Related papers: A Computer-Assisted Stability Proof for a Stationa…
In this paper we establish the orbital stability of periodic traveling waves for a general class of dispersive equations. We use the Implicit Function Theorem to guarantee the existence of smooth solutions depending of the corresponding…
We present a class of new explicit and stable numerical algorithms to solve the spatially discretized linear heat or diffusion equation. After discretizing the space and the time variables like conventional finite difference methods, we do…
We carry out a comprehensive linear stability analysis of active Brownian particle systems around a constant homogeneous state. These scalar models, being important prototypes for the continuous description of active matter, are…
Large time dynamics of reaction-diffusion systems modeling some irreversible reaction networks are investigated. Depending on initial masses, these networks possibly possess boundary equilibria, where some of the chemical concentrations are…
Convection-diffusion equations arise in a variety of applications such as particle transport, electromagnetics, and magnetohydrodynamics. Simulation of the convection-dominated regime for these problems, even with high-fidelity techniques,…
We study a fractional reaction-diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides…
A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic…
The aim of this paper is to contribute to the understanding of the pattern formation phenomenon in reaction-diffusion equations coupled with ordinary differential equations. Such systems of equations arise, for example, from modeling of…
We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized…
The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit…
In this paper we present a fast algorithm for the numerical solution of systems of reaction-diffusion equations, $\partial_t u + a \cdot \nabla u = \Delta u + F (x, t, u)$, $x \in \Omega \subset \mathbf{R}^3$, $t > 0$. Here, $u$ is a…
In this study a stabilized finite element method for solving advection-diffusion-reaction equation with spatially variable coefficients has been carried out. Here subgrid scale approach along with algebraic approximation to the sub-scales…
In this paper, extending previous results of \cite{J1}, we obtain pointwise nonlinear stability of periodic traveling reaction-diffusion waves, assuming spectral linearized stability, under nonlocalized perturbations. More precisely, we…
We deal with a mass-conserved three-component reaction-diffusion system which is proposed by a model describing the dynamics of wavelike actin polymerization in the macropinocytosis and numerically exhibits dynamical patterns such as…
We investigate a nonlinear parabolic reaction-diffusion equation describing the oxygen concentration in encapsulated pancreatic cells with a general core-shell geometry. This geometry introduces a discontinuous diffusion coefficient as the…
We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable travelling wave solutions to the deterministic system retain their orbital stability if the…
We study analytically and numerically a bistable reaction-diffusion equation on an arbitrary finite network. We prove that stable fixed points (multi-fronts) exist for any configuration as long as the diffusion is small. We also study fold…
Radiative transfer is essential in astronomy, both for interpreting observations and simulating various astrophysical phenomena. However, self-consistent line radiative transfer is computationally expensive, especially in 3D. To reduce the…
Mathematical modeling of fluid dynamics for computer graphics requires high levels of theoretical rigor to ensure visually plausible and computationally efficient simulations. This paper presents an in-depth theoretical framework analyzing…
Normally hyperbolic invariant manifolds theory provides an efficient tool for proving diffusion in dynamical systems. In this paper we develop a methodology for computer assisted proofs of diffusion in a-priori chaotic systems based on this…