47 research outputs found
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
Theoretical error estimates for computing the matrix logarithm by Pad\'e-type approximants
In this article, we focus on the error that is committed when computing the
matrix logarithm using the Gauss--Legendre quadrature rules. These formulas can
be interpreted as Pad\'e approximants of a suitable Gauss hypergeometric
function. Empirical observation tells us that the convergence of these
quadratures becomes slow when the matrix is not close to the identity matrix,
thus suggesting the usage of an inverse scaling and squaring approach for
obtaining a matrix with this property. The novelty of this work is the
introduction of error estimates that can be used to select a priori both the
number of Legendre points needed to obtain a given accuracy and the number of
inverse scaling and squaring to be performed. We include some numerical
experiments to show the reliability of the estimates introduced
Fast and accurate approximations to fractional powers of operators
In this paper we consider some rational approximations to the fractional
powers of self-adjoint positive operators, arising from the Gauss-Laguerre
rules. We derive practical error estimates that can be used to select a priori
the number of Laguerre points necessary to achieve a given accuracy. We also
present some numerical experiments to show the effectiveness of our approaches
and the reliability of the estimates
Rational approximations to fractional powers of self-adjoint positive operators
We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in approximation theory involving Pad\ue9 approximants. The analysis improves some existing results and the numerical experiments proves its accuracy
EFFICIENT COMPUTATION OF THE WRIGHT FUNCTION AND ITS APPLICATIONS TO FRACTIONAL DIFFUSION-WAVE EQUATIONS
In this article, we deal with the efficient computation of the Wright function in the cases of
interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the Laplace transform of a particular expression of the Wright function for which we discuss in detail the error analysis. We also present a code package that implements the algorithm proposed here in different programming languages. The analysis and implementation are accompanied by an extensive set of numerical experiments that validate both the theoretical estimates of the error and the applicability of the proposed method for representing the solutions of fractional differential equations
Efficient computation of the sinc matrix function for the integration of second-order differential equations
This work deals with the numerical solution of systems of oscillatory
second-order differential equations which often arise from the
semi-discretization in space of partial differential equations. Since these
differential equations exhibit pronounced or highly) oscillatory behavior,
standard numerical methods are known to perform poorly. Our approach consists
in directly discretizing the problem by means of Gautschi-type integrators
based on matrix functions. The novelty contained here is
that of using a suitable rational approximation formula for the
matrix function to apply a rational Krylov-like
approximation method with suitable choices of poles. In particular, we discuss
the application of the whole strategy to a finite element discretization of the
wave equation
On the construction and properties of m-step methods for FDEs
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