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In this paper we apply a finite difference lattice Boltzmann model to study the phase separation in a two-dimensional liquid-vapor system. Spurious numerical effects in macroscopic equations are discussed and an appropriate numerical scheme…
In this paper a finite difference/local discontinuous Galerkin method for the fractional diffusion-wave equation is presented and analyzed. We first propose a new finite difference method to approximate the time fractional derivatives, and…
We study a 12-parameter stochastic process involving particles with two-site interaction and hard-core repulsion on a $d$-dimensional lattice. In this model, which includes the asymmetric exclusion process, contact processes and other…
Continuous Time Random Walk models (CTRW) of anomalous diffusion are studied, where the anomalous exponent $\beta(x) \in (0,1)$ varies in space. This type of situation occurs e.g. in biophysics, where the density of the intracellular matrix…
Starting from a continuous time random walk (CTRW) model of particles that may evanesce as they walk, our goal is to arrive at macroscopic integro-differential equations for the probability density for a particle to be found at point r at…
In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the…
Existing deterministic variational inference approaches for diffusion processes use simple proposals and target the marginal density of the posterior. We construct the variational process as a controlled version of the prior process and…
Lattice Boltzmann models are briefly introduced together with references to methods used to predict their ability for simulations of systems described by partial differential equations that are first order in time and low order in space…
Stochastic diffusion equations are crucial for modeling a range of physical phenomena influenced by uncertainties. We introduce the generalized finite difference method for solving these equations. Then, we examine its consistency,…
A class of Laplace transforms is examined to show that particular cases of this class are associated with production-destruction and reaction-diffusion problems in physics, study of differences of independently distributed random variables…
We present a Bayesian non-parametric way of inferring stochastic differential equations for both regression tasks and continuous-time dynamical modelling. The work has high emphasis on the stochastic part of the differential equation, also…
For the regime-switching diffusion process with and without advection term we propose an integro-differential equation describing the densities of states continuously distributed over a segment. We demonstrate that there exists a…
A diffusive lattice gas is characterized by the diffusion coefficient depending only on the density. The Green-Kubo formula for diffusivity can be represented as a variational formula, but even when the equilibrium properties of a lattice…
This paper is concerned with the large deviation principle of the stochastic reaction-diffusion lattice systems defined on the N-dimensional integer set, where the nonlinear drift term is locally Lipschitz continuous with polynomial growth…
The diffusion system with time-fractional order derivative is of great importance mathematically due to the nonlocal property of the fractional order derivative, which can be applied to model the physical phenomena with memory effects. We…
In this paper, we study numerical methods for the homogenization of linear second-order elliptic equations in nondivergence-form with periodic diffusion coefficients and large drift terms. Upon noting that the effective diffusion matrix can…
In this work we investigate an inverse problem of identifying a spatially variable order in the one-dimensional subdiffusion model from the boundary flux measurement. The model involves a generalized Caputo derivative in time, and arises in…
A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations can be interpreted as probability densities evolving in time…
We consider a time-space fractional diffusion equation with a variable coefficient and investigate the inverse problem of reconstructing the source term, after regularizing the problem with the quasiboundary value method to mitigate the…
We consider the transport of conserved charges in spatially inhomogeneous quantum systems with a discrete lattice symmetry. We analyse the retarded two point functions involving the charge and the associated currents at long wavelengths,…