相关论文: A fast solver for systems of reaction-diffusion eq…
We present an efficient integral equation approach to solve the heat equation, $u_t (\x) - \Delta u(\x) = F(\x,t)$, in a two-dimensional, multiply connected domain, and with Dirichlet boundary conditions. Instead of using integral equations…
We discuss several qualitative properties of the solutions of reaction-diffusion systems and equations of the form $u_t = \epsilon^2 D \Delta u + f(u,x,\epsilon t)$, that are used in modeling pattern formation. We analyze the diffusion…
The first part of this paper is devoted to the derivation of a technical result, related to the stability of the solution of a reaction-diffusion equation $u_t-\Delta u = f(x,u)$ on $(0,\infty)\times \mathbb{R}^N$, where the initial datum…
A condition is identified that implies that solutions to the stochastic reaction-diffusion equation $\frac{\partial u}{\partial t} = \mathcal{A} u + f(u) + \sigma(u) \dot{W}$ on a bounded spatial domain never explode. We consider the case…
We introduce and analyze a numerical approximation of the porous medium equation with fractional potential pressure introduced by Caffarelli and V\'azquez: \[ \partial_t u = \nabla \cdot (u^{m-1}\nabla (-\Delta)^{-\sigma}u) \qquad…
The main subject of this paper is a computer assisted stability proof for a stationary solution of reaction diffusion equations in one dimensional space. We use Nakao's numerical verification method to enclose a stationary solution of…
In this paper, we develop fast procedures for solving linear systems arising from discretization of ordinary and partial differential equations with Caputo fractional derivative w.r.t time variable. First, we consider a finite difference…
Heat transfer in a fluid can be greatly enhanced by natural convection, giving rise to the nuanced relationship between the Nusselt number and Rayleigh number that has been a focus of modern fluid dynamics. Our work explores convection in…
We advance an exact, explicit form for the solutions to the fractional diffusion-advection equation. Numerical analysis of this equation shows that its solutions resemble power-laws.
This paper develops a two-level fourth-order scheme for solving time-fractional convection-diffusion-reaction equation with variable coefficients subjected to suitable initial and boundary conditions. The basis properties of the new…
Our aim is to study the limit of the solution of reaction-diffusion porous medium equation with linear drift $\displaystyle\partial_t u -\Delta u^m +\nabla \cdot (u \: V)=g(t,x,u) $, as $m\to\infty.$ We study the problem in bounded domain…
In this paper, we define the transition semi-wave solution of the following reaction diffusion equation with free boundaries \begin{equation}\label{0.1} \left\{ \begin{aligned} u_{t}=u_{xx}+f(t,x,u),\ \ &t\in\Real, x<h(t), u(t,h(t))=0,\ \…
In this note, a numerical method based on finite differences to solve a class of nonlinear advection-diffusion fractional differential equation is proposed. The fractional operator considered here is the fractional Riemann-Liouville…
In this paper, we present a method based on Radial Basis Function (RBF)-generated Finite Differences (FD) for numerically solving diffusion and reaction-diffusion equations (PDEs) on closed surfaces embedded in $\mathbb{R}^d$. Our method…
We will look at reaction-diffusion type equations of the following type, $$\partial^\beta_tV(t,x)=-(-\Delta)^{\alpha/2} V(t,x)+I^{1-\beta}_t[V(t,x)^{1+\eta}].$$ We first study the equation on the whole space by making sense of it via an…
Fractional calculus has been used to describe physical systems with complexity. Here, we show that a fractional calculus approach can restore or include complexity in any physical systems that can be described by partial differential…
In this paper we consider efficient algorithms for solving the algebraic equation ${\mathcal A}^\alpha {\bf u}={\bf f}$, $0< \alpha <1$, where ${\mathcal A}$ is a symmetric and positive definite matrix obtained form finite difference or…
We investigate a reaction-diffusion-advection equation of the form $u_t-u_{xx}+\beta u_x=f(u)$ $(t>0,\,0<x<h(t))$ with mixed boundary condition at $x=0$ and a free boundary condition at $x=h(t)$. Such a model may be applied to describe the…
Systems of reaction-diffusion equations are commonly used in biological models of food chains. The populations and their complicated interactions present numerous challenges in theory and in numerical approximation. In particular,…
Diffuse scattering is a rich source of information about disorder in crystalline materials, which can be modelled using atomistic techniques such as Monte Carlo and molecular dynamics simulations. Modern X-ray and neutron scattering…