<p style='text-indent:20px;'>We consider an initial and Dirichlet boundary value problem for a logarithmic Schrödinger equation over a two dimensional rectangular domain. We construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization, with or without regularizing the logarithmic term. We develop a convergence analysis yielding a new almost second order a priori error estimates in the discrete <inline-formula><tex-math id="M1">\begin{document}$ L_t^{\infty}(L_x^2) $\end{document}</tex-math></inline-formula> norm, and we show results from numerical experiments exposing the efficiency of the method proposed. It is the first time in the literature where an error estimate for a numerical method applied to the logarithmic Schrödinger equation is provided, without regularizing its nonlinear term.</p>
We formulate an initial and Dirichlet boundary value problem for a semilinear heat equation with logarithmic nonlinearity over a two-dimensional rectangular domain. We approximate its solution by employing the standard second-order finite difference method for space discretization, and a linearized backward Euler method, or, a linearized BDF2 method for time stepping. For the linearized backward Euler finite difference method, we derive an almost optimal order error estimate in the discrete L t ∞ ( L x ∞ ) L^{\infty}_{t}(L^{\infty}_{x}) -norm without imposing mesh conditions, and for the linearized BDF2 finite difference method, we establish an almost optimal order error estimate in the discrete L t ∞ ( H x 1 ) L^{\infty}_{t}(H^{1}_{x}) -norm, allowing a mild mesh condition to be satisfied. Finally, we show the efficiency of the numerical methods proposed, by exposing results from numerical experiments. It is the first time in the literature where numerical methods for the approximation of the solution to the heat equation with logarithmic nonlinearity are applied and analysed.
Covid-19 is the most recent strain from the corona virus family that its rapid spread across the globe has caused a pandemic, resulting in over 200,000,000 infections and over 4,000,000 deaths so far. Many countries had to impose full lockdowns, with serious effects in all aspects of everyday life (economic, social etc.). In this paper, a computational framework is introduced, aptly named COVID-LIBERTY, in order to assist the study of the pandemic in Europe. Initially, the mathematics and details of the computational engine of the framework, a feed-forward, back-propagation Artificial Neural Network are presented. 5 European countries with similar population numbers were chosen and we examined the main factors that influence the spread of the virus, in order to be taken into consideration in the simulations. In this way lockdown, seasonal variability and virus effective reproduction were considered. The effectiveness of lockdown in the spread of the virus was examined and the Lockdown Index was introduced. Moreover, the relation of Covid- 19 to seasonal variability was demonstrated and the parametrization of seasonality presented.
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