Abstract. It is proved that the eigenvalues of the Jacobi Tau method for the second derivative operator with Dirichlet boundary conditions are real, negative and distinct for a range of the Jacobi parameters. Special emphasis is placed on the symmetric case of the Gegenbauer Tau method where the range of parameters included in the theorems can be extended and characteristic polynomials given by...

Abstract Many physical phenomena can be modelled through nonlocal boundary value problems whose conditions involve integral terms. In this work we propose a numerical algorithm, by combining second-order Crank–Nicolson schema for the temporal discretization and Legendre–Chebyshev pseudo-spectral method (LC–PSM) space discretization, to solve class of parabolic integrodifferential equations subj...

Common practice is to recommend evaluation of polynomials by Horner’s rule. Here’s an example where it is fast but doesn’t work nearly as accurately as another fairly easy method. Can a method for Legendre polynomials be both fast and accurate? 1 1 Legendre Polynomials A substantial literature has grown up around the uses for orthonormal polynomials. Here we look at the example of Legendre poly...

The fractional inverse M (real γ > 0) of a matrix M is expanded in a series of Gegenbauer polynomials. If the spectrum of M is confined to an ellipse not including the origin, convergence is exponential, with the same rate as for Chebyshev inversion. The approximants can be improved recursively and lead to an iterative solver for Mx = b in Krylov space. In case of γ = 1/2, the expansion is in t...

A Maple algorithm for the computation of the zeros of orthogonal polynomials (OPs) and special functions (SFs) in a given interval [x1, x2] is presented. The program combines symbolic and numerical calculations and it is based on fixed point iterations. The program uses as inputs the analytic expressions for the coefficients of the three-term recurrence relation and a differencedifferential rel...

We construct an orthogonal wavelet basis for the interval using a linear combination of Legendre polynomials. The coefficients are taken as appropriate roots of Chebyshev polynomials of the second kind. The one-dimensional transform is applied to analytical data and appropriate definitions of a scalogram as well as local and global spectra are presented. The transform is then extended to the mu...

Let μ be a probability measure on the real line with finite moments of all orders. Apply the Gram-Schmidt orthogonalization process to the system {1, x, x, . . . , xn, . . . } to get orthogonal polynomials Pn(x), n ≥ 0, which have leading coefficient 1 and satisfy (x − αn)Pn(x) = Pn+1(x) + ωnPn−1(x). In general it is almost impossible to use this process to compute the explicit form of these po...

In this work, the convection-diffusion integro-differential equation with a weakly singular kernel is discussed. The  Legendre spectral tau method is introduced for finding the unknown function. The proposed method is based on expanding the approximate solution as the elements of a shifted Legendre polynomials. We reduce the problem to a set of algebraic equations by using operational matrices....

We introduce a new and efficient Chebyshev-Legendre Galerkin method for elliptic problems. The new method is based on a Legendre-Galerkin formulation, but only the Chebyshev-Gauss-Lobatto points are used in the computation. Hence, it enjoys advantages of both the LegendreGalerkin and Chebyshev-Galerkin methods.