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Search Results (6)

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Authors = Felice Iavernaro

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28 pages, 3168 KiB  
Review
FDE-Testset: Comparing Matlab© Codes for Solving Fractional Differential Equations of Caputo Type
by Luigi Brugnano, Gianmarco Gurioli, Felice Iavernaro and Mikk Vikerpuur
Fractal Fract. 2025, 9(5), 312; https://doi.org/10.3390/fractalfract9050312 - 13 May 2025
Cited by 1 | Viewed by 604
Abstract
Fractional differential equations (FDEs) have attracted more and more attention in the last years; among them, equations of Caputo type allow for “more natural” initial conditions when the order is greater than one. As a result, many numerical methods have been devised and [...] Read more.
Fractional differential equations (FDEs) have attracted more and more attention in the last years; among them, equations of Caputo type allow for “more natural” initial conditions when the order is greater than one. As a result, many numerical methods have been devised and investigated for approximating their solution: the Matlab© codes of some of them are also available. The aim of this paper is a systematic comparison of such codes on a selected set of test problems. The obtained results are available on the web. Full article
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17 pages, 1386 KiB  
Review
Continuous-Stage Runge–Kutta Approximation to Differential Problems
by Pierluigi Amodio, Luigi Brugnano and Felice Iavernaro
Axioms 2022, 11(5), 192; https://doi.org/10.3390/axioms11050192 - 21 Apr 2022
Cited by 11 | Viewed by 2770
Abstract
In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge–Kutta methods. In [...] Read more.
In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge–Kutta methods. In this review paper, we recall this aspect and extend it to higher-order differential problems. Full article
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15 pages, 1369 KiB  
Article
A Fourth Order Symplectic and Conjugate-Symplectic Extension of the Midpoint and Trapezoidal Methods
by Felice Iavernaro and Francesca Mazzia
Mathematics 2021, 9(10), 1103; https://doi.org/10.3390/math9101103 - 13 May 2021
Cited by 3 | Viewed by 2151
Abstract
The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for [...] Read more.
The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems)
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28 pages, 1480 KiB  
Review
Line Integral Solution of Hamiltonian PDEs
by Luigi Brugnano, Gianluca Frasca-Caccia and Felice Iavernaro
Mathematics 2019, 7(3), 275; https://doi.org/10.3390/math7030275 - 18 Mar 2019
Cited by 14 | Viewed by 3634
Abstract
In this paper, we report on recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs) by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger [...] Read more.
In this paper, we report on recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs) by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger equation, and the Korteweg–de Vries equation, to illustrate the main features of this novel approach. Full article
(This article belongs to the Special Issue Geometric Numerical Integration)
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3 pages, 200 KiB  
Editorial
Advanced Numerical Methods in Applied Sciences
by Luigi Brugnano and Felice Iavernaro
Axioms 2019, 8(1), 16; https://doi.org/10.3390/axioms8010016 - 31 Jan 2019
Cited by 1 | Viewed by 3284
Abstract
The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which [...] Read more.
The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
28 pages, 407 KiB  
Review
Line Integral Solution of Differential Problems
by Luigi Brugnano and Felice Iavernaro
Axioms 2018, 7(2), 36; https://doi.org/10.3390/axioms7020036 - 1 Jun 2018
Cited by 95 | Viewed by 5872
Abstract
In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is [...] Read more.
In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is taken into account that a suitable numerical quadrature is used. The methods, at first devised for the numerical solution of Hamiltonian problems, have been later generalized along several directions and, actually, the research is still very active. In this paper we collect the main facts about line integral methods, also sketching various research trends, and provide a comprehensive set of references. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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