On the approximation of particular solution of nonhomogeneous linear differential equation with Legendre series

Abstract

In this study, we propose an approximation method for particular solutions of the nonhomogeneous second-order differential equations by truncated Legendre series. Particulary, the govern problem is a linear differential equation with constant coefficients. The choice of series solutions depends upon the complementary solutions and the approximate nonhomogeneous terms. An upper bound for the approximation error is formulated. Some examples are presented to demonstrate the validity of the proposed method.Keywords: nonhomogeneous differential equations, particular solutions, Legendre polynomials, series, error bound IntroductionMany problems in natural science such as physics, engineering and mathematical modeling are governed by differential equations (Jung et al., 2014). Solving such equations will lead to understanding the behaviors of the systems. Although solutions are known to be exist, there is an only few problems that can be solved for analytic solution. Several attempts are devoted to numerical method or approximation techniques to obtain the high accuracy of approximations (Jung et al., 2014).Taylor series and orthogonal functions such as Chebyshev and Legendre polynomials are powerful tools for functions approximations in terms of polynomials (Gulsu et al., 2006; Wang and Xiang, 2012;Patanarapeelert and Varnasavang, 2013). As a by-product they can be used for approximating the solution of ordinary differential equations. Sezer and Gulsu (2010) proposed a numerical method based on the hybrid Legendre and Taylor polynomials for solving the high-order linear differential equations. Olagunju and Olaninejum (2012) formulated a trial solution for nonhomogeneous differential equations where Legendre polynomials are used as basis functions. Recently, Jung et al. (2014) proposed the method to solutions of second-order differential equations by using Tau method based on Legendre operational matrix.In this paper, we present a method for approximating the particular solutions of nonhomogenous linear second-order differential equations. Rather than approximating as a whole we focus on in part, approximating particular solution by which the complementary solution is prior known. In doing so, we transfer the original problem into an approximate one