‎This article develops a direct method for solving numerically‎ ‎multi delay-fractional differential and integro-differential equations‎. ‎A Galerkin method based on Legendre polynomials is implemented for solving‎ ‎linear and nonlinear of equations‎. ‎The main characteristic behind this approach is that it reduces such problems to those of‎ ‎solving a system of algebraic equations‎. ‎A conver...

In this paper an accurate method to construct the bidiagonal factorization of collocation and Wronskian matrices Jacobi polynomials is obtained used compute with high relative accuracy their eigenvalues, singular values inverses. The particular cases Legendre polynomials, Gegenbauer Chebyshev first second kind rational are considered. Numerical examples included.

Keywords: Trigonometric functions Hurwitz zeta function Legendre chi function Lerch zeta function Bernoulli polynomials Euler polynomials a b s t r a c t In this sequel to our recent note [D. Cvijović, Values of the derivatives of the cotangent at rational multiples of π, Appl. Math. Lett. it is shown, in a unified manner, by making use of some basic properties of certain special functions, suc...

The parameters of experimentally obtained exponentials are usually found by least-squares fitting methods. Essentially, this is done by minimizing the mean squares sum of the differences between the data, most often a function of time, and a parameter-defined model function. Here we delineate a novel method where the noisy data are represented and analyzed in the space of Legendre polynomials. ...

Motivated by an expression Persson and Strang on integral involving Legendre polynomials, stating that the square of $P_{2n+1}(x)/x$ integrated over $[-1,1]$ is always $2$, we present analog results for Hermite, Chebyshev, Laguerre Gegenbauer polynomials as well original polynomial with even index.

We analyze transient heat conduction in a thick functionally graded plate by using a higher-order plate theory and a meshless local Petrov-Galerkin (MLPG) method. The temperature field is expanded in the thickness direction by using Legendre polynomials as basis functions. For temperature prescribed on one or both major surfaces of the plate, modified Lagrange polynomials are used as basis and ...

Abstract. Motivated by questions on the preconditioning of spectral methods, and independently of the extensive literature on the approximation of zeroes of orthogonal polynomials, either by the Sturm method, or by the descent method, we develop a stationary phase-like technique for calculating asymptotics of Legendre polynomials. The difference with the classical stationary phase method is tha...

The recent numerical implementation by Fornberg and collaborators of the so-called unified method to linear elliptic PDEs in polygonal domains involves the computation of the finite Fourier transform of the Legendre polynomials. A variation of this approach, introduced by two of the authors, also involves the same computation. Here, instead of expressing the finite Fourier transform of the Lege...

In this paper we describe an efficient algorithm for computing the potentials of the form r−λ where λ ≥ 1. This treecode algorithm uses spherical harmonics to compute multipole coefficients that are used to evaluate these potentials. The key idea in this algorithm is the use of Gegenbauer polynomials to represent r−λ in a manner analogous to the use of Legendre polynomials for the expansion of ...

The factorization of the Legendre polynomial of degree (p− e)/4, where p is an odd prime, is studied over the finite field Fp. It is shown that this factorization encodes information about the supersingular elliptic curves in Legendre normal form which admit the endomorphism √ −2p, by proving an analogue of Deuring’s theorem on supersingular curves with multiplier √ −p. This is used to count th...