Related papers: Effective Approximation for a Stochastic System wi…
We consider the stochastic Swift-Hohenberg equation on a large domain near its change of stability. We show that, under the appropriate scaling, its solutions can be approximated by a periodic wave, which is modulated by the solutions to a…
We study the following ultraparabolic equation \[ \frac{\partial}{\partial t}u\left(t,s\right)+\frac{\partial}{\partial…
The optimized expansion is used to formulate a systematic approximation scheme to the probability distribution of a stochastic system. The first order approximation for the one-dimensional system driven by noise in an anharmonic potential…
We consider an elliptic equation with purely imaginary, highly heterogeneous, and large random potential with a sufficiently rapidly decaying correlation function. We show that its solution is well approximated by the solution to a…
We construct a probabilistic representation of a system of fully coupled parabolic equations arising as a model describing spatial segregation of interacting population species. We derive a closed system of stochastic equations such that…
This paper provides a provably quasi-optimal preconditioning strategy of the linear Schr\"odinger eigenvalue problem with periodic potentials for a possibly non-uniform spatial expansion of the domain. The quasi-optimality is achieved by…
In this work, simple exact results are presented for summations in two-particle potential with long-range interactions. Polygamma function is used to evaluate summations. Results are found when a periodic media is consider. Periodic…
Stochastic systems in biology often exhibit substantial variability within and between cells. This variability, as well as having dramatic functional consequences, provides information about the underlying details of the system's behaviour.…
We develop an essentially optimal finite element approach for solving ergodic stochastic two-scale elliptic equations whose two-scale coefficient may depend also on the slow variable. We solve the limiting stochastic two-scale homogenized…
A systematic procedure to derive exact solutions of the associated Lame equation for an arbitrary value of the energy is presented. Supersymmetric transformations in which the seed solutions have factorization energies inside the gaps are…
In this paper we investigate the porous medium equation with a fractional temporal derivative. We justify that the resulting equation emerges when we consider the waiting-time (or trapping) phenomenon that can happen in the medium. Our…
In this paper, we construct a type of interacting particle systems to approximate a class of stochastic different equations whose coefficients depend on the conditional probability distributions of the processes given partial observations.…
We present a stochastic numerical method for solving fully non-linear free boundary problems of parabolic type and provide a rate of convergence under reasonable conditions on the non-linearity.
Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system…
We present a space-time multiscale method for a parabolic model problem with an underlying coefficient that may be highly oscillatory with respect to both the spatial and the temporal variables. The method is based on the framework of the…
A large system of ordinary differential equations is approximated by a parabolic partial differential equation with dynamic boundary condition and a different one with Robin boundary condition. Using the theory of differential operators…
We consider a mixed dimensional elliptic partial differential equation posed in a bulk domain with a large number of embedded interfaces. In particular, we study well-posedness of the problem and regularity of the solution. We also propose…
In this paper, a nonlinear system of fractional ordinary differential equations with multiple scales in time is investigated. We are interested in the effective long-term computation of the solution. The main challenge is how to obtain the…
A new tool for modeling electrochemical kinetics is presented. An extension of the Stochastic Simulation Algorithm framework to electrochemical systems is proposed. The physical justifications and constraints for the derivation of a…
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…