Lidia Aceto

Rational Approximation to the Fractional Laplacian Operator in Reaction-Diffusion Problems

This paper provides a new numerical strategy to solve fractional in space reaction-diffusion equations on bounded domains under homogeneous Dirichlet boundary conditions. Using the matrix transform method the fractional Laplacian operator is replaced by a matrix which, in general, is dense. The approach here presented is based on the approximation of this matrix by the product of two suitable banded matrices. This leads to a semi-linear initial value problem in which the matrices involved are sparse. Numerical results are presented to verify the effectiveness of the proposed solution strategy.

On the Construction and Properties of

In this paper we consider the numerical solution of fractional differential equations by means of

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-step Methods for FDEs

-step recursions. The construction of such formulas can be obtained in many ways. Here we study a technique based on the rational approximation of the generating functions of fractional backward differentiation formulas (FBDFs). Accurate approximations lead to the definition of methods which simulate the underlying FBDF, with important computational advantages. Numerical experiments are presented.

A unified matrix approach to the representation of Appell polynomials

In this paper, we propose a unified approach to matrix representations of different types of Appell polynomials. This approach is based on the creation matrix – a special matrix which has only the natural numbers as entries and is closely related to the well-known Pascal matrix. By this means, we stress the arithmetical origins of Appell polynomials. The approach also allows to derive, in a simplified way, the properties of Appell polynomials by using only matrix operations.

Boundary Value Methods for the reconstruction of Sturm-Liouville potentials

Abstract In this paper we present numerical procedures for solving the two inverse Sturm–Liouville problems known in the literature as the two-spectra and the half inverse problems. The method proposed looks for a continuous approximation of the unknown potential belonging to a suitable function space of finite dimension. In order to compute such an approximation a sequence of direct problems has to be solved. This is done by applying one of the Boundary Value Methods, generalizing the classical Numerov scheme, recently introduced by the authors. Numerical results confirming the effectiveness of the approach proposed are also reported.

An Algebraic Procedure for the Spectral Corrections Using the Miss-Distance Functions in Regular and Singular Sturm-Liouville Problems

A general method based on the evaluation of the zeros of a suitable polynomial is suggested in order to have an estimation of the spectral error in the numerical treatment of Sturm-Liouville problems. The method is strictly concerned with the miss-distance function arising in the shooting algorithm for eigenvalues. The error correcting procedure derived from the method is particularly helpful when difficulties arise in the numerical integration. Two kinds of Sturm-Liouville problems are considered: the standard regular problems on a closed interval and the problems where an eigenvalue is nonlinearly involved and embedded in an essential spectrum giving origin to an inner singularity. Numerical experiments clearly highlight the efficaciousness of the proposed method both in the regular and singular case.

Matrix methods for radial Schrödinger eigenproblems defined on a semi-infinite domain

In this paper, we discuss numerical approximation of the eigenvalues of the one-dimensional radial Schrodinger equation posed on a semi-infinite interval. The original problem is first transformed to one defined on a finite domain by applying suitable change of the independent variable. The eigenvalue problem for the resulting differential operator is then approximated by a generalized algebraic eigenvalue problem arising after discretization of the analytical problem by the matrix method based on high order finite difference schemes. Numerical experiments illustrate the performance of the approach.

The Performances of the Code TOM on the Holt Problem

The code TOM, for the solution of boundary value problems, is based on linear multistep methods used as BVMs [5, 6, 16, 17]. Among the peculiar features of this code, the mesh selection strategy, based on two measures of conditioning of the problem, seems the most interesting. In this paper the application of the code to the classical Holt problem, one of the most famous and difficult BVP, will permit to stress the effectiveness of such approach along with the utility of the additional information provided by the two measures.

Efficient Implementation of Rational Approximations to Fractional Differential Operators

This paper deals with some numerical issues about the rational approximation to fractional differential operators provided by the Padé approximants. In particular, the attention is focused on the fractional Laplacian and on the Caputo’s derivative which, in this context, occur into the definition of anomalous diffusion problems and of time fractional differential equations (FDEs), respectively. The paper provides the algorithms for an efficient implementation of the IMEX schemes for semi-discrete anomalous diffusion problems and of the short-memory-FBDF methods for Caputo’s FDEs.