22,877 research outputs found
Gravitational lensing in fourth-order gravity
No full textGravitational lensing is investigated in the weak field limit of fourth order gravity in which the Lagrangian of the gravitational field is modified by replacing the Ricci scalar curvature R with an analytical expression f(R). Considering the case of a pointlike lens, we study the behavior of the deflection angle in the case of power-law Lagrangians, i.e. with f(R)Rn. In order to investigate possible detectable signatures, the position of the Einstein ring and the solutions of the lens equation are evaluated considering the change with respect to the standard case. Effects on the amplification of the images and the Paczynski curve in microlensing experiments are also estimated. © 2006 The American Physical Society
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)
Low surface brightness galaxies rotation curves in the low energy limit of R**n gravity: no need for dark matter?
No full textLow surface brightness galaxy rotation curves in the low energy limit of Rn gravity: No need for dark matter?
No full textWe investigate the possibility that the observed flatness of the rotation curves of spiral galaxies is not evidence for the existence of dark matter haloes, but rather a signal of the breakdown of General Relativity. To this aim, we consider power-law fourth-order theories of gravity obtained by replacing the scalar curvature R with f(R) = f0 Rn in the gravity Lagrangian. We show that, in the low energy limit, the gravitational potential generated by a point-like source may be written as φ(r) α r -1[1 + (r/rc)β] with β a function of the slope n of the gravity Lagrangian and rc a scalelength depending on the gravitating system properties. In order to apply the model to realistic systems, we compute the modified potential and the rotation curve for spherically symmetric and for thin disc mass distributions. It turns out that the potential is still asymptotically decreasing, but the corrected rotation curve, although not flat, is higher than the Newtonian one, thus offering the possibility to fit rotation curves without dark matter. To test the viability of the model, we consider a sample of 15 low surface brightness galaxies with combined H I and Hα measurements of the rotation curve extending in the putative dark matter dominated region. We find a very good agreement between the theoretical rotation curve and the data using only stellar disc and interstellar gas when the slope n of the gravity Lagrangian is set to the value n = 3.5 (giving β = 0.817) obtained by fitting the Type la supernova Hubble diagram with the assumed power-law f(R) model and no dark matter. The excellent agreement between theoretical and observed rotation curves and the values of the stellar mass-to-light ratios in agreement with the predictions of population synthesis models make us confident that Rn gravity may represent a good candidate to solve both the dark energy problem on cosmological scales and the dark matter one on galactic scales with the same value of the slope n of the higher-order gravity Lagrangian. © 2007 RAS
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