Showing papers by "Erkan Nane published in 2017"

Asymptotic properties of some space-time fractional stochastic equations

Abstract: Consider non-linear time-fractional stochastic heat type equations of the following type, $$\begin{aligned} \partial ^\beta _tu_t(x)=- u (-\Delta )^{\alpha /2} u_t(x)+I^{1-\beta }_t[\lambda \sigma (u)\mathop {F}\limits ^{\cdot }(t,x)] \end{aligned}$$ in \((d+1)\) dimensions, where \( u >0, \beta \in (0,1), \alpha \in (0,2]\). The operator \(\partial ^\beta _t\) is the Caputo fractional derivative while \(-(-\Delta )^{\alpha /2} \) is the generator of an isotropic stable process and \(I^{1-\beta }_t\) is the Riesz fractional integral operator. The forcing noise denoted by \(\mathop {F}\limits ^{\cdot }(t,x)\) is a Gaussian noise. And the multiplicative non-linearity \(\sigma :\mathbb {R}\rightarrow \mathbb {R}\) is assumed to be globally Lipschitz continuous. Mijena and Nane (Stochastic Process Appl 125(9):3301–3326, 2015) have introduced these time fractional SPDEs. These types of time fractional stochastic heat type equations can be used to model phenomenon with random effects with thermal memory. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter \(\lambda \). In particular, our results are significant extensions of those in Ann Probab (to appear), Foondun and Khoshnevisan (Electron J Probab 14(21): 548–568, 2009), Mijena and Nane (2015) and Mijena and Nane (Potential Anal 44:295–312, 2016). Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.

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

Abstract: We consider the following fractional stochastic partial differential equation on a bounded, open subset B of R d for d ≥ 1 ∂ t u t ( x ) = L u t ( x ) + ξ σ ( u t ( x ) ) F ˙ ( t , x ) , where ξ is a positive parameter and σ is a globally Lipschitz continuous function. The stochastic forcing term F ˙ ( t , x ) is white in time but possibly colored in space. The operator L is fractional Laplacian which is the infinitesimal generator of a symmetric α-stable Levy process in R d . We study the behaviour of the solution with respect to the parameter ξ.We show that under zero exterior boundary conditions, in the long run, the pth-moment of the solution grows exponentially fast for large values of ξ. However when ξ is very small we observe eventually an exponential decay of the pth-moment of this same solution.

A random regularized approximate solution of the inverse problem for the Burgers' equation

Abstract: In this paper, we find a regularized approximate solution for an inverse problem for the Burgers' equation. The solution of the inverse problem for the Burgers' equation is ill-posed, i.e., the solution does not depend continuously on the data. The approximate solution is the solution of a regularized equation with randomly perturbed coefficients and randomly perturbed final value and source functions. To find the regularized solution, we use the modified quasi-reversibility method associated with the truncated expansion method with nonparametric regression. We also investigate the convergence rate.

Regularized solutions for some backward nonlinear parabolic equations with statistical data

Abstract: In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed, i.e., the solution does not depend continuously on the data. To regularize the instable solution, we develop some new methods to construct some new regularized solution. We also investigate the convergence rate between the regularized solution and the solution of our equations. In particular, we establish results for several equations with constant coefficients and time dependent coefficients. The equations with constant coefficients include heat equation, extended Fisher-Kolmogorov equation, Swift-Hohenberg equation and many others. The equations with time dependent coefficients include Fisher type Logistic equations, Huxley equation, Fitzhugh-Nagumo equation. The methods developed in this paper can also be applied to get approximate solutions to several other equations including 1-D Kuramoto-Sivashinsky equation, 1-D modified Swift-Hohenberg equation, strongly damped wave equation and 1-D Burger's equation with randomly perturbed operator.

On a backward problem for multidimensional Ginzburg-Landau equation with random data

Abstract: In this paper, we consider a backward in time problem for Ginzburg-Landau equation in multidimensional domain associated with some random data. The problem is ill-posed in the sense of Hadamard. To regularize the instable solution, we develop a new regularized method combined with statistical approach to solve this problem. We prove a upper bound, on the rate of convergence of the mean integrated squared error in $L^2 $ norm and $H^1$ norm.

Regularized solutions for some backward nonlinear partial differential equations with statistical data

Abstract: In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed, i.e., the solution does not depend continuously on the data. To regularize the instable solution, we develop some new methods to construct some new regularized solution. We also investigate the convergence rate between the regularized solution and the solution of our equations. In particular, we establish results for several equations with constant coefficients and time dependent coefficients. The equations with constant coefficients include heat equation, extended Fisher-Kolmogorov equation, Swift-Hohenberg equation and many others. The equations with time dependent coefficients include Fisher type Logistic equations, Huxley equation, Fitzhugh-Nagumo equation. The methods developed in this paper can also be applied to get approximate solutions to several other equations including 1-D Kuramoto-Sivashinsky equation, 1-D modified Swift-Hohenberg equation, strongly damped wave equation and 1-D Burger's equation with randomly perturbed operator.

Convergence rates in expectation for a nonlinear backward parabolic equation with Gaussian white noise

Abstract: The main purpose of this paper is to study the problem of determining initial condition of nonlinear parabolic equation from noisy observations of the final condition. We introduce a regularized method to establish an approximate solution. We prove an upper bound on the rate of convergence of the mean integrated squared error.