Related papers: Quasi-stationary Stefan problem and computer simul…
Nowadays, fractional differential equations are a well established tool to model phenomena from the real world. Since the analytical solution is rarely available, there is a great effort in constructing efficient numerical methods for their…
A numerical approach for the approximation of inertial manifolds of stochastic evolutionary equations with multiplicative noise is presented and illustrated. After splitting the stochastic evolutionary equations into a backward and a…
Phase field equations describe the novel approach to the Stefan problems. We calculate these equations numerically performed in two-dimensions. We take full advantage of the phase field parameter $\varphi$ to track the interface on which…
Random invariant manifolds often provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the…
An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary…
This work deals with the one-dimensional Stefan problem with a general time-dependent boundary condition at the fixed boundary. Stochastic solutions are obtained using discrete random walks, and the results are compared with analytic…
We consider a linear partial integro-differential equation that arises in the modeling of various physical and biological processes. We study the problem in a spatial periodic domain. We analyze numerical stability and numerical convergence…
We review applications of theory of classical and quantum integrable systems to the free-boundary problems of fluid mechanics as well as to corresponding problems of statistical mechanics. We also review important exact results obtained in…
We study multi-phase Stefan problem with increasing Riemann initial data and with generally negative latent specific heats for the phase transitions. We propose the variational formulation of self-similar solutions, which allows to find…
We study numerical algorithms to solve a specific Partial Differential Equation (PDE), namely the Stefan problem, using Physics Informed Neural Networks (PINNs). This problem describes the heat propagation in a liquid-solid phase change…
We study a space-fractional Stefan problem, where the non-local diffusion flux is modeled by the Caputo derivative. We obtain the unique existence of classical solution to this problem.
A number of recent studies have proposed that linear representations are appropriate for solving nonlinear dynamical systems with quantum computers, which fundamentally act linearly on a wave function in a Hilbert space. Linear…
In this work, we present a novel approach for solving stochastic shape optimization problems. Our method is the extension of the classical stochastic gradient method to infinite-dimensional shape manifolds. We prove convergence of the…
We study a space-fractional Stefan problem with the Dirichlet boundary conditions. It is a model that describes superdiffusive phenomena. Our main result is the existence of the unique classical solution to this problem. In the proof we…
Quantum computing is gaining increased attention as a potential way to speed up simulations of physical systems, and it is also of interest to apply it to simulations of classical plasmas. However, quantum information science is…
So far the problem of interface behavior upon phase transition has not yet acquired a satisfactory mathematical formulation due to a variety of the physical phenomena involved. Analytical solutions exist only for elementary problems…
In this work we introduce a new system of partial differential equations as a simplified model for the evolution of reversible martensitic transformations under thermal cycling in low hysteresis alloys. The model is developed in the context…
Most heat transfer models for bulk crystal growth rely on the classical Stefan formulation to evaluate interface motion during phase change. However, when the interface is non-smooth the use of the classical Stefan formulation may lead to…
Moving boundary problems allow to model systems with phase transition at an inner boundary. Driven by problems in economics and finance, in particular modeling of limit order books, we consider a stochastic and non-linear extension of the…
We study the numerical solution of nonlinear partially observed optimal stopping problems. The system state is taken to be a multi-dimensional diffusion and drives the drift of the observation process, which is another multi-dimensional…