%0 Journal Article
%T Direct Computation of Operational Matrices for Polynomial Bases
%A Osvaldo Guimar£¿es
%A Jos¨¦ Roberto C. Piqueira
%A Marcio Lobo Netto
%J Mathematical Problems in Engineering
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/139198
%X Several numerical methods for boundary value problems use integral and differential operational matrices, expressed in polynomial bases in a Hilbert space of functions. This work presents a sequence of matrix operations allowing a direct computation of operational matrices for polynomial bases, orthogonal or not, starting with any previously known reference matrix. Furthermore, it shows how to obtain the reference matrix for a chosen polynomial base. The results presented here can be applied not only for integration and differentiation, but also for any linear operation. 1. Introduction One of the main characteristics of the use of polynomial bases is to reduce the solving process of differential or integral equations to systems of algebraic equations, expressing the solution by truncated series approximations, up to order [1¨C4], such that The choice of the polynomial basis is normally one of the orthogonal bases belonging to the Hilbert space of functions, in order to ensure that the expansion of the series to a higher order does not affect the coefficients previously calculated, being applicable to classical methods, as the Runge-Kutta, for instance [5]. However, it is also possible to use nonorthogonal bases, as done in [6, 7], where ,£¿£¿ Considering the line vector that contains the coefficients and the column vector that contains the base polynomials , expression (1.1) can be written as considering: [8]. The central idea when working with operational matrices is to write the integral or differential of the elements of the basis as a linear combination of the same base elements, transforming the integral and differential operations of into matrix operations in a Hilbert space [8]. So, defining as the operational integration matrix and as the operational differential matrix, it is possible to obtain the line vector containing the coefficients of the series that represent the integrated function or the differentiated function by and . Consequently, Recently, Doha and Bhrawy [1] presented a method to obtain the operational matrices of integration considering the Jacobi polynomials. Here, a simpler and more direct way to get the operational matrix, by using Theorem 2.1 from the next section is presented. Additionally, a way to extend it to any polynomial basis, by using Theorem 3.1, presented in Section 3, is also developed. In spite of the fact that those theorems are applied to integration and differentiation operations, the result is valid to any linear operation, as shown ahead. 2. Obtaining the Operational Matrix Theorem 2.1. Considering a square
%U http://www.hindawi.com/journals/mpe/2010/139198/