An exponential matrix method for solving systems of linear differential equations

Abstract

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.

Exponential spline for the numerical solutions of linear Fredholm integro-differential equations

“…Moreover, for second-order impulsive integro-differential equations, a class of three-point boundary value problems in Banach space have been developed in [19]. Yüzbaşi et al in [50][51][52][53][54][55][56] used the non-polynomial functions to solve differential equations that have been based on non-polynomial functions set {1, e -t , e -2t , . .…”

An exponential method to solve linear Fredholm–Volterra integro-differential equations and residual improvement

“…In recent years, integro-differential equations have solved semianalytical methods such as the homotopy perturbation method [7,26], the Taylor collocation method [11], the Haar functions method, [14,15], He's variational iteration technique [8], the power series method [24], the Chebyshev technique [22], the Legendre-spectral method [9], the Tau method [20], the Legendre multiwavelets method [13], the finite-difference scheme [5], the variational iteration method [21], the CAS wavelet operational matrix method [3], the trigonometric wavelets method [12], the Legendre matrix method [25], the Taylor polynomial approach [18], the Adomian method [1], the differential transformation method [4], the Galerkin method [16], the Bessel matrix method [30], the Legendre collocation method [32], the improved homotopy perturbation method [27], the modified homotopy perturbation method [10], and the moving least square method [6,17]. Yübaşı and Sezer [31] gave a matrix method based on exponential polynomials for solutions of systems of differential equations.…”

Exponential Collocation Method for Solutions of Singularly Perturbed Delay Differential Equations

“…Recently, Yüzbaşı and Sezer have studied the exponential polynomial solutions of the systems of linear differential equations in [26].…”