A solution method for a nonlocal problem for a system of linear differential equations

For a system of linear ordinary differential equations supplemented by a linear nonlocal condition specified by the Stieltjes integral, a solution method is examined. Unlike the familiar methods for solving problems of this type, the proposed method does not use any specially chosen auxiliary boundary conditions. This method is numerically stable if the original problem is numerically stable.     

Self-correcting approximate solution by the iterative method for linear and nonlinear stochastic differential equations

The iterative method of the first author makes possible a self-correcting scheme for the approximate solution obtained. This scheme accelerates the convergence and substantially decreases the number of terms necessary in computation for applications involving either linear or nonlinear stochastic systems. A “feedback” term compensates for the approximation of the system inverse operator by a partial sum. Further, errors are determined in calculating the mean solution 〈y〉 and the correlation $\text{R}{}^{\mathrm{y}}\text{(t}{}^1\text{,t}{}^2\text{) = 〈y(t}{}^1\text{)}\text{y}\text{1}\text{(t}{}^2\text{)〉}$ by using the approximations 〈φ_n〉 and 〈φⁿ(t¹)φⁿ(t²)〉 , where φ_n represents n terms of the solution by the iterative method for stochastic differential equations.     

A polynomial method of approximate centers for linear programming

We present a path-following algorithm for the linear programming problem with a surprisingly simple and elegant proof of its polynomial behaviour. This is done both for the problem in standard form and for its dual problem. We also discuss some implementation strategies.This author completed this work under the support of the research grant No. 1467086 of the Fonds National Suisses de la Recherche Scientifique.     

An exponential matrix method for solving systems of linear differential equations

This paper presents an exponential matrix method for the solutions of systems of high‐order linear differential equations with variable coefficients. The problem is considered with the mixed conditions. On the basis of the method, the matrix forms of exponential functions and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown coefficients are determined and thus the approximate solutions are obtained. Also, an error estimation based on the residual functions is presented for the method. The approximate solutions are improved by using this error estimation. To demonstrate the efficiency of the method, some numerical examples are given and the comparisons are made with the results of other methods. Copyright © 2012 John Wiley & Sons, Ltd.     

A note on the use of Adomian decomposition method for high-order and system of nonlinear differential equations

This paper extends an earlier work [Hosseini MM, Nasabzadeh H. Modified Adomian decomposition method for specific second order ordinary differential equations. Appl Math Comput 2007;186:117–23] to high order and system of differential equations. Solution of these problems is considered by proposed modification of Adomian decomposition method. Furthermore, with providing some examples, the aforementioned cases are dealt with numerically.     

On conjugacy of high-order linear ordinary differential equations

It is shown that the differential equation

A Hermite collocation method for the approximate solutions of high‐order linear Fredholm integro‐differential equations

In this study, a Hermite matrix method is presented to solve high‐order linear Fredholm integro‐differential equations with variable coefficients under the mixed conditions in terms of the Hermite polynomials. The proposed method converts the equation and its conditions to matrix equations, which correspond to a system of linear algebraic equations with unknown Hermite coefficients, by means of collocation points on a finite interval. Then, by solving the matrix equation, the Hermite coefficients and the polynomial approach are obtained. Also, examples that illustrate the pertinent features of the method are presented; the accuracy of the solutions and the error analysis are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1707–1721, 2011     

Sensitivity analysis of the differential matrix Riccati equation based on the associated linear differential system

Perturbation bounds are given for the solution of the nth order differential matrix Riccati equation using the associated linear 2nth order differential system. The new bounds are alternative to those existing in the literature and are sharper in some cases. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Taylor matrix method for the solution of a two-dimensional linear hyperbolic equation

A Taylor matrix method is proposed for the numerical solution of the two-space-dimensional linear hyperbolic equation. This method transforms the equation into a matrix equation and the unknown of this equation is a Taylor coefficients matrix. Solutions are easily acquired by using this matrix equation, which corresponds to a system of linear algebraic equations. As a result, the finite Taylor series approach with three variables is obtained. The accuracy of the proposed method is demonstrated with one example.     

An optimum iterative method for solving any linear system with a square matrix

A method is presented to solveAx=b by computing optimum iteration parameters for Richardson's method. It requires some information on the location of the eigenvalues ofA. The algorithm yields parameters well-suited for matrices for which Chebyshev parameters are not appropriate. It therefore supplements the Manteuffel algorithm, developed for the Chebyshev case. Numerical examples are described.     

A multistage differential transformation method for approximate solution of Hantavirus infection model

The goal of this study is presented a reliable algorithm based on the standard differential transformation method (DTM), which is called the multi-stage differential transformation method (MsDTM) for solving Hantavirus infection model. The results obtanied by using MsDTM are compared to those obtained by using the Runge-Kutta method (R-K-method). The proposed technique is a hopeful tool to solving for a long time intervals in this kind of systems.     

An approximate method for integration of ordinary differential equations

An approximate analytic method of solving a Cauchy problem for normal systems of ordinary differential equations is considered. The method is based on the approximation of the solution by partial sums of shifted Chebyshev series. The coefficients of the series are determined by an iterative process using Markov quadrature formulas.     

A new differential equation method for finding the Perron root of a positive matrix

A basic problem in linear algebra is the determination of the largest eigenvalue (Perron root) of a positive matrix. In the present paper a new differential equation method for finding the Perron root is given. The method utilizes the initial value differential system developed in a companion paper for individually tracking the eigenvalue and corresponding right eigenvector of a parametrized matrix.