2,202 research outputs found
A MATLAB Code for Fractional Differential Equations Based on Two-Step Spline Collocation Methods
No full textWe illustrate a MATLAB implementation of two-step spline collocation methods for the numerical solution of fractional differential equations, introduced by Cardone, Conte and Paternoster in (Discrete Dyn. Syst. Ser. B 23(7) 2709--2725 (2018)). The computational tasks include the evaluation of fractional integrals, a suitable starting procedure, and the efficient computation of the coefficients of certain polynomials. The whole algorithm is described in detail. Some numerical experiments show the performances of the proposed algorithm
Adapted discretization of evolutionary problems by non-polynomially fitted numerical methods
No full textThe talk is devoted to the discretization of selected evolutionary problems generating periodic
wavefronts [5] and aims to explain the benefits gained by adapting the numerical scheme to the
problem. Such an adaptation is carried out by merging the a-priori known qualitative information
on the problem, as well as the structure of the vector field itself, into the numerical scheme.
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Particular emphasis will be given to advection-reaction-diffusion problems, for which the adaptation
in space is developed by means of a finite difference scheme based on trigonometrical basis
functions [3], rather than on algebraic polynomials which could strongly reduce the stepsize in
order to accurately reproduce the prescribed oscillations of the exact solution. The adaptation
in time takes into account that the spatially discretized problem is characterized by a vector
field consisting in stiff and nonstiff terms, hence it makes sense to adopt an implicit-explicit
(IMEX) time integration, which implicitly integrate only the stiff constituents, while the nonstiff
part is computed explicitly. Clearly, the employ of non-polynomial basis functions makes
the coefficients of the numerical method dependent on unknown parameters (i.e. the frequency
of the oscillations), which need to be properly estimated [4]; the proposed estimation relies
on a minimization procedure of the local truncation error that is carried out a-priori, without
affecting the computational cost of the integration. A rigorous analysis on the stability and
accuracy properties of the overall method is presented, together with some numerical tests, in
order to highlight the effectiveness of the approach. The introduced technique also covers the
case of periodic dynamics generated by evolutionary problems with memory [1, 2], discretized
in terms of non-polynomially fitted quadrature methods able to accurately reproduce the oscillatory
behavior with a reduced computational cost with respect to their analogous polynomial
version, when a good estimate of the unknown frequency is provided. Stability issues for such
a discretization are also addressed. References
[1] Cardone, A., Ixaru, L.Gr. and Paternoster, B. Exponential fitting direct quadrature methods
for Volterra integral equations, Numer. Algorithms 55(4), 467–480 (2010).
[2] Cardone, A., Ixaru, L.Gr., Paternoster, B. and Santomauro, G. Ef-gaussian direct quadrature
methods for Volterra integral equations with periodic solution, Math. Comput. Simul.,
110, 125–143 (2015).
[3] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. Adapted numerical methods for
advection-reaction-diffusion problems generating periodic wavefronts. Comput. Math. Appl.
74(5), 1029–1042 (2017).
[4] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. Parameter estimation in IMEXtrigonometrically
fitted methods for the numerical solution of reaction-diffusion problems.,
Comput. Phys. Commun. 226, 55–66 (2018).
[5] Perumpanani, A.J., Sherratt, J.A. and Maini, P.K. Phase differences in reaction-diffusionadvection
systems and applications to morphogenesis, J. Appl. Math. 55, 19–33 (1995)
Dissecting the physiology and pathophysiology of glucagon-like peptide-1
Get PDFCopyright © 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
Rizzo, G., Buccione, R., Paternoster, M., 2021. Natural sources of trace elements in the atmosphere and their influence on human health. Dust 2021, IV International Conference on Atmospheric Dust, Monopoli, Italy, 4-7 October 2021.
No full text
Adapted numerical methods for advection-reaction-diffusion problems generating periodic wavefronts
Get PDFWe present an adapted method of lines for advection-reaction-diffusion problems generating periodic wavefronts [3], by exploiting the a-priori known information about the qualitative behaviour of the solution.
Since the dynamics exhibits a non-polynomial character, classical finite difference methods could require a very small stepsize because they are constructed in order to be exact (within round-off error) on polynomials up to a certain degree. In our approach, the employ of non-polynomially fitted finite differences may guarantee a better balance between accuracy and efficiency requirements.
Once a advection-reaction-diffusion problem is discretized in space, the vector field of the resulting system of ordinary differential equations results to be split in two different terms, a stiff term and a nonlinear one. Hence, we propose an implicit-explicit (IMEX) method that implicitly integrates only stiff components and explicitly integrates the nonlinear part, with a significant benefit in terms of efficiency. For the overall numerical scheme, combining the non-polynomial fitting strategy with the IMEX time integration, accuracy and stability properties are rigorously studied, also in comparison with the classical polynomial case [1]. Moreover, since the adapted method has non-constant coefficients depending on unknown parameters linked to the solution, we propose an estimation strategy based on minimization of the leading term of the local discretization error [2].
This is a joint work with Raffaele D'Ambrosio and Beatrice Paternoster (University of Salerno).
[1] D'Ambrosio, R., Moccaldi, M., Paternoster, B., Adapted IMEX numerical methods for reaction-diffusion problems, Appl. Numer. Math. (submitted)
[2] D'Ambrosio, R., Moccaldi, M., Paternoster, B., Parameter estimation in adapted numerical methods for reaction-diffusion problems, J. Sci. Comput. (submitted)
[3] Perumpanani, A.J., Sherratt, J.A., Maini, P.K., Phase differences in reaction-diffusion-advection systems and applications to morphogenesis, IMA J. Appl. Math. 55, 19--33 (1995)
Adapted numerical approximation of advection-reaction-diffusion problems
No full textWe present an adapted numerical method for the approximate solution of advection-reaction-diffusion problems on a bidimensional spatial domain and generating periodic wavefronts [3]. In particular, we propose to merge into the numerical scheme the a-priori known information about the qualitative behaviour of the exact solution and the structure of the problem. Traditional finite difference methods could impose a severe reduction of the stepsize in order to accurately follow the oscillations because they are developed in order to achieve exactness (within round-off error) on algebraic polynomials up to a certain degree. Extending the ideas described in [1,2], we develop an adapted method of lines based on trigonometrically fitted finite differences, whose coefficients depend on unknown parameters characterising the exact solution. We deal with the more challenging issue of estimating these parameters by properly manipulating the leading term of the local truncation error a-priori.
The vector field of the resulting system of ordinary differential equations is composed by stiff and non-stiff terms, so we suggest to employ an implicit-explicit (IMEX) time method, which implicitly integrates only stiff components and explicitly integrates the others, obtaining advantages in terms of efficiency and stability. The stability and accuracy properties of the overall scheme are rigorously investigated and some numerical tests are presented to show its effectiveness.
This is a joint work with Raffaele D’Ambrosio from University of L’Aquila and Beatrice Paternoster from University of Salerno.
[1] D’Ambrosio, R., Moccaldi, M., Paternoster, B., Adapted numerical methods for advection-reaction-
diffusion problems generating periodic wavefronts, Comput. Math. Appl. 74(5), 1029–1042, 2017.
[2] D’Ambrosio, R., Moccaldi, M., Paternoster, B., Parameter estimation in IMEX-trigonometrically
fitted methods for the numerical solution of reaction-diffusion problems, Comput. Phys. Commun.
226, 55–66, 2018.
[3] Hundsdorfer, W., Verwer, J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction
Equations, Springer-Verlag, 2003
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
Adapted time-integration of partial differential equations generating periodic wavefronts
No full textThe talk focuses on the numerical solution of advection-reaction-diffusion problems
by adapted finite difference schemes. In other terms, the numerical scheme is developed
in order to exploit the a-priori knowledge of the qualitative behaviour of the solution,
gaining advantages in terms of efficiency and accuracy with respect to classical schemes
already known in literature, which mostly rely on algebraic polynomials. The adaptation
is carried out by the so-called trigonometrical fitting technique for the space-discretization,
giving rise to a system of ODEs whose vector field contains both stiff and non-stiff terms.
Due to this mixed nature of the vector field, an Implicit-Explicit (IMEX) method is employed for the time-integration. The coefficients of the introduced numerical scheme depend on unknown parameters which have to be properly estimated: such an estimate is performed by an efficient offline minimization of the leading term of the local truncation error. The effectiveness of this problem-oriented approach is shown through a rigorous theoretical analysis and selected numerical experiments.
References
[1] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. 2017 Adapted numerical methods for advection-reaction-diffusion problems generating periodic wavefronts. Comput. Math. Appl. 74(5), 1029–1042.
[2] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. 2018 Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction-diffusion problems., Comput. Phys. Commun. 226, 55–66.
[3] Perumpanani, A.J., Sherratt, J.A., Maini, P.K. 1995 Phase differences in reaction–diffusion–advection systems and applications to morphogenesis, J. Appl. Math. 55, 19–33
Buccione, R., Rizzo G., Paternoster M., Mongelli G., 2024. LucAS project: Evaluation of geochemical-mineralogical background in environmental matrices for the assessment of health risk Abstract. 2nd SOGEI Conference Perugia, 1 ǀ 4 July 2024.
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