%0 Journal Article
%T Solving Operator Equation Based on Expansion Approach
%A A. Aminataei
%A S. Ahmadi-Asl
%A M. Pakbaz
%J International Journal of Computational Mathematics
%D 2014
%R 10.1155/2014/671965
%X To date, researchers usually use spectral and pseudospectral methods for only numerical approximation of ordinary and partial differential equations and also based on polynomial basis. But the principal importance of this paper is to develop the expansion approach based on general basis functions (in particular case polynomial basis) for solving general operator equations, wherein the particular cases of our development are integral equations, ordinary differential equations, difference equations, partial differential equations, and fractional differential equations. In other words, this paper presents the expansion approach for solving general operator equations in the form , with respect to boundary condition , where , and are linear, nonlinear, and boundary operators, respectively, related to a suitable Hilbert space, is the domain of approximation, is an arbitrary constant, and is an arbitrary function. Also the other importance of this paper is to introduce the general version of pseudospectral method based on general interpolation problem. Finally some experiments show the accuracy of our development and the error analysis is presented in norm. 1. Introduction Approximation theory is an important field of mathematics which has pure and applied aspects. This field tries to approximate the complicated functions with simplified representments. Some important branches of this field are interpolation, extrapolation, best approximation, and so forth. Also, expansion series have high applications in this field (approximation theory). Expansion series try to represent a function using suitable basis functions such as standard polynomials (nonperiodic case) and trigonometric polynomials (periodic case). Taylor series and Fourier series in classical and nonclassical forms are two high applicable versions of expansions. These series are used in the numerical solution of ordinary, partial differential, and integral equations. Weighted residual methods are classes of methods which use expansion series for solving ordinary, partial differential, and integral equations. In these methods, by substituting a suitable expansion of exact solution of ordinary, partial differential, and integral equations in its equation, the residuals are obtained and then the residuals are minimized in a certain way. This minimization leads to specific methods such as Galerkin, collocation, and Tau formulations. In this paper, using Fourier series and interpolation cases as expansions, we implement the spectral and pseudospectral methods for solving the general operator equation ,
%U http://www.hindawi.com/journals/ijcm/2014/671965/