Maria Carmela De Bonis

University of Basilicata | UniBas · Department of Mathematics, Computer Science and Economics

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α(Dαf)(y)=1Γ(m−α)∫0y(y−x)m−α−1f(m)(x)dx,y>0 with m−1<α≤m,m∈N. The numerical procedure is based on approximating f(m) by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitable chosen t...

This paper deals with the global approximation of the solutions of Boundary Value Problems (BVPs) of second order on the real line. We first reduce the BVP to an equivalent Fredholm integral equation of the second kind and then approximate its solution by a Nyström type method based on a suitable product quadrature rule. Such quadrature formula is...

This paper deals with a quadrature rule for the numerical evaluation of hypersingular integrals of highly oscillatory functions on the positive semiaxis. The rule is of product type and consists in approximating the density function f by a truncated interpolation process based on the zeros of generalized Laguerre polynomials and an additional point...

In this paper we consider a generalized metastatic tumor growth model that describes the primary tumor growth by means of an Ordinary Differential Equation (ODE) and the evolution of the metastatic density using a transport Partial Differential Equation (PDE). The numerical method is based on the resolution of a linear Volterra integral equation (V...

In order to solve Prandtl-type equations we propose a collocation-quadrature method based on de la Vallée Poussin (briefly VP) filtered interpolation at Chebyshev nodes. Uniform convergence and stability are proved in a couple of Hölder-Zygmund spaces of locally continuous functions. With respect to classical methods based on Lagrange interpolation...

A Nyström method for linear second kind Volterra integral equations on unbounded intervals, with sufficiently smooth kernels, is described. The procedure is based on the use of a truncated Lagrange interpolation process and of a truncated Gaussian quadrature formula. The stability and the convergence of the method in suitable weighted spaces of fun...

The paper deals with an integral equation arising from a problem in mathematical biology. We propose approximating its solution by Nyström methods based on Gaussian rules and on product integration rules according to the smoothness of the kernel function. In particular, when the latter is weakly singular we propose two Nyström methods constructed b...

The paper deals with the approximate solution of integro-differential equations of Prandtl’s type. Quadrature methods involving “optimal” Lagrange interpolation processes are proposed and conditions under which they are stable and convergent in suitable weighted spaces of continuous functions are proved. The efficiency of the method has been tested...

In order to solve Prandtl-type equations we propose a collocation-quadrature method based on VP filtered interpolation at Chebyshev nodes. Uniform convergence and stability are proved in a couple of Holder - Zygmund spaces of locally continuous functions. With respect to classical methods based on Lagrange interpolation at the same collocation node...

The paper deals with the approximate solution of integro-differential equations of Prandtl's type. Quadrature methods involving ``optimal'' Lagrange interpolation processes are proposed and conditions under which they are stable and convergent in suitable weighted spaces of continuous functions are proved. The efficiency of the method has been test...

In the present paper we propose a product integration rule for hypersingular integrals on the positive semi-axis. The rule is based on an approximation of the density function f by a suitable truncated Lagrange polynomial. We discuss theoretical aspects by proving stability and convergence of the procedure for density functions f belonging to weigh...

This paper deals with the numerical solution of second kind integral equations with fixed singularities of Mellin convolution type. The main difficulty in solving such equations is the proof of the stability of the chosen numerical method, being the noncompactness of the Mellin integral operator the chief theoretical barrier. Here, we propose a Nys...

La teoria dell’approssimazione è un capitolo della Matematica presente in diversi contesti scientifici e in Analisi numerica costituisce la base teorica per la costruzione di varie procedure stabili e convergenti. Il volume, rivolto agli studenti della laurea magistrale in Matematica e agli studiosi della disciplina, tratta l’approssimazione trigon...

This paper deals with a special Hermite–Fejér interpolation process based at the zeros of generalized Freud polynomials which are orthogonal with respect to the weight w(x)=|x|αe-|x|β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackag...

In the present paper we consider hypersingular integrals of the following type (Formula presented) where the integral is understood in the Hadamard finite part sense, p is a positive integer, w α (x) = e −x x α is a Laguerre weight of parameter α ≥ 0 and t > 0. In [6] we proposed an efficient numerical algorithm for approximating (1), focusing our...

We consider integral equations of the second kind with fixed singularities of Mellin type. According to the behavior of the Mellin kernel, we first determine suitable weighted Lp spaces where we look for the solution. Then, for its approximation, we propose a numerical method of Nyström type based on a Gauss–Jacobi quadratura formula. Actually, a s...

The authors propose a numerical method for computing Hilbert and Hadamard transforms on (0;\infty) by a simultaneus approximation process involving a suitable Lagrange polynomial of degree s and " truncated" Gaussian rule of order m, with s<< m. The proposed procedure is convergent and pointwise error estimates are given. Finally, some numerical te...

In this paper we propose some different strategies to approximate hypersingular integrals on the real semiaxis. Hadamard Finite Part integrals (shortly FP integrals), regarded as p th derivative of Cauchy principal value integrals, are of interest in the solution of hypersingular BIE, which model many different kind of Physical and Engineering prob...

In this paper we propose a global method to approximate the derivatives of the weighted Hilbert transform of a given function f Hp(fw?,t)=dpdtp?0+?f(x)x?tw?(x)dx=p!?0+?f(x)(x?t)p+1w?(x)dx, where p?{1,2,?}, t>0, and w?(x)=e?xx? is a Laguerre weight. The right-hand integral is defined as the finite part in the Hadamard sense. The proposed numerical a...

In the present paper the authors propose two numerical methods to approximate Hadamard transforms on the real line. One of the procedures employed here is based on a simple tool like the “truncated” Gaussian rule conveniently modified to remove numerical cancellation and overflow phenomena. The second approach is a process of simultaneous approxima...

The aim of this paper is to propose a new modified Nyström method for the approximation of the solutions of second kind integral equations with fixed singularities of Mellin convolution type. The stability and the convergence are proved in L^2 spaces and error estimates in L^2 norm are given. Finally, numerical tests showing the effectiveness of th...

In this paper the author extends the mapping properties of some singular integral operators in Zygmund spaces equipped with uniform norm. As a by-product quadrature methods for solving CSIE having indices 0 and 1 are proposed. Their stability and convergence are proved and error estimates in Zygmund norm are given. Some numerical tests are also sho...

In this paper the authors propose a Nystr\"om method based on a ``truncated" Gaussian rule to solve systems of Fredholm integral equations on the real line. They prove that it is stable and convergent and that the matrices of the solved linear systems are well-conditioned. Moreover, they give error estimates in weighted uniform norm and show some n...

This paper deals with the numerical solution of a class of systems of Cauchy singular integral equations with constant coefficients. The proposed procedure consists of two basic steps: the first one is to consider a modified problem equivalent to the original one under suitable conditions, the second one is to approximate its solution by means of a...

The aim of this paper is to propose a numerical method approximating the solutions of a system of CSIE. The stability and the convergence of the method are proved in weighted L2 spaces. An application to the numerical resolution of CSIE on curves is also given. Finally, some numerical tests confirming the error estimates are shown.

We study the behavior of some “truncated” Gaussian rules based on the zeros of Pollaczek-type polynomials. These formulas are stable and converge with the order of the best polynomial approximation in suitable function spaces. Moreover, we apply these results to the related Lagrange interpolation process and to prove the stability and the convergen...

� In this papers the authors propose two projection methods to solve CSIE having smooth or weakly singular kernels. They prove their stability and convergence in Zygmund spaces equipped with uniform norm. Some numerical examples illustrating the accuracy of the methods are given.

In this paper, the authors introduce a quadrature rule to approximate integrals of highly oscillatory functions. They prove that it is stable and convergent. Moreover they give applications to the approximation of integrals of singular highly oscillatory functions on bounded and unbounded intervals. Finally they show some numerical tests in which t...

In this paper, the authors propose a Nyström method to approximate the solutions of Cauchy singular integral equations with constant coefficients having a negative index. They consider the equations in spaces of continuous functions with weighted uniform norm. They prove the stability and the convergence of the method and show some numerical tests...

In this paper, the authors introduce a Nyström method for solving systems of Fredholm integral equations on the real semiaxis. They prove that the method is stable and convergent. Moreover, they show some numerical tests that confirm the error estimates. Finally, they discuss a specific application to an inverse scattering problem for the Schröding...

In this paper the authors propose numerical methods to approximate the solutions of systems of second kind Fredholm integral equations. They prove that such methods are stable and convergent. Error estimates in weighted Lp norm, 1<=�p<=� +∞, are given and some numerical tests are shown.

We study mapping properties, boundedness and invertibility of some Cauchy singular integral operators in a scale of pairs of Besov type subspaces of C(−1, 1). Our results include all those already available in the literature. Mathematics Subject Classification (2000). Primary 44A15; Secondary 41A10.

In this paper the authors propose a numerical method for the approximate solution of some classes of Fredholm and Cauchy integral equations including the “discrete collocation” and “collocation” methods.

The authors propose a simple numerical method to approximate the solution of CSIE.The convergence and the stability of the procedure are proved and some numerical examples are shown.

The authors propose a numerical method to approximate the solu- tions of particular Cauchy singular integral equations (CSIE). It is based on interpolation processes and it is stable and convergent. Error estimates and numerical tests are shown.

The authors propose some simple algorithms for computing the Cauchy principal value integrals on the real line, using zeros of Markov-Sonin polynomials. Error estimates are proved and some numerical tests are shown.

The authors estimate the error of best polynomial approximation in Lp-weighted spaces with weights having zeros inside [−1, 1], using a suit- able modulus of smoothness. The Jackson and Stechkin inequalities are given. Moreover some estimates of the derivatives of the polynomials of best approx- imation are proved.

The authors propose two new algorithms for the computation of Cauchy principal value integrals on the real semiaxis. The proposed quadrature rules use zeros of Laguerre polynomials. Theoretical error estimates are proved and some numerical examples are showed.

The authors study the Hilbert Transform on the real line. They introduce some polynomial approximations and some algorithms for its numerical evaluation. Error estimates in uniform norm are given.

The authors characterize the best weighted polynomial approximation on the real semiaxis by means of a new modulus of smoothness and the related K-functional. The Jackson and Stechkin type inequalities are stated. Moreover some connected functional spaces are introduced.

An algorithm is proposed to compute Cauchy principal value integrals on the real line when the integrating functions have an exponential decay at infinity. The procedure is essentially based on the Hermite-gaussian quadrature formula.

The author proposes an algorithm for the numerical evaluation of two-dimensional Cauchy principal value integrals with respect to generalized Ditzian-Totik weight functions. Convergence results and some numerical examples are given.

This papers proposes an algorithm for the numerical evaluation of two-dimensional Cauchy principal value integrals with respect to Jacobi weight functions. Convergence results and several numerical examples are given.