of Solutions of Nekrasov’s Integral Equation

This paper describes numerical verification of solutions of Nekrasov’s integral equation which is a mathematical model of two-dimensional water waves. This nonlinear and periodic integral equation includes a logarithmic singular kernel which is typically found in some two-dimensional potential problems. We propose the verification method using some properties of the singular integral for trigonometric polynomials and Schauder’s fixed point theorem in the periodic Sobolev space. A numerical example shows effectiveness of the present method.     

Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations

In this paper, the numerical solution of nonlinear Fredholm integral equations of the second kind is considered by two methods. The methods are developed by means of the Sinc approximation with the single exponential (SE) and double exponential (DE) transformations. These numerical methods combine a Sinc collocation method with the Newton iterative process that involves solving a nonlinear system of equations. We provide an error analysis for the methods. So far approximate solutions with polynomial convergence have been reported for this equation. These methods improve conventional results and achieve exponential convergence. Some numerical examples are given to confirm the accuracy and ease of implementation of the methods.     

Solutions of the Cahn–Hilliard equation

In this paper, we found some exact solutions of the Cahn–Hilliard equation and the system of the equations by considering a modified extended tanh function method. A numerical solution to a Cahn–Hilliard equation is obtained using a homotopy perturbation method (HPM) combined with the Adomian decomposition method (ADM). The comparisons are given in the tables.     

A Taylor Collocation Method for the Solution of Linear Integro-Differential Equations

In this study, a matrix method called the Taylor collocation method is presented for numerically solving the linear integro-differential equations by a truncated Taylor series. Using the Taylor collocation points, this method transforms the integro-differential equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. Also the method can be used for linear differential and integral equations. To illustrate the method, it is applied to certain linear differential, integral, and integro-differential equations and the results are compared.     

Systems of linear difference equations with bounded and robustly bounded solutions

In this paper, we investigate higher-order systems of linear difference equations where the associated characteristic matrix polynomial is self-inversive. We consider classes of equations with bounded solutions. It is known that stability properties of higher-order systems of linear difference equations are determined by the characteristic values of the corresponding matrix polynomials. All solutions are bounded (in both time directions) if the spectrum of the corresponding matrix polynomial lies on the unit circle, and moreover if the characteristic values of modulus 1 are semisimple. If the corresponding matrix polynomial is self-inversive, then one can use the inner radius of the numerical range to obtain a criterion for boundedness of solutions. We show that all solutions are bounded if the inner radius is greater than 1. In the case of matrix polynomials with positive definite coefficient matrices, we derive a computable lower bound for the inner radius and we obtain a criterion for robust boundedness.     

Stability problem of systems with multiple delay channels

A new form of coupled differential-difference equations with one delay in each channel is proposed to model systems with multiple delays. This formulation has significant advantage over the traditional differential-difference equations or the prevailing form of coupled differential-difference equations with multiple delays. Fundamental solutions, general solutions, and the construction of the Lyapunov-Krasovskii functional are discussed. For systems with a large number of state variables with multiple low-dimensional delay elements, this formulation allows a drastic reduction of computational cost as compared to the traditional differential-difference equation formulation when the discretized Lyapunov-Krasovskii functional method is used.     

一类广义Riccati矩阵方程对称解的双迭代算法

研究了一类广义系统控制理论导出的Riccati矩阵方程对称解的数值计算方法．运用牛顿算法将Riccati矩阵方程的对称解问题转化为线性矩阵方程的对称解或者对称最小二乘解问题,采用修正共轭梯度法解决导出的线性矩阵方程的对称解问题,可建立求Riccati矩阵方程对称解的双迭代算法．数值算例表明,双迭代算法是有效的．