79 research outputs found
Fast Collocation methods for Volterra Integral equations of convolution type
No full textIn this paper we present fast discrete collocation methods forVolterra integral equations of Hammerstein type, where the Laplace
transform of the kernel is known a priori. To compute the numerical solution over N time steps, the constructed methods require
O(N log(N )) operations, O(log(N )) memory and preserve the order of accuracy of the corresponding exact collocation methods.
The numerical experiments confirm the expected accuracy and the computational cost
Fast Runge-Kutta methods for nonlinear convolution systems of Volterra Integral equations
No full textIn this paper fast implicit and explicit Runge–Kutta methods for systems of Volterra
integral equations of Hammerstein type are constructed. The coefficients of the methods
are expressed in terms of the values of the Laplace transform of the kernel. These
methods have been suitably constructed in order to be implemented in an efficient way,
thus leading to a very low computational cost both in time and in space. The order of
convergence of the constructed methods is studied. The numerical experiments confirm
the expected accuracy and computational cost
Efficient numerical methods for Volterra integral equations of Hammerstein type
No full textVolterra integral equations (VIEs) are the mathematical model of many evolutionary problems with memory arising from biology, chemistry, physics, engineering. It is known that the numerical treatment of VIEs has an high computational cost, due mainly to the computation of the ``lag term'' or ``tail term'' which contains the past history of the phenomenon. Since it depends on time, the ``lag term'' has to be computed for each time step and its cost increases when time passes. Among the Volterra equations, the Hammerstein type ones, are particularly interesting for the applications.
The aim of this thesis is the construction of numerical methods for VIEs of Hammerstein type which produce accurate solution at a low computational cost and ``catch'' the qualitative behaviour of the exact solution.
The study developed has been concerned at first with the construction and analysis of efficient methods for the numerical treatment of VIEs of Hammerstein type where the Laplace transform of the kernel rather than the convolution kernel itself is a priori known. This is not an anomalous or restricting situation, as a matter of fact these kind of problems arise in chemical absorption kinetics in the determination of non reflecting boundary conditions, and in general in situations when Laplace transform tecnique are used to reduce systems of ordinary or partial differential equations in VIEs.
It is known that a classical numerical method for computing the numerical solution of such equations over Nt time steps requires O(N2t) operations and O(Nt) memory space.
In this thesis we construct two classes of fast numerical methods based on collocation and Runge-Kutta formulas respectively. These methods have a computational cost of O(NtlogNt) operations, O(logNt) memory requirement and they have an high order of accuracy. In both cases the knowledge of the Laplace transform of the kernel and the convolution nature of the kernel itself are exploited in order to obtain a fast computation of the lag term. This is possible by using an opportune inverse Laplace transform approximation formula for computing the kernel evaluations.
The fast numerical methods constructed in this thesis tend to the corresponding classical methods when the inverse Laplace transfrom approximation formula is exact. The convergence analysis of the fast collocation and Runge-Kutta methods shows that their order of convergence coincides with the order of the corresponding classical methods.
We also analyse the stability properties of the fast Runge-Kutta methods with respect to test equations.
We prove that the stability regions depend on the approxiamation of the inverse Laplace transform and that the stability properties of the classical Runge-Kutta methods are obtained when the error of the inverse Laplace transform approximation formula tends to zero.
The numerical experiments on some significant problems taken from the ``Test Set'' collection project confirm the expected accuracy, computational cost and the stability properties of the constructed methods.
The second part of the thesis concerns with the numerical treatment of problems of SIS epidemic diffusion with periodic immigration flow. The mathematical model of such problems is represented by an Hammerstein type VIE with convolution kernel.
We consider problems caracterized by the relapse of the epidemic which implies that the VIE has an asymptotically periodic solution.
It is clear that an efficient numerical method has to reproduce the asymptotically periodic solution whenever applied to equations that show this behaviour.
For this reason we analyse the discrete Volterra equation (DVE) corresponding to such VIEs and we prove a theorem which establishes the existence and the uniqueness of the asymptotically periodic solution of the DVE.
Moreover we consider SIS epidemic models with periodic immigration flow and constant contact rate. Also in this case we prove, for the DVE corresponding to the problem, the existence and the uniqueness of the asymptotically periodic solution when the DVE satisfies some significant hypothesis depending only on its kernel and forcing term.
In order to analyse if the existing numerical methods satisfy these conditions, that is if they are AP-stable, we consider the class of θ-methods and we prove that they are AP-stable if the integration step satisfies an inequality depending only on some parameters that are characteristic of the problem
High performance parallel numerical methods for Volterra equations with weakly singular kernels
No full textNon-stationary discrete time waveform relaxation methods for Abel systems of Volterra integral equations using fractional linear multistep formulae are introduced. Fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed. A significant expression of the error is proved, which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance. The numerical experiments confirm the theoretical expectations
Asymptotic periodicity of nonlinear discrete Volterra equations and applications
No full textSufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations
Imsejhin ghall-qadi tal-komunita` Nisranija permezz tal-ministeru sacerdotali
No full textThe article focuses upon the call to service through the exercise of the priestly ministry. The point of departure of the study is the Johannine pericope which describes the washing of the disciples' feet by Jesus (Jn 13:1-20). The article describes how the celebration of the liturgy is a fount of vocations to the ordained ministry. The author then studies three documents of the Second Vatican Council -- Optatam Totius, Presbyterorum Ordinis and Christus Dominus -- in order to depict the centrality of service to priestly ministry. A number of evocative texts are referred to, as well as formation documents from the local Church of Malta. Furthermore, the respective contributions of Bishop Tonino Bello, Pope Benedict XVI and Pope Francis are also given pride of place, as more light is thrown on the theme of the article.peer-reviewe
Il prete dentro la Chiesa e la cultura: Tra immagini del passato recente e dinamiche generative per oggi
No full textStarting from a description of some kinds of priest in the recent history, the author highlights the underlying logic. With the collected elements, the article suggests a way to think and practice the processes that today contribute, as regards the Church’s and culture historicity, to generate the experience and the way to understand the ministry and the identity of the priest. The result is the emerging of some concrete spaces for reflection and training, to meet and bring up figures of priests according to the purposes of the Church, and the real needs of the world today.Partendo dalla descrizione di alcune tipologie di prete nella storia recente, vengono evidenziate le logiche che stanno alla base. Con gli elementi raccolti, viene poi proposto un modo di pensare e praticare i processi che oggi contribuiscono, sul versante della storicità della Chiesa e della cultura, a generare l’esperienza e i modi di comprendere il ministero e l’identità del prete. Ne emergono degli spazi concreti di riflessione e di formazione, da frequentare per accogliere e generare delle figure di preti consone con l’intenzione della Chiesa, e le esigenze autentiche del mondo di oggi
EFFICIENT NUMERICAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS OF HANMERSTEIN TYPE
No full textVolterra integral equations (VIEs) are the mathematical model of many evolutionary problems with memory arising from biology, chemistry, physics, engineering. It is known that the numerical treatment of VIEs has an high computational cost, due mainly to the computation of the ``lag term'' or ``tail term'' which contains the past history of the phenomenon. Since it depends on time, the ``lag term'' has to be computed for each time step and its cost increases when time passes. Among the Volterra equations, the Hammerstein type ones, are particularly interesting for the applications. The aim of this thesis is the construction of numerical methods for VIEs of Hammerstein type which produce accurate solution at a low computational cost and ``catch'' the qualitative behaviour of the exact solution. The study developed has been concerned at first with the construction and analysis of efficient methods for the numerical treatment of VIEs of Hammerstein type where the Laplace transform of the kernel rather than the convolution kernel itself is a priori known. This is not an anomalous or restricting situation, as a matter of fact these kind of problems arise in chemical absorption kinetics in the determination of non reflecting boundary conditions, and in general in situations when Laplace transform tecnique are used to reduce systems of ordinary or partial differential equations in VIEs. It is known that a classical numerical method for computing the numerical solution of such equations over Nt time steps requires O(N^2t) operations and O(Nt) memory space. In this thesis we construct two classes of fast numerical methods based on collocation and Runge-Kutta formulas respectively. These methods have a computational cost of O(NtlogNt) operations, O(logNt) memory requirement and they have an high order of accuracy. In both cases the knowledge of the Laplace transform of the kernel and the convolution nature of the kernel itself are exploited in order to obtain a fast computation of the lag term. This is possible by using an opportune inverse Laplace transform approximation formula for computing the kernel evaluations. The fast numerical methods constructed in this thesis tend to the corresponding classical methods when the inverse Laplace transfrom approximation formula is exact. The convergence analysis of the fast collocation and Runge-Kutta methods shows that their order of convergence coincides with the order of the corresponding classical methods. We also analyse the stability properties of the fast Runge-Kutta methods with respect to test equations. We prove that the stability regions depend on the approxiamation of the inverse Laplace transform and that the stability properties of the classical Runge-Kutta methods are obtained when the error of the inverse Laplace transform approximation formula tends to zero. The numerical experiments on some significant problems taken from the ``Test Set'' collection project confirm the expected accuracy, computational cost and the stability properties of the constructed methods. The second part of the thesis concerns with the numerical treatment of problems of SIS epidemic diffusion with periodic immigration flow. The mathematical model of such problems is represented by an Hammerstein type VIE with convolution kernel. We consider problems caracterized by the relapse of the epidemic which implies that the VIE has an asymptotically periodic solution. It is clear that an efficient numerical method has to reproduce the asymptotically periodic solution whenever applied to equations that show this behaviour. For this reason we analyse the discrete Volterra equation (DVE) corresponding to such VIEs and we prove a theorem which establishes the existence and the uniqueness of the asymptotically periodic solution of the DVE. Moreover we consider SIS epidemic models with periodic immigration flow and constant contact rate. Also in this case we prove, for the DVE corresponding to the problem, the existence and the uniqueness of the asymptotically periodic solution when the DVE satisfies some significant hypothesis depending only on its kernel and forcing term. In order to analyse if the existing numerical methods satisfy these conditions, that is if they are AP-stable, we consider the class of θ-methods and we prove that they are AP-stable if the integration step satisfies an inequality depending only on some parameters that are characteristic of the problem
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