Fractional diffusions with time-varying coefficients

This paper is concerned with the fractionalized diffusion equations governing the law of the fractional Brownian motion $B_H(t)$. We obtain solutions of these equations which are probability laws extending that of $B_H(t)$. Our analysis is based on McBride fractional operators generalizing the hyper-Bessel operators $L$ and converting their fractional power $L^{\alpha}$ into Erd\'elyi--Kober fractional integrals. We study also probabilistic properties of the r.v.'s whose distributions satisfy space-time fractional equations involving Caputo and Riesz fractional derivatives. Some results emerging from the analysis of fractional equations with time-varying coefficients have the form of distributions of time-changed r.v.'s

Similarity transformations for Nonlinear Schrodinger Equations with time varying coefficients: Exact results

In this paper we use a similarity transformation connecting some families of Nonlinear Schrodinger equations with time-varying coefficients with the autonomous cubic nonlinear Schrodinger equation. This transformation allows one to apply all known results for that equation to the non-autonomous case with the additional dynamics introduced by the transformation itself. In particular, using stationary solutions of the autonomous nonlinear Schrodinger equation we can construct exact breathing solutions to multidimensional non-autonomous nonlinear Schrodinger equations. An application is given in which we explicitly construct time dependent coefficients leading to solutions displaying weak collapse in three-dimensional scenarios. Our results can find physical applicability in mean field models of Bose-Einstein condensates and in the field of dispersion-managed optical systems

Homogenization of a singular random one dimensional parabolic PDE with time varying coefficients

The paper studies homogenization problem for a non-autonomous parabolic equation with a large random rapidly oscillating potential in the case of one dimensional spatial variable. We show that if the potential is a statistically homogeneous rapidly oscillating function of both temporal and spatial variables then, under proper mixing assumptions, the limit equation is deterministic and the convergence in probability holds. To the contrary, for the potential having a microstructure only in one of these variables, the limit problem is stochastic and we only prove the convergence in law

Flexible Modelling of Discrete Failure Time Including Time-Varying Smooth Effects

Discrete survival models have been extended in several ways. More flexible models are obtained by including time-varying coefficients and covariates which determine the hazard rate in an additive but not further specified form. In this paper a general model is considered which comprises both types of covariate effects. An additional extension is the incorporation of smooth interaction between time and covariates. Thus in the linear predictor smooth effects of covariates which may vary across time are allowed. It is shown how simple duration models produce artefacts which may be avoided by flexible models. For the general model which includes parametric terms, time-varying coefficients in parametric terms and time-varying smooth effects estimation procedures are derived which are based on the regularized expansion of smooth effects in basis functions

A semiparametric recurrent events model with time-varying coefficients

We consider a recurrent events model with time-varying coefficients motivated by two clinical applications. We use a random effects (Gaussian frailty) model to describe the intensity of recurrent events. The model can accommodate both time-varying and time-constant coefficients. We use the penalized spline method to estimate the time-varying coefficients. We use Laplace approximation to evaluate the penalized likelihood without a closed form. We estimate the smoothing parameters in a similar way to variance components. We conduct simulations to evaluate the performance of the estimates for both time-varying and time-independent coefficients. We apply this method to analyze two data sets: a stroke study and a child wheeze study