This paper provides a new numerical strategy for solving fractional-in-space reaction-diffusion equations on bounded domains under homogeneous Dirichlet boundary conditions. Using the matrix transfer technique 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 semilinear initial value problem in which the matrices involved are sparse. Numerical results are presented to verify the effectiveness of the proposed solution strategy.

The Pascal matrix has been known since ancient times,and it was mentioned in Chinese mathematical texts dating from 1303. Nevertheless, it has been studied carefully only recently. It arises in probability, numerical analysis,surface reconstruction, combinatorics; we came across it while studying stability properties of numerical methods for solving ordinary differential equations. Our goal is to share some of the beauty of the Pascal matrix and show how it is related to other matrices associated with great names such as Vandermonde, Frobenius, Stirling,etc

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.

In this paper we consider the numerical solution of fractional differential equations by means of $m$-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.