Burgers’ equation is a fundamental partial differential equation in fluid mechanics. This paper reports a new space-time spectral algorithm for obtaining an approximate solution for the space-time fractional Burgers’ equation (FBE) based on spectral shifted Legendre collocation (SLC) method in combination with the shifted Legendre operational matrix of fractional derivatives. The fractional der...

The problem of inversion the integral Laplace transform, which belongs to class ill-posed problems, is considered. Integral equations are reduced ill-conditioned systems linear algebraic (SLAE), in unknowns either expansion coefficients a series terms shifted Legendre polynomials, or approximate values desired original at number points. first step reduction SLAE apply quadrature formulas that p...

In this paper, two collocation methods based on the shifted Legendre polynomials are proposed for solving system of nonlinear Fredholm-Volterra integro-differential equations. The equation considered in paper involves derivative unknown functions integral term, which makes its numerical solution more complicated. We first introduce a single-step method interval [0, 1]. Next, multi-step version ...

-The discrete Legendre transform is compared to the discrete cosine transform (DCT), which is based on Chebyshev polynomials, in terms of image compression efficiency. Using standard test images in various image compression configurations, the DCT is found to perform marginally better than the discrete Legendre transform in all cases examined. A simplified fundamental matrix theory for construc...

A general framework is introduced to analyze the approximation properties of mapped Legendre polynomials and of interpolations based on mapped Legendre–Gauss–Lobatto points. Optimal error estimates featuring explicit expressions on the mapping parameters for several popular mappings are derived. These results not only play an important role in numerical analysis of mapped Legendre spectral and ...

These are notes on a preliminary follow-up to a question of Jim Propp, about cyclic sieving of cyclic codes. We show that two of the Mahonian polynomials are cyclic sieving polynomials for certain Dual Hamming Codes: X and X inv for q = 2, 3 and q = 2, respectively.

We give explicit formulas for the number of distinct elliptic curves over a finite field, up to isomorphism, in the families of Legendre, Jacobi, Hessian and Edwards curves.

For a graph G we denote by dG(u, v) the distance between vertices u and v in G, by dG(u) the degree of vertex u. The Hosoya polynomial of G is H(G) = ∑ {u,v}⊆V (G) x dG (u,v). For any positive numbers m and n, the partial Hosoya polynomials of G are Hm(G) = ∑ {u, v} ⊆ V (G) dG (u) = dG (v) = m xdG (u,v), Hmn(G) = ∑ {u, v} ⊆ V (G) dG (u) = m, dG (v) = n xdG (u,v). It has been shown that H(G1) − ...