116 research outputs found
Il problema della stabilità per metodi numerici per ODEs
No full textIn this paper we introduce and analyze some relations between the Pascal matrix and a new class of numerical methods for differential equations obtained generalizing the Adams methods. In particular, we shall prove that these methods are suitable for solving stiff problems since their absolute stability regions contain the negative half complex plane
Some applications of the Pascal matrix to the study of numerical methods for differential equations
No full textIn this paper we introduce and analyze some relations between the Pascal matrix and a new class of numerical methods for differential equations obtained generalizing the Adams methods. In particular, we shall prove that these methods are suitable for solving stiff problems since their absolute stability regions contain the negative half complex plane
Efficient implementation of rational approximations to fractional differential operators
No full textThis 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
Efficient computation of the Wright function and its applications to fractional diffusion-wave equations
No full textIn 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
No full textThis 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
Exponentially convergent trapezoidal rules to approximate fractional powers of operators
No full textIn this paper we are interested in the approximation of fractional powers of
self-adjoint positive operators. Starting from the integral representation of
the operators, we apply the trapezoidal rule combined with a single-exponential
and a double-exponential transform of the integrand function. For the first
approach our aim is only to review some theoretical aspects in order to refine
the choice of the parameters that allow a faster convergence. As for the double
exponential transform, in this work we show how to improve the existing error
estimates for the scalar case and also extend the analysis to operators. We
report some numerical experiments to show the reliability of the estimates
obtained
Rational approximations to fractional powers of self-adjoint positive operators
No full textWe 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\'{e} approximants. The analysis
improves some existing results and the numerical experiments proves its
accuracy
Computing the action of the matrix generating function of Bernoulli polynomials on a vector with an application to non-local boundary value problems
No full textThis paper deals with efficient numerical methods for computing the action of the matrix generating function of Bernoulli polynomials, say q(τ, A), on a vector when A is a large and sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of q(τ, w) have already been addressed in the scientific literature. The contribution of this paper is twofold. First, we place these methods in the classical framework of Krylov-Lanczos
(polynomial-rational) techniques for accelerating Fourier series. This allows us to apply the convergence results developed in this context to our function. Second, we
design a new acceleration scheme. Some numerical results are presented to show the
effectiveness of the proposed algorithms
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