1,721,033 research outputs found
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
The matrices of Pascal and other greats
No full textThe 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
The stability problem for linear multistep methods: old and new results
No full textThe paper reviews results on rigorous proofs for stability properties of classes of linear multistep methods (LMMs) used either as IVMs or as BVMs. The considered classes are not only the well-known classical ones (BDF, Adams, ...) along with their BVM correspondent, but also those which were considered unstable as IVMs, but stable as BVMs. Among the latter we find two classes which deserve attention because of their peculiarity: the TOMs (top order methods) which have the highest order allowed to a LMM and the Bs-LMMs which have the property to carry with each method its natural continuous extension
Symmetric schemes, time reversal symmetry and conservative methods for Hamiltonian systems
No full textIt is important, when integrating numerically Hamiltonian problems, that the numerical methods retain some properties of the continuous problem such as the constants of motion and the time reversal symmetry. This may be a difficult task for multistep numerical methods. In the present paper we discuss the problem in the case of linear autonomous Hamiltonian systems and we show the equivalence among the symmetry of the numerical methods and the above-mentioned requirements. In particular, the analysis is carried out for the class of methods known as boundary value methods (BVMs) (Brugnano, Trigiante. Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publishers, Amsterdam, 1998)
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
Theoretical analysis of the stability for Extended Trapezoidal Rules
No full textIn this paper a rigorous analysis of the linear stability for two classes of methods, both generalizing the trapezoidal rule, is done. The information on the coefficients of the methods, useful for the analysis, are obtained by an extensive use of properties of the Pascal matrix, some of which are derived in this work
Il problema della stabilità per metodi numerici per ODEs
Full text linkIn 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
Fast and accurate approximations to fractional powers of operators
Full text linkIn 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
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