urysohn integral equation is one of the most applicable topics in both pure and applied mathematics. the main objective of this paper is to solve the urysohn type fredholm integral equation. to do this, we approximate the solution of the problem by substituting a suitable truncated series of the well known legendre polynomials instead of the known function. after discretization of the problem o...

The Hosoya polynomial triangle is a triangular arrangement of polynomials where each entry is a product of two polynomials. The geometry of this triangle is a good 1 tool to study the algebraic properties of polynomial products. In particular, we find closed formulas for the alternating sum of products of polynomials such as Fibonacci polynomials, Chebyshev polynomials, Morgan-Voyce polynomials...

In this paper we apply hybrid functions of general block-pulse‎ ‎functions and Legendre polynomials for solving linear and‎ ‎nonlinear multi-order fractional differential equations (FDEs)‎. ‎Our approach is based on incorporating operational matrices of‎ ‎FDEs with hybrid functions that reduces the FDEs problems to‎ ‎the solution of algebraic systems‎. ‎Error estimate that verifies a‎ ‎converge...

In this paper a unified approach via block-pulse functions (BPFs) or shifted Legendre polynomials (SLPs) is presented to solve the linear-quadratic-Gaussian (LQG) control problem. Also a recursive algorithm is proposed to solve the above problem via BPFs. By using the elegant operational properties of orthogonal functions (BPFs or SLPs) these computationally attractive algorithms are developed....

We construct an orthogonal wavelet basis for the interval using a linear combination of Legendre polynomial functions. The coefficients are taken as appropriate roots of Chebyshev polynomials of the second kind, as has been proposed in reference [1]. A multi-resolution analysis is implemented and illustrated with analytical data and real-life signals from turbulent flow fields.

Abstract. An O(N2) algorithm for the convolution of compactly supported Legendre series is described. The algorithm is derived from the convolution theorem for Legendre polynomials and the recurrence relation satisfied by spherical Bessel functions. Combining with previous work yields an O(N2) algorithm for the convolution of Chebyshev series. Numerical results are presented to demonstrate the ...