1,647 research outputs found
Exponentially-fitted quadrature methods for evolution problems with periodic solution
No full textThe interest for numerical solution of physical and biological problems with oscillating and/or periodic
behaviour requires the use of special-purpose methods. Examples include the electromagnetic scattering, the response of nonlinear circuits to a periodic input and the evolution of an age-structuredpopulation. These problems are characterized by either infinite integrals where the integrand function is oscillatory function, or by Volterra integral equations of type with periodic solution. By exploting the Exponential Fitting theory [1, 2, 3, 4,
5], a new class of quadrature rules, that are a generalization of the usual Gauss-Laguerre formulae, for
problem (1) and a new direct quadrature (DQ) method for problem (2) are derived, respectively. Two extra problems appear in the context of building the exponentially-fitted (ef ) DQ method. The first one is the construction of a two-nodes ef quadrature rule of Gaussian type, that is a generalization of the usual two-nodes Gauss-Legendre formula, on which the DQ method is based. The second problem is the building of a suitable ef interpolation technique on four points which preserves the order of
convergence of the overall method. These works are in collaboration with L. Gr. Ixaru (National Institute of Physics and Nuclear Engineering, Bucharest, Romania), B. Paternoster, A. Cardone and D. Conte (University of Salerno).
References
[1] Ixaru, L.Gr.; Vanden Berghe, G., Exponential fitting, Kluwer Academic Publishers, Dordrecht
(2004).
[2] A. Cardone, B. Paternoster, G. Santomauro, Exponential fitting quadrature rule for functional equations, AIP Conference Proceedings 1479 (2012) 1169-1172.
[3] A. Cardone, L. Gr. Ixaru, B. Paternoster, G. Santomauro, Ef-Gaussian direct quadrature methods
for Volterra integral equations with periodic solution, Mathematics and Computers in Simulation (submitted).
[4] D. Conte, B. Paternoster, G. Santomauro, An exponentially fitted quadrature rule over unbounded
intervals, AIP Conference Proceedings 1479 (2012) 1173-1176.
[5] D. Conte, L. Gr. Ixaru, B. Paternoster, G. Santomauro, Gauss-Laguerre quadrature rule for oscillatory integrands, Computational and Applied Mathematics (submitted)
Exponentially fitted numerical methods for differential systems with equation dependent coefficients
No full textThe derivation of special purpose numerical methods for differential systems, i.e. adapted to accurately solve problems whose qualitative behaviour is supposed to be known a-priori, is usually carried out by means of non-polynomial fitting techniques. Exponential fitting (compare [2] and references therein) is certainly one of the most spread out techniques to obtain special purpose formulae in many fields of numerical analysis.
In the context of numerical methods for ordinary differential equations, exponentially fitted Runge-Kutta formulae have been considered by many authors (see [3] for an updated state-of-art on the topic). The issue we want to revisit in this talk is the way of deriving the coefficients of such methods: we decide, indeed, to take into account the effect of the error inherited from the computation of the internal stages. Such contribution has always been neglected in previous version of exponentially fitted Runge-Kutta methods: on the contrary, we aim to make the propagation of the error along the stages visible. The revised technique is illustrated for hybrid methods and Runge-Kutta methods [1], for which we obtain new expressions of the coefficients, explicitly depending on the form of the system to be solved. The version obtained in this way is then compared for accuracy and stability with that achieved by means of the standard exponential fitting technique. Acknowledgments The authors express their gratitude to prof. Liviu Gr. Ixaru for the profitable discussions we had on the topic.
References
[1] R. D’Ambrosio, L. Gr. Ixaru, B. Paternoster, Construction of the EF-based Runge-Kutta methods revisited, Comp. Phys. Commun. 182, 322-329 (2011).
[2] L. Gr. Ixaru and G.Vanden Berghe, Exponential Fitting, Kluwer Academic Publishers, Dordrecht (2004).
[3] B. Paternoster, Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70-th anniversary, submitted
STABILITY OF COLLOCATION METHODS FOR FRACTIONAL DIFFERENTIAL EQUATIONS
No full textCollocation methods for fractional differential equations have been introduced by Blank [1] and later on rigorously analyzed by Pedas and Tamme [3,4]. Recently Cardone, Conte and Paternoster [2] introduced two step collocation methods, which raise the order of convergence, by using additional information from the past, without increasing the computational cost. Here
we study the stability of both classes of methods, in order to nd methods with unbounded stability regions.
This is a joint work with D. CONTE and B. PATERNOSTER from UNIVERSITY OF SALERNO.
[1 ] L. Blank, Numerical treatment of differential equations of fractional order, Nonlinear World, 4 (1997), 473-491.
[2 ] A. Cardone, D. Conte, B. Paternoster, Two-step collocation methods forfractional dierential equations, to appear in Discrete Cont.Dyn.-B. [3 ] A. Pedas and E. Tamme, On the convergence of spline collocation methods for solving fractional differential equations, J. Comput. Appl.Math., 235 (2011), 3502-3514.
[4 ] A. Pedas and E. Tamme, Numerical solution of nonlinear fractional differential equations by spline collocation methods, J. Comput. Appl.Math., 255 (2014), 216-230
ADAPTED NUMERICAL METHODS FOR ADVECTION DIFFUSION PROBLEMS
No full textWe present exponentially fitted two step peer methods for the numerical solution of systems of ordinary differential equations having oscillatory solutions (2; 3). Such equations arise for example in the semi-discretization in space of advection-diffusion problems whose solution exhibits an oscillatory behaviour, such as the Boussinesq equation (1). Exponentially fitted methods are able to exploita-prioriknowninformationaboutthequalitativebehaviourofthesolutionin order to efficiently furnish an accurate solution. Moreover peer methods are very suitable for a parallel implementation, which may be necessary when the number ofspatialpointsincreases. Theeffectivenessofthisproblem-orientedapproachis shown through numerical tests on well-known problems.
References
[1] A. Cardone, R. D’Ambrosio, B. Paternoster. (2017). Exponentially fitted IMEX methods for advectiondiffusion problems, J. Comput. Appl. Math. (316), 100–108.
[2] D. Conte, R. D’Ambrosio, M. Moccaldi, B. Paternoster. (2018). Adapted explicit two-step peer methods, J. Numer. Math., in press.
[3] D. Conte, L. Moradi, B. Paternoster. (2017). Adapted implicit two-step peer methods, in preparation
Dissecting the physiology and pathophysiology of glucagon-like peptide-1
No full textCopyright © 2018 Paternoster and Falasca. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. An aging world population exposed to a sedentary life style is currently plagued by chronic metabolic diseases, such as type-2 diabetes, that are spreading worldwide at an unprecedented rate. One of the most promising pharmacological approaches for the management of type 2 diabetes takes advantage of the peptide hormone glucagon-like peptide-1 (GLP-1) under the form of protease resistant mimetics, and DPP-IV inhibitors. Despite the improved quality of life, long-term treatments with these new classes of drugs are riddled with serious and life-threatening side-effects, with no overall cure of the disease. New evidence is shedding more light over the complex physiology of GLP-1 in health and metabolic diseases. Herein, we discuss the most recent advancements in the biology of gut receptors known to induce the secretion of GLP-1, to bridge the multiple gaps into our understanding of its physiology and pathology
Exponentially fitted peer methods for advection diffusion problems
No full textWe consider advection-diffusion problems whose solution exhibits an oscillatory behaviour, such as the Boussinesq equation [1]. The semi-discretization in space of such equation gives rise to a system of ordinary differential equations, whose dimension depends on the number of spatial points. We present a general class of exponentially fitted two step peer methods for the numerical integration of ordinary differential equations having oscillatory solutions [2, 3]. Such methods are able to exploit a-priori known information about the qualitative behaviour of the solution in order to efficiently furnish an accurate solution. Moreover peer methods are very suitable for a parallel implementation, which may be necessary when the number of spatial points increases. The effectiveness of this problemoriented approach is shown through numerical tests on well-known problems.
References
[1] A. Cardone, R. D’Ambrosio, B. Paternoster (2017). Exponentially fitted IMEX methods for advectiondiffusion problems, J. Comput. Appl. Math (316), 100–108.
[2] D. Conte, R. D’Ambrosio, M. Moccaldi, B. Paternoster (2018). Adapted explicit two-step peer methods, J. Numer. Math., in press.
[3] D. Conte, L. Moradi, B. Paternoster (2017). Adapted implicit two-step peer methods, in preparation
About stability of nonlinear stochastic difference equations
No full textUsing the method of Lyapunov functionals construction, it is shown that investigation of stability in probability of nonlinear stochastic difference equation with order of nonlinearity more than one can be reduced to the investigation of asymptotic mean square stability of the linear part of this equation
Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations
No full textIt is supposed that the fractional difference equation xn+1=(μ+∑j=0kajxn−j)/(λ+∑j=0kbjxn−j), n=0,1,…, has an equilibrium point x^ and is exposed to additive stochastic perturbations type of Ã(xn−x^)ξn+1 that are directly proportional to the deviation of the system state xn from the equilibrium point x^. It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted
Problem oriented discretizations for a vegetation model
No full textThe use of functional equations represents the most common strategy for modeling real phenomena. In particular, we are interested in the numerical solution of models of Partial Differential Equations (PDEs) coming from applications in real contexts, such as corrosion [3, 7], sustainability [8], vegetation [6]. The numerical treatment of these problems is not trivial, since they are often characterized by high stiffness, which requires the use of very dense spatial and temporal discretizations. This obviously leads to unacceptable computing times. Furthermore, a numerical method is not always able to exploit a-priori known properties of the problem, such as any positivity or oscillating trend of the solution, the asymptotic stability, and so on.
In this talk, we focus on a reaction-diffusion vegetation model that has been introduced to investigate the coexistence of two different plant species in arid environments, characterized by scarce presence of water [6]. The considered system of PDEs is an extension of the well-known Klausmeier model. The latter investigates the growth of a single type of plant with varying water availability. The first, on the other hand, constitutes a generalization of the Klausmeier model, as it considers the competition of two different species of plants, which must somehow try to survive by sharing the same limiting resource. The model under investigation has high stiffness, and is characterized by positivity and oscillating behavior in space. We therefore show numerical techniques capable of dealing with the stiffness of the problem, and also of preserving the a-priori known properties of the exact solution for each choice of the spatial and temporal discretization steps [4]. In particular, to preserve positivity, we extend non-standard finite differences using exponential integrators and TASE operators, which have been recently introduced to stabilize explicit Runge-Kutta methods [1]. This also helps to deal with the stiffness of the problem. Furthermore, to preserve the spatial oscillations of the solution, we integrate the exponential fitting framework within the non-standard discretizations. To further improve the efficiency of the proposed approaches, we show the use of adapted parallel peer methods for the considered vegetation problem [2, 5]. Finally, numerical tests are shown to confirm the effectiveness of the proposed techniques.
References
[1] M. Bassenne, L. Fu, and A. Mani. Time-accurate and highly-stable explicit operators for stiff differential equations. , J. Comput. Phys., 424:Paper No. 109847, 24, 2021.
[2] D. Conte, P. De Luca, A. Galletti, G. Giunta, L. Marcellino, G. Pagano and B. Paternoster. First Experiences on Parallelizing Peer Methods for Numerical Solution of a Vegetation Model., Lect. Notes Comput. Sci., 13376, 384–394, 2022.
[3] D. Conte and G. Frasca-Caccia. A Matlab code for the computational solution of a phase field model for pitting corrosion., Dolomites Res. Notes Approx., 15:47–65, 2022.
[4] D. Conte, G. Pagano, and B. Paternoster. Nonstandard finite differences numerical methods for a vegetation reaction-diffusion model., J. Comput. Appl. Math., 419:Paper No. 114790, 17, 2023.
[5] D. Conte, G. Pagano, and B. Paternoster. Time-accurate and highly-stable explicit peer methods for stiff differential problems. , Commun. Nonlinear Sci. Numer. Simul., 119:Paper No. 107136, 20, 2023.
[6] L. Eigentler and J. A. Sherratt. Metastability as a coexistence mechanism in a model for dryland vegetation patterns., Bull. Math. Biol., 81:2290–2322, 2019.
[7] H. Gao, L. Ju, R. Duddu, and H. Li. An efficient second-order linear scheme for the phase field model of corrosive dissolution., J. Comput. Appl. Math., 367:112472, 16 pp., 2020.
[8] B. Maldon and N. Thamwattana. Review of diffusion models for charge-carrier densities in dye-sensitized solar cells., J. Phys. Commun., 4:Paper No. 082001, 2020
ALGEBRAICALLY STABLE TWO-STEP RUNGE-KUTTA METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
No full textWe investigate algebraic stability of two-step Runge-Kutta methods [2] for ordinary differential equations using the criterion proposed by Hewitt and Hill [1] for general linear methods. This criterion is based on suitable transformations on the coefficient matrices of the methods under consideration, in such a way that the G-matrix of algebraically stable formulae is the identity matrix. This gives a remarkable improvement, since the determination of the G-matrix is, in general, a nontrivial task. Examples of algebraically stable two-step Runge-Kutta methods possessing the above feature are presented. This work is in collaboration with Zdzislaw Jackiewicz (Arizona State University), Beatrice Paternoster (University of Salerno) and Dajana Conte (University of Salerno).
REFERENCES [1] L. L. Hewitt, A. T. Hill. Algebraically stable diagonally implicit general linear methods. Appl. Numer. Math., 60 (6):629–636, 2010.
[2] Z. Jackiewicz. General linear methods for ordinary differential equations. John Wiley & Sons, 2009
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