Anna Napoli

New Formulas Involving Fibonacci and Certain Orthogonal Polynomials

Symmetry

In this paper, new formulas for the Fibonacci polynomials, including high-order derivatives and r... more In this paper, new formulas for the Fibonacci polynomials, including high-order derivatives and repeated integrals of them, are derived in terms of the polynomials themselves. The results are then used to solve connection problems between the Fibonacci and orthogonal polynomials. The inverse cases are also studied. Finally, new results for the linear products of the Fibonacci and orthogonal polynomials are determined using the earlier result for the moments formula of Fibonacci polynomials.

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Matrix Calculus-Based Approach to Orthogonal Polynomial Sequences

Mediterranean Journal of Mathematics, Jun 25, 2020

In this paper, an approach to orthogonal polynomials based on matrix calculus is proposed. Known ... more In this paper, an approach to orthogonal polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations and determinant forms. New algorithms, similar, but not identical, to the Chebyshev one, for practical calculation of the polynomials are presented. The cases of monic and symmetric orthogonal polynomial sequences and the case of orthonormal polynomial sequences have been considered. Some classical and non-classical examples are given. The work is framed in a broader perspective, already started by the authors. It provides for the determination of properties of a general sequence of polynomials and, therefore, their applicability to special classes of the most important polynomials.

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A global method for approximation of the Cauchy differential problem by Chebyshev polynomials of the second kind

Polynomial Sequences and Their Applications

Mathematics, Dec 16, 2022

A new spectral method for a class of linear boundary value problems

Journal of Computational and Applied Mathematics, 2016

In this paper we use Bernoulli polynomials to derive a new spectral method to find the numerical ... more In this paper we use Bernoulli polynomials to derive a new spectral method to find the numerical solutions of second order linear boundary value problems. The proposed method has been applied to several BVPs with Dirichlet, Neumann and mixed types boundary conditions. Numerical examples provide favorable comparisons with other existing methods.

Collocation for high order differential equations with two-points Hermite boundary conditions

Applied Numerical Mathematics, 2015

ABSTRACT For the numerical solution of high even order differential equations with two-points Her... more ABSTRACT For the numerical solution of high even order differential equations with two-points Hermite boundary conditions a general collocation method is derived and studied. Computation of the integrals which appear in the coefficients are generated by a recurrence formula and no integrals are involved in the calculation. An application to the solution of the beam problem is given. Numerical experiments provide favorable comparisons with other existing methods.

A multipoint Birkhoff type boundary value problem

Journal of Numerical Mathematics, 2015

A multipoint boundary value problem is considered. The existence and uniqueness of solution is pr... more

Polynomial sequences: elementary basic methods and application hints. A survey

Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, May 2, 2019

The present paper contains a small survey on the principal and elementary methods for a systemati... more The present paper contains a small survey on the principal and elementary methods for a systematic study of general polynomial sequences. For the set of polynomial sequences the algebraic structure of a group is given. Matricial forms, recurrence relations and conjugate sequences of polynomials are examined. For every element of a polynomial sequence determinant forms are determined by suitable Hessenberg matrices. Generating functions and derivation matrix are derived. Then, associated differential operator and Sheffer classification are considered. As an application, a general interpolation problem is hinted at. For a real valued function, the generalized Taylor polynomial and series are given. As an illustrative example, the shifted (with respect to the degree) Fibonacci polynomial sequence is considered.

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Recurrence relations and determinant forms for general polynomial sequences. Application to Genocchi polynomials

Integral Transforms and Special Functions, Oct 23, 2018

In this paper the set of general polynomial sequences is considered. An elementary systematic app... more In this paper the set of general polynomial sequences is considered. An elementary systematic approach is proposed. In fact a structure of group is given and for every element of this group recurrence relations and determinant forms are derived. Applications of the derived determinant forms are considered. In particular, the general linear interpolation and bounds of the zeros of each polynomial of the sequence are sketched. Finally, as an illustrative example, the shifted (with respect to the degree) Genocchi polynomial sequence is analyzed.

A class of Birkhoff–Lagrange-collocation methods for high order boundary value problems

Applied Numerical Mathematics, Jun 1, 2017

A general procedure to determine collocation methods for high order boundary value problems is pr... more A general procedure to determine collocation methods for high order boundary value problems is presented. These methods provide globally continuous differentiable solution in the form of polynomial functions and also numerical solution on a set of discrete points. Some special cases are considered. Numerical experiments are presented which support theoretical results and provide favorable comparisons with other existing methods.

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Solving IVPs and BVPs Using Taylor Series by Mathematica

A Matlab code for general second order boundary value problems

A Mathematica package for the solution of Ordinary Differential Equations using Taylor Series method

Economical Runge-Kutta Methods with Weak Second Order for Stochastic Differential Equations

Explicit economical three-stage Runge-Kutta methods for the numerical solution of stochastic diff... more

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A method for global approximation of the Cauchy differential problem by Chebyshev polynomials of first kind

A Mathematica package for the solution of ordinary differential equations using Taylor series method

On a class of stochastic Runge Kutta methods

For the numerical solution of stochastic differential equations explicit economical three-stage R... more

Il concetto di integrale con supporto di MatCos

Economical Runge-Kutta Methods with Weak Second Order for Stochastic Differential Equations

Int. J. Contemp. Math. Sciences, 2010

Explicit economical three-stage Runge-Kutta methods for the numerical solution of stochastic diff... more

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A method for global approximation of the Cauchy differential problem

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