In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of second kind with a weakly singular kernel. We use some function transformation and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1, 1], so that the solution of the new equation possesses better regularity and the Jacobi or...

The recent interest in isolating real roots of polynomials has revived interest in computing sharp upper bounds on the values of the positive roots of polynomials. Until now Cauchy’s method was the only one widely used in this process. Ştefănescu’s recently published theorem offers an alternative, but unfortunately is of limited applicability as it works only when there is an even number of sig...

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 ...

We first give a method to get multidimensional Leja sequences by considering intertwining from one-dimensional ones. An application is the existence of explicit for closed unit polydisc. Next, we deal with some applications in bidimensional Lagrange interpolation sequences. These results also require an formula associated fundamental polynomials uniform estimates.

Fuzzy Hermite interpolation of 5th degree generalizes Lagrange interpolation by fitting a polynomial to a function f that not only interpolates f at each knot but also interpolates two number of consecutive Generalized Hukuhara derivatives of f at each knot. The provided solution for the 5th degree fuzzy Hermite interpolation problem in this paper is based on cardinal basis functions linear com...

A t-out-of-n secret-sharing scheme allows an honest dealer D to distribute a secret s among n players, such that any subset of t players has no information about s, but every set of t + 1 players can collaboratively reconstruct the secret. The most famous secret-sharing scheme is Shamir’s Sharing Scheme [Sha79] (cf. Section 7.1.2). It uses polynomials to obtain the desired properties. Before pr...

Extending the rigorous presentation of the “classical umbral calculus” [28], the so-called partition polynomials are interpreted with the aim to point out the umbral nature of the Poisson random variables. Among the new umbrae introduced, the main tool is the partition umbra that leads also to a simple expression of the functional composition of the exponential power series. Moreover a new shor...

We seek pointwise error estimates for interpolants, on scattered data, constructed using a basis of conditionally positive deenite functions of order m, and polynomials of degree not exceeding m-1. Two diierent approaches to the analysis of such interpolation are considered. The former uses distributions and reproducing kernel ideas, whilst the latter is based on a Lagrange function approach. E...

We present formulas for computations involving companion matrix pencils as may arise in considering polynomial eigenvalue problems. In particular, we provide explicit companion matrix pencils for matrix polynomials expressed in a variety of polynomial bases including monomial, orthogonal, Newton, Lagrange, and Bernstein/Bézier bases. Additionally, we give a pair of explicit LU factors associate...