Numerical Approaches of the Generalized Time-Fractional Burgers’ Equation with Time-Variable Coefficients

The generalized time-fractional, one-dimensional, nonlinear Burgers equation with time-variable coefficients is numerically investigated. The classical Burgers equation is generalized by considering the generalized Atangana-Baleanu time-fractional derivative. The studied model contains as particular cases the Burgers equation with Atangana-Baleanu, Caputo-Fabrizio, and Caputo time-fractional derivatives. A numerical scheme, based on the finite-difference approximations and some integral representations of the two-parameter Mittag-Leffler functions, has been developed. Numerical solutions of a particular problem with initial and boundary values are determined by employing the proposed method. The numerical results are plotted to compare solutions corresponding to the problems with time-fractional derivatives with different kernels.

Towards Generic Simulation for Demanding Stochastic Processes

We outline and test a new methodology for genuine simulation of stochastic processes with any dependence structure and any marginal distribution. We reproduce time dependence with a generalized, time symmetric or asymmetric, moving-average scheme. This implements linear filtering of non-Gaussian white noise, with the weights of the filter determined by analytical equations, in terms of the autocovariance of the process. We approximate the marginal distribution of the process, irrespective of its type, using a number of its cumulants, which in turn determine the cumulants of white noise, in a manner that can readily support the generation of random numbers from that approximation, so that it be applicable for stochastic simulation. The simulation method is genuine as it uses the process of interest directly, without any transformation (e.g., normalization). We illustrate the method in a number of synthetic and real-world applications, with either persistence or antipersistence, and with non-Gaussian marginal distributions that are bounded, thus making the problem more demanding. These include distributions bounded from both sides, such as uniform, and bounded from below, such as exponential and Pareto, possibly having a discontinuity at the origin (intermittence). All examples studied show the satisfactory performance of the method.

Quadratic First Integrals of Time-dependent Dynamical Systems of the Form q¨a=-Γbcaq˙bq˙c-ω(t)Qa(q)

We consider the time-dependent dynamical system q¨a=−Γbcaq˙bq˙c−ω(t)Qa(q) where ω(t) is a non-zero arbitrary function and the connection coefficients Γbca are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) I we assume that I=Kabq˙aq˙b+Kaq˙a+K where the unknown coefficients Kab,Ka,K are tensors depending on t,qa and impose the condition dIdt=0. This condition leads to a system of partial differential equations (PDEs) involving the quantities Kab,Ka,K,ω(t) and Qa(q). From these PDEs, it follows that Kab is a Killing tensor (KT) of the kinetic metric. We use the KT Kab in two ways: a. We assume a general polynomial form in t both for Kab and Ka; b. We express Kab in a basis of the KTs of order 2 of the kinetic metric assuming the coefficients to be functions of t. In both cases, this leads to a new system of PDEs whose solution requires that we specify either ω(t) or Qa(q). We consider first that ω(t) is a general polynomial in t and find that in this case the dynamical system admits two independent QFIs which we collect in a Theorem. Next, we specify the quantities Qa(q) to be the generalized time-dependent Kepler potential V=−ω(t)rν and determine the functions ω(t) for which QFIs are admitted. We extend the discussion to the non-linear differential equation x¨=−ω(t)xμ+ϕ(t)x˙(μ≠−1) and compute the relation between the coefficients ω(t),ϕ(t) so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane–Emden equation.

A generalized kinetic model of the advection-dispersion process in a sorbing medium

A new time-fractional derivative with Mittag-Leffler memory kernel, called the generalized Atangana-Baleanu time-fractional derivative is defined along with the associated integral operator. Some properties of the new operators are proved. The new operator is suitable to generate by particularization the known Atangana-Baleanu, Caputo-Fabrizio and Caputo time-fractional derivatives. A generalized mathematical model of the advection-dispersion process with kinetic adsorption is formulated by considering the constitutive equation of the diffusive flux with the new generalized time-fractional derivative. Analytical solutions of the generalized advection-dispersion equation with kinetic adsorption are determined using the Laplace transform method. The solution corresponding to the ordinary model is compared with solutions corresponding to the four models with fractional derivatives.

Towards Generic Simulation for Demanding Stochastic Processes

We outline and test a new methodology for genuine simulation of stochastic processes with any dependence and any marginal distribution. We reproduce time dependence with a generalized, time symmetric or asymmetric, moving-average scheme. This implements linear filtering of non-Gaussian white noise, with the weights of the filter determined by analytical equations in terms of the autocovariance of the process. We approximate the marginal distribution of the process, irrespective of its type, using a number of its cumulants, which in turn determine the cumulants of white noise in a manner that can readily support the generation of random numbers from that approximation, so that it be applicable for stochastic simulation. The simulation method is genuine as it uses the process of interest directly without any transformation (e.g. normalization). We illustrate the method in a number of synthetic and real-world applications with either persistence or antipersistence, and with non-Gaussian marginal distributions that are bounded, thus making the problem more demanding. These include distributions bounded from both sides, such as uniform, and bounded form below, such as exponential and Pareto, possibly having a discontinuity at the origin (intermittence). All examples studied show the satisfactory performance of the method.

Probability of infectious disease in humans during epidemic

Popular SIR models and their modifications used to generate predictions about epidemics and, specifically, the COVID-19 pandemic, are inadequate. The aim of this study was to find the laws describing the probability of infection in a biological object. Using theoretical methods of research based on the probability theory, we constructed the laws describing the probability of infection in a human depending on the infective dose and considering the temporal characteristics of a given infection. The so-called generalized time-factor law, which factors in the time of onset and the duration of an infectious disease, was found to be the most general. Among its special cases are the law describing the probability of infection developing by some point in time t, depending on the infective dose, and the law that does not factor in the time of onset. The study produced a full list of quantitative characteristics of pathogen virulence. The laws described in the study help to solve practical tasks and should lie at the core of mathematical epidemiological modeling.