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).     

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).     

Taylor polynomial solutions of systems of linear differential equations with variable coefficients

A Taylor collocation method has been presented for numerically solving systems of high-order linear ordinary, differential equations with variable coefficients. Using the Taylor collocation points, this method transforms the ODE system and the given conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equation, a new system of equations corresponding to the system of linear algebraic equations is gained. Hence by finding the Taylor coefficients, the Taylor polynomial approach is obtained. Also, the method can be used for the linear systems in the normal form. To illustrate the pertinent features of the method, examples are presented and results are compared.     

A method for the numerical solution of singular integral equations with logarithmic singularities

A method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed. This method is based on the reduction of these equations to equivalent singular integral equations with Cauchy-type singularities in their kernels and the application to the latter of the methods of numerical solution, based on the use of an appropriate numerical integration rule for the reduction to a system of linear algebraic equations. The aforementioned method is presented in two forms giving slightly different numerical results. Furthermore, numerical applications of the proposed methods are made. Some further possibilities are finally investigated     

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 Taylor polynomial approach for solving generalized pantograph equations with nonhomogenous term

A numerical method for solving the generalized (retarded or advanced) pantograph equation with constant and variable coefficients under mixed conditions is presented. The method is based on the truncated Taylor polynomials. The solution is obtained in terms of Taylor polynomials. The method is illustrated by studying an initial value problem. IIIustrative examples are included to demonstrate the validity and applicability of the technique. The results obtained are compared to the known results.     

Semiconvergence of P-regular splittings for solving singular linear systems

We investigate necessary and sufficient conditions for semiconvergence of a splitting for solving singular linear systems, where the coefficient matrix A is a singular EP matrix. When A is a singular Hermitian matrix, necessary and sufficient conditions for semiconvergence of P-regular splittings are given, which generalize known results. As applications, the necessary and sufficient conditions for semiconvergence of block AOR and SSOR iterative methods are derived. A numerical example is given to illustrate the theoretical results. The work is supported by the National Natural Science Foundation of China under grant 10371056, the Foundation for the Authors of the National Excellent Doctoral Thesis Award of China under grant 200720 and the Natural Science Foundation of Jiangsu Province of China under grant BK2006725.     

Solutions to the generalized Sylvester matrix equations by a singular value decomposition

In this paper, solutions to the generalized Sylvester matrix equations AX-XF=BY and MXN-X=TY with A,M ∈R ,B,T∈R, F,N∈R and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n×(n + pr). The algorithm proposed in this paper for the euqation AX-XF = BY does not require the controllability of matrix pair (A,B) and the restriction that A,F don’t have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.     

Solutions to the generalized Sylvester matrix equations by a singular value decomposition

In this paper,solutions to the generalized Sylvester matrix equations AX-XF=BY and MXN-X=TY with A,M∈Rn×n,B,T∈Rn×n,F,N∈Rp×p and the matrices N,F being in companion form,are established by a singular value decomposition of a matrix with dimensions n×(n pr).The algorithm proposed in this paper for the euqation AX-XF=BY does not require the controllability of matrix pair(A,B)andthe restriction that A,F do not have common eigenvalues.Since singular value decomposition is adopted,the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations,and can perform important functions in many design problems in control systems theory.     

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.     

Artificial neural networks based modeling for solving Volterra integral equations system

Properly designing an artificial neural network is very important for achieving the optimal performance. This study aims to utilize an architecture of these networks together with the Taylor polynomials, to achieve the approximate solution of second kind linear Volterra integral equations system. For this purpose, first we substitute the Nth truncation of the Taylor expansion for unknown functions in the origin system. Then we apply the suggested neural net for adjusting the numerical coefficients of given expansions in resulting system. Consequently, the reported architecture using a learning algorithm that based on the gradient descent method, will adjust the coefficients in given Taylor series. The proposed method was illustrated by several examples with computer simulations. Subsequently, performance comparisons with other developed methods was made. The comparative experimental results showed that this approach is more effective and robust.     

Gradient-based maximal convergence rate iterative method for solving linear matrix equations

This paper is concerned with numerical solutions to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Gradient based iterative algorithm is proposed to approximate the exact solution. A necessary and sufficient condition guaranteeing the convergence of the algorithm is presented. A sufficient condition that is easy to compute is also given. The optimal convergence factor such that the convergence rate of the algorithm is maximized is established. The proposed approach not only gives a complete understanding on gradient based iterative algorithm for solving linear matrix equations, but can also be served as a bridge between linear system theory and numerical computing. Numerical example shows the effectiveness of the proposed approach.     

On the error analysis of the numerical solution of linear Volterra integral equations of the second kind

In the numerical solution of linear Volterra integral equations, two kinds of errors occur. If we use the collocation method, these errors are the collocation and numerical quadrature errors. Each error has its own effect in the accuracy of the obtained numerical solution. In this study we obtain an error bound that is sum of these two errors and using this error bound the relation between the smoothness of the kernel in the equation and also the length of the integration interval and each of these two errors are considered. Concluded results also are observed during the solution of some numerical examples.     

改进的求解线性方程组的并行Arnoldi方法

以Galerkin原理为基础,提出了求解循环块三对角线性方程组的并行算法。根据系数矩阵的稀疏性,选取适当的子空间的基,使算法不但不会发生中断,并从理论上证明了当系数矩阵对称正定时,该并行算法收敛。最后,在HP rx2600集群上进行的数值实验结果表明,该算法的并行效率很高,理论和实际计算相一致。     

A new artificial neural network structure for solving high-order linear fractional differential equations

Artificial neural networks afford great potential in learning and stability against small perturbations of input data. Using artificial intelligence techniques and modelling tools offers an ever-greater number of practical applications. In the present study, an iterative algorithm, which was based on the combination of a power series method and a neural network approach, was used to approximate a solution for high-order linear and ordinary differential equations. First, a suitable truncated series of the solution functions were substituted into the algorithm's equation. The problem considered here had a solution as a series expansion of an unknown function, and the proper implementation of an appropriate neural architecture led to an estimate of the unknown series coefficients. To prove the applicability of the concept, some illustrative examples were provided to demonstrate the precision and effectiveness of this method. Comparing the proposed methodology with other available traditional techniques showed that the present approach was highly accurate.     

An algorithm based on Q.I. modified splines for singular integral models

In this paper, we propose an algorithm supporting an approximation using quasi-interpolatory (q.i.) splines for the numerical solution of integro-differential equations with Cauchy singular kernel. Some different choices of parameters, defining the numerical model, are analyzed and compared in view of the algorithm efficiency.     

Fast algorithms for high-order numerical methods for space-fractional diffusion equations

In this paper, fast numerical methods for solving space-fractional diffusion equations are studied in two stages. Firstly, a fast direct solver for an implicit finite difference scheme proposed by Hao et al. [A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), pp. 787–805], which is fourth-order accurate in space and second-order accurate in time, is developed based on a circulant-and-skew-circulant (CS) representation of Toeplitz matrix inversion. Secondly, boundary value method with spatial discretization of Hao et al. [A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), pp. 787–805] is adopted to produce a numerical solution with higher order accuracy in time. Particularly, a method with fourth-order accuracy in both space and time can be achieved. GMRES method is employed for solving the discretized linear system with two preconditioners. Based on the CS representation of Toeplitz matrix inversion, the two preconditioners can be applied efficiently, and the convergence rate of the preconditioned GMRES method is proven to be fast. Numerical examples are given to support the theoretical analysis.     

块三对角线性方程组的并行直接解法

提出了分布式环境下求解块三对角线性方程组的一种并行算法,该算法充分利用系数矩阵结构的特殊性,通过对系数矩阵进行适当分解及近似处理,使算法只在相邻处理机间通信两次。并从理论上给出了算法有效的一个充分条件。最后,在HP rx2600集群上进行了数值实验,结果表明,实算与理论是一致的,并行性也很好。     

Iterative algorithm for minimal norm least squares solution to general linear matrix equations

This paper is concerned with minimal norm least squares solution to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Two iterative algorithms are proposed to solve this problem. The first method is based on the gradient search principle for solving optimization problem and the second one can be regarded as its dual form. For both algorithms, necessary and sufficient conditions guaranteeing the convergence of the algorithms are presented. The optimal step sizes such that the convergence rates of the algorithms are maximized are established in terms of the singular values of some coefficient matrix. It is believed that the proposed methods can perform important functions in many analysis and design problems in systems theory.