arXiv subjects

Ngartelbaye Guerngar

Publications and source records attributed to Ngartelbaye Guerngar.

7 recordsLinked to original sources

arXiv2025

Space-time fractional SPDEs with locally Lipschitz coefficients: well-posedness

In this article, we study the space-time SPDE $$ \partial_t^βu=-(-Δ)^{α/2} u+I_t^{1-β}[b(u)+σ(u)\dot{W}],$$ where $u=u(t,x)$ is defined for $(t,x)\in\mathbb{R}_+\times \mathbb{R},$ $β\in(0,1), α\in(0,2)$ and $\dot{W}$ denotes a space-time white noise. It has long been conjectured that this equation has a unique solution with finite moments under the minimal assumptions of locally Lipschitz coefficients $b$ and $σ$ with linear growth. We prove that this SPDE is well-posed under the assumptions that the initial condition $u_0$ is bounded and measurable, and the functions $b$ and $σ$ are locally Lipschitz and have at-most linear growth and some conditions on the Lipschitz constants on the truncated versions of $b$ and $σ$. Our results generalize the work of Foondun et al.(2025) to a space-time fractional setting.

math.PR↗

arXiv2024

Propagation of high peaks for the space-time fractional stochastic partial differential equations

We study the space-time nonlinear fractional stochastic heat equation driven by a space-time white noise, \begin{align*} \partial_t^βu(t,x)=-(-Δ)^{α/2}u(t,x)+I_t^{1-β}\Big[σ(u(t,x))\dot{W}(t,x)\Big],\ \ t>0, \ x\in \mathbb{R} , \end{align*} where $σ:\mathbb{R}\rightarrow\mathbb{R}$ is a globally Lipschitz function and the initial condition is a measure on $\mathbb{R}.$ Under some growth conditions on $σ,$ we derive two important properties about the moments of the solution: (i) For $p\geq 2,$ the $p^{\text{th}}$ absolute moment of the solution to the equation above grows exponentially with time. (ii) Moreover, the distances to the origin of the farthest high peaks of these moments grow exactly exponentially with time. Our results provide an extension of the work of Chen and Dalang (Stoch PDE: Anal Comp (2015) 3:360-397) to a time-fractional setting. We also show that condition (i) holds when we study the same equation for $x\in\mathbb{R}^d.$

math.PR↗

arXiv2022

Distributed-order space-time fractional diffusions in bounded domains

We provide explicit classical solutions and stochastic analogues for distributed-order space-time fractional diffusion equations on bounded domains with zero exterior boundary conditions. We also show that our results still hold when the mixing measure in the distributed-order time-derivative is singular.

math.AP↗

arXiv2020

A uniqueness determination of the fractional exponents in a three-parameter fractional diffusion

In this article, we consider the space-time Fractional (nonlocal) diffusion equation $$\partial_t^βu(t,x)={\mathtt{L}_D^{α_1,α_2}} u(t,x), \ \ t\geq 0, \ x\in D, $$ where $\partial_t^β$ is the Caputo fractional derivative of order $β\in (0,1)$ and the differential operator ${\mathtt{L}_D^{α_1,α_2}}$ is the generator of a Lévy process, sum of two symmetric independent $α_1-$stable and $α_2-$stable processes and ${D}$ is the open unit interval in $\mathbb{R}$. We consider a nonlocal inverse problem and show that the fractional exponents $β$ and $α_i, \ i=1,2$ are determined uniquely by the data $u(t, 0) = g(t),\ 0 < t < T.$ The uniqueness result is a theoretical background for determining experimentally the order of many anomalous diffusion phenomena, which are important in many fields, including physics and environmental engineering. We also discuss the numerical approximation of the inverse problem as a nonlinear least-squares problem and explore parameter sensitivity through numerical experiments.

math.AP↗

arXiv2020

Moment bounds of a class of stochastic heat equations driven by space-time colored noise in bounded domains

We consider the fractional stochastic heat type equation \begin{align*} \frac{\partial}{\partial t} u_t(x)=-(-Δ)^{α/2}u_t(x)+ξσ(u_t(x))\dot{F}(t,x),\ \ \ x\in D, \ \ t>0, \end{align*} with nonnegative bounded initial condition, where $α\in (0,2]$, $ξ>0$ is the noise level, $σ:\mathbb{R}\rightarrow\mathbb{R}$ is a globally Lipschitz function satisfying some growth conditions and the noise term behaves in space like the Riez kernel and is possibly correlated in time and $D$ is the unit open ball centered at the origin in $\mathbb{R}^d$. When the noise term is not correlated in time, we establish a change in the growth of the solution of these equations depending on the noise level $ξ$. On the other hand when the noise term behaves in time like the fractional Brownian motion with index $H\in (1/2,1)$, We also derive explicit bounds leading to a well-known intermittency property.

math.PR↗

arXiv2019

Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation $\partial_t^βu= -\big(-Δ\big)^{α/2}u-\big(-Δ\big)^{γ/2}u$

In this article, we consider the space-time fractional (nonlocal) equation characterizing the so-called "double-scale" anomalous diffusion $$\partial_t^βu(t, x) = -(-Δ)^{α/2}u(t,x) - (-Δ)^{γ/2}u(t,x) \ \ t> 0, \ -1<x<1, $$ where $\partial_t^β$ is the Caputo fractional derivative of order $β\in (0,1)$ and $0<α\leq γ<2.$ We consider a nonlocal inverse problem and show that the fractional exponents $β$, $α$ and $γ$ are determined uniquely by the data $u(t, 0) = g(t), \ 0 < t \leq T.$ The existence of the solution for the inverse problem is proved using the quasi-solution method which is based on minimizing an error functional between the output data and the additional data. In this context, an input-output mapping is defined and its continuity is established. The uniqueness of the solution for the inverse problem is proved by means of eigenfunction expansion of the solution of the forward problem and some basic properties of fractional Laplacian. A numerical method based on discretization of the minimization problem, namely the steepest descent method and a least squares approach, is proposed for the solution of the inverse problem. The numerical method determines the fractional exponents simultaneously. Finally, numerical examples with noise-free and noisy data illustrate applicability and high accuracy of the proposed method.

math.AP↗

arXiv2016

Some properties of non-linear fractional stochastic heat equations on bounded domains

Consider the following stochastic partial differential equation, \begin{equation*} \partial_t u_t(x)= \mathcal{L}u_t(x)+ ξσ(u_t(x)) \dot F(t,x), \end{equation*} where $ξ$ is a positive parameter and $σ$ is a globally Lipschitz continuous function. The stochastic forcing term $\dot F(t,x)$ is white in time but possibly colored in space. The operator $\mathcal{L}$ is a non-local operator. We study the behaviour of the solution with respect to the parameter $ξ$, extending the results in \cite{FoonNual} and \cite{Bin}

math.PR↗