2,098 research outputs found
Grant Cardone International Sales Expert, Author & Coach
No full textGrant Cardone International Sales Expert, Author & Coach
BOUND STATES OF A CONVERGING QUANTUM WAVEGUIDE
No full textWe consider a two-dimensional quantum waveguide composed of two semi-strips of width 1 and 1 - epsilon, where epsilon > 0 is a small real parameter, i.e. the waveguide is gently converging. The width of the junction zone for the semi-strips is 1 + O(root epsilon). We will present a sufficient condition for the existence of a weakly coupled bound state below pi(2), the lower bound of the continuous spectrum. This eigenvalue in the discrete spectrum is unique and its asymptotics is constructed and justified when epsilon -> 0(+). RI Cardone, Giuseppe/I-2998-2012 OI Cardone, Giuseppe/0000-0002-5050-890
Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics
No full textWe consider a magnetic Schrodinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for the first band functions, as well as the complete asymptotics expansion (up to an exponentially small term) for the bottom of the band spectrum. RI Cardone, Giuseppe/I-2998-2012 OI Cardone, Giuseppe/0000-0002-5050-890
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)
A conservative numerical method for a time fractional diffusion equation
No full textGeometric numerical integration, the branch of numerical analysis with the goal of finding approximate solutions of differential equations that preserve some structure of the continuous problem, is a well established field of research [5]. In particular, requiring that invariants or conservation laws are preserved, on one hand, applies on the approximations some constraints that are satisfied also by the exact solutions. On the other hand, it guarantees a better propagation of the error over long integration times [3].
In the last two decades, new techniques for finding conservation laws of fractional differential equations have been derived by suitably generalising methods for PDEs [4, 6]. However, the numerical preservation of conservation laws of time fractional differential
equations is a research topic still at an embryonic state. This talk deals with the numerical solution of diffusion equations in the form
D^α_t u = D^2_x K(u), α ∈ R,
where D_x is the partial derivative in space, K is an arbitrary regular function, and D^α_t
denotes the Riemann-Liouville fractional derivative of order α.
The proposed numerical method combines a finite difference scheme in space with a spectral time integrator and preserves discrete versions of the conservation laws of the original differential equation [1, 2].
The conservative and convergence properties of the proposed method are verified by the computational solution of some numerical experiments.
References
[1] K. Burrage, A. Cardone, R. D’Ambrosio, B. Paternoster. Numerical solution of time fractional diffusion systems. Appl. Numer. Math., 116 (2017), 82–94.
[2] A. Cardone, G. Frasca-Caccia. Numerical conservation laws of time fractional diffusion PDEs. arXiv.2203.01966, (2022).
[3] A. Dur ́an, J. M. Sanz-Serna. The numerical integration of relative equilibrium solutions. Geometric theory. Nonlinearity, 11, 1547–1567, (1998).
[4] G. S. F. Frederico, D. F. M. Torres. Fractional conservation laws in optimal control theory. Nonlinear Dyn., 53 (2008), 215–222.
[5] E. Hairer, C. Lubich, G. Wanner. Geometric Numerical Integration. Structure Preserving Algorithms for Ordinary Differential Equations, volume 31 of Springer Series in Computational Mathematics. Springer, Berlin, second edition, 2006.
[6] S. Y. Lukashchuk. Conservation laws for time-fractional subdiffusion and diffusionwave equations. Nonlinear Dyn., 80 (2015), 791–80
Averaging procedure in variable-G cosmologies
No full textPrevious work in the literature had built a formalism for spatially averaged
equations for the scale factor, giving rise to an averaged Raychaudhuri equation
and averaged Hamiltonian constraint, which involve a backreaction source term. The
present paper extends these equations to include modelswith variableNewton parameter
and variable cosmological term, motivated by the nonperturbative renormalization
program for quantum gravity based upon the Einstein–Hilbert action.We focus on the
Brans–Dicke form of the renormalization-group improved action functional. The coupling
between backreaction and spatially averaged three-dimensional scalar curvature
is found to survive, and a variable-G cosmic quintet is found to emerge. Interestingly,
under suitable assumptions, an approximate solution can be found where the early universe
tends to a Friedmann–Lemaitre–Robertson–Walker model, while keeping track
of the original inhomogeneities through three effective fluids. The resulting qualitative
picture is that of a universe consisting of baryons only, while inhomogeneities average
out to give rise to the full dark-side phenomenology
Towards an averaging procedure in variable-G cosmologies
No full textPrevious work in the literature had built a formalism for spatially averaged equations
for the scale factor, giving rise to an averaged Raychaudhuri equation and averaged Hamiltonian
constraint, which involve a backreaction source term. The present paper extends these equations
to include models with variable Newton parameter and variable cosmological term, motivated
by the nonperturbative renormalization program for quantum gravity based upon the Einstein -
Hilbert action. We focus on the Brans-Dicke form of the renormalization-group improved action
functional. The coupling between backreaction and spatially averaged three-dimensional scalar
curvature is found to survive, and a variable-G cosmic quintet is found to emerge. Interestingly,
under suitable assumptions, an approximate solution can be found where the early universe tends
to a FLRW model, while keeping track of the original inhomogeneities through three effective
fluids. The resulting qualitative picture is that of a universe consisting of baryons only, while
inhomogeneities average out to give rise to the full dark-side phenomenology
Numerical schemes specially tuned for some evolutionary problems
No full textThe effective numerical integration of evolutionary problems arising from real-life applications requires the analysis of the characteristics of the phenomenon and of the corresponding mathematical model. The resulting numerical methods will therefore be able to reproduce the behavior of the analytical solution and to exploit the knowledge on the problem to reduce the computational effort. This approach has been developed for some classes of differential systems and for some classes of problems with memory modeled by integral or fractional equations.
Problems like advection-diffusion or reaction-diffusion problems are usually solved by a semidiscretization along space, which gives raise to (large) systems of ordinary differential systems characterized by a stiff part and a non-stiff one. IMEX methods treat implicitly the stiff part and explicitly the non-stiff one, in order to have strong stability properties and to reduce the computational cost. We introduce a class of IMEX general linear methods which have no coupling order conditions, do not suffer of the order reduction phenomenon thanks to the high stage order, and have optimal stability properties.
Periodic phenomena with memory, like the spread of seasonal diseases, are modeled by Volterra integral equations with periodic solution. Classical methods require a small stepsize to follow the oscillations.We apply the exponential fitting technique [8] to derive direct quadrature methods with parameters depending on an estimate of the frequency. The error is smaller than the error of classical methods, when periodic problems are treated; the numerical stability is not affected by the accuracy of the estimate of the frequency.
Fractional models can represent memory effects of natural processes and also the anomalous kinetics of some processes in physics, chemistry, pharmacokinetis. Here we focus on the numerical solution of time-fractional reaction-diffusion systems, by a spectral technique along time and a finite difference scheme along space, which are specially designed to reproduce the behavior of the analytical solution and to simplify the overall computation.
The results presented here have been obtained by various collaborations, with K. Burrage, R. D’Ambrosio, L.Gr. Ixaru, Z. Jackiewicz, B. Paternoster, A. Sandu, G. Santomauro, H. Zhang.
References
[1] Ascher, U.M., Ruuth, S.J., Spiteri, R.J. Implicit-explicit Runge-Kutta methods for timedependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997).
[2] Cardone, A., Jackiewicz, Z., Sandu, A., Zhang, H., Extrapolated implicit-explicit Runge-Kutta methods. Math. Model. Anal. 19, 18–43 (2014).
[3] Cardone, A., Jackiewicz, Z., Sandu, A., Zhang, H., Extrapolation-based implicit-explicit general linear methods. Numer. Algorithms 65, 377–399 (2014).
[4] Cardone, A., Jackiewicz, Z., Sandu, A., Zhang, H., Construction of highly stable implicit-explicit general linear methods, accepted for publication in Discrete Contin. Dyn. Systs.
[5] A. Cardone, L. Gr. Ixaru, and B. Paternoster, Exponential fitting direct quadrature methods for Volterra integral equations, Numer. Algorithms 55, no. 4, 467-480 (2010).
[6] A. Cardone, L.Gr. Ixaru, B. Paternoster, and G. Santomauro, Ef-gaussian direct quadrature methods for Volterra integral equations with periodic solution, Math. Comput. Simul., in press.
[7] V. Gafiychuk, B. Datsko, and V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems, J. Comput. Appl. Math. 220(1-2), 215-225 (2008).
[8] L.Gr. Ixaru, G. Vanden Berghe, (2004) Exponential Fitting. Kluwer Academic Publishers, Dordrecht.
[9] L. Pareschi, G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput. 25(1-2), 129–155 (2005)
Efficient general linear methods for non-stiff differential equations
No full textThe aim of our research is the construction and analysis of efficient general linear methods (GLM), which achieve a good balance between accuracy and stability properties.
In order to reach our goal we consider the class of GLMs with quadratic stability (QS), i.e. methods whose stability function has only two non-zero roots [4]. This property simplifies the study of stability and the search for methods with high order and good stability properties. In this talk we describe the conditions which guarantee the QS property and the construction of explicit Nordsieck methods with QS and maximum area of the region of
absolute stability [2]. The search for these methods with high order is realized by various optimization routines [3], and the analogous search for another class of GLMs has been carried out in [1]. Examples of methods which compare favorably with respect to existing explicit GLM are presented, up to order six. Some issues concerning the implementation of our methods in a variable-step algorithm are addressed, such as the estimate of the
local error and the computation of the input vector for the next step. This is a joint work with G. Izzo from Università di Napoli 'Federico II' and Z. Jackiewicz from Arizona State University.
[1] M. Bras, A. Cardone, Construction of Efficient General Linear Methods for Non-Stiff Differential Systems, Math. Model. Anal. 17, 171-189 (2012).
[2] A. Cardone, Z. Jackiewicz, Explicit Nordsieck methods with quadratic stability, Numer. Algorithms, 60 (1), 1{25 (2012).
[3] A. Cardone, Z. Jackiewicz, H. D. Mittelmann, Optimization-based search for Nordsieck methods of high order with quadratic stability, to appear in Math. Model. Anal.
[4] Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations, John Wiley, Hoboken, New Jersey, 2009
High order exponentially fitted methods for periodic Volterra integral equations
No full textVolterra integral equations with periodic solution model a number of periodic phenomena with memory, e.g. the spread of seasonal epidemics. An efficient and accurate numerical solution of these equations may be found by means of special purpose methods, which exploit the a priori knowledge of the qualitative behavior of the solution. On this direction, we propose exponentially-fitted direct quadrature methods of high order. The coefficients of these methods depend on the frequency of the problem, to reduce the error when
periodic problems are treated and an estimate of the frequency is available. In this poster we present the construction and analysis of these methods, and illustrate their performances on some significant test problems.
REFERENCES
[1] A. Cardone, L.Gr. Ixaru and B. Paternoster. Exponential fitting direct quadrature methods for Volterra integral equations. Numer. Algorithms, 55 (4), 2010, 467 – 480.
[2] A. Cardone, L.Gr. Ixaru, B. Paternoster and G. Santomauro. Ef-Gaussian direct quadrature methods for Volterra integral equations with periodic solution. Math. Comput. Simul., 110 2015, 125 – 143.
[3] L.Gr. Ixaru and G. Vanden Berghe. Exponential fitting. Kluwer Academic Publishers, Dordrecht, 2004
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