arXiv subjects
Publications and source records attributed to Mokhtar Kirane.
This paper studies the solutions of a reaction--diffusion system with nonlinearities that generalise the Lengyel--Epstein and FitzHugh--Nagumo nonlinearities. Sufficient conditions are derived for the global asymptotic stability of the system's solutions and confirmed through numerical Examples.
In this paper, we propose a new concept of derivative with respect to an arbitrary kernel-function. Several properties related to this new operator, like inversion rules, integration by parts, etc. are studied. In particular, we introduce the notion of conjugate kernels, which will be useful to guaranty that the proposed derivative operator admits a right inverse. The proposed concept includes as special cases Riemann-Liouville fractional derivatives, Hadamard fractional derivatives, and many other fractional operators. Moreover, using our concept, new fractional operators involving certain special functions are introduced, and some of their properties are studied. Finally, an existence result for a boundary value problem involving the introduced derivative operator is proved.
In this paper we obtain new estimates of the Hadamard fractional derivatives of a function at its extreme points. The extremum principle is then applied to show that the initial-boundary-value problem for linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution depends continuously on the initial and boundary conditions. The extremum principle for an elliptic equation with a fractional Hadamard derivative is also proved.
In this paper, a maximum principle for the one-dimensional sub-diffusion equation with Atangana-Baleanu fractional derivative is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Atangana-Baleanu fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.
A class of inverse problems for a heat equation with involution perturbation is considered using four different boundary conditions, namely, Dirichlet, Neumann, periodic and anti-periodic boundary conditions. Proved theorems on existence and uniqueness of solutions to these problems are presented. Solutions are obtained in the form of series expansion using a set of appropriate orthogonal basis for each problem. Convergence of the obtained solutions is also discussed.
In this work, we obtain a Lyapunov-type and a Hartman-Wintner-type inequalities for a linear and a nonlinear fractional differential equation with generalized Hilfer operator subject to Dirichlet-type boundary conditions. We prove existence of positive solutions to a nonlinear fractional boundary value problem. As an application, we obtain a lower bound for the eigenvalues of corresponding equations.
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.
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.
In this paper we study some boundary value problems for a fractional analogue of second order elliptic equation with an involution perturbation in a rectangular domain. Theorems on existence and uniqueness of a solution of the considered problems are proved by spectral method.
We deal with the problem of reconstructing material coefficients from the farfields they generate. By embedding small (single) inclusions to these media, located at points $z$ in the support of these materials, and measuring the farfields generated by these deformations we can extract the values of the total field generated by these media at the points $z$. The second step is to extract the values of the material coefficients from these internal values of the total field. The main difficulty in using internal fields is the treatment of their possible zeros. In this work, we propose to deform the medium using multiple (precisely double) and close inclusions instead of only single ones. By doing so, we derive from the asymptotic expansions of the farfields the internal values of the Green function, in addition to the internal values of the total fields. This is possible because of the deformation of the medium with multiple and close inclusions which generates scattered fields due to the multiple scattering between these inclusions. Then, the values of the index of refraction can be extracted from the singularities of the Green function. Hence, we overcome the difficulties arising from the zeros of the internal fields. We test these arguments for the acoustic scattering by a refractive index in presence of inclusions modeled by the impedance type small obstacles.
We consider a new model for shunting inhibitory cellular neural networks, retarded functional differential equations with piecewise constant argument. The existence and exponential stability of almost periodic solutions are investigated. An illustrative example is provided.
We deal with the scattering of an acoustic medium modeled by an index of refraction $n$ varying in a bounded region $Ω$ of $\mathbb{R}^3$ and equal to unity outside $Ω$. This region is perforated with an extremely large number of small holes $D_m$'s of maximum radius $a$, $a<<1$, modeled by surface impedance functions. Precisely, we are in the regime described by the number of holes of the order $M:=O(a^{β-2})$, the minimum distance between the holes is $d\sim a^t$ and the surface impedance functions of the form $λ_m \sim λ_{m,0} a^{-β}$ with $β>0$ and $λ_{m,0}$ being constants and eventually complex numbers. Under some natural conditions on the parameters $β, t$ and $λ_{m,0}$, we characterize the equivalent medium generating, approximately, the same scattered waves as the original perforated acoustic medium. We give an explicit error estimate between the scattered waves generated by the perforated medium and the equivalent one respectively, as $a \rightarrow 0$. As applications of these results, we discuss the following findings: 1. If we choose negative valued imaginary surface impedance functions, attached to each surface of the holes, then the equivalent medium behaves as a passive acoustic medium only if it is an acoustic metamaterial with index of refraction $\tilde{n}(x)=-n(x),\; x \in Ω$ and $\tilde{n}(x)=1,\; x \in \mathbb{R}^3\setminus{\overlineΩ}$. This means that, with this process, we can switch the sign of the index of the refraction from positive to negative values. 2. We can choose the surface impedance functions attached to each surface of the holes so that the equivalent index of refraction $\tilde{n}$ is $\tilde{n}(x)=1,\; x \in \mathbb{R}^3$. This means that the region $Ω$ modeled by the original index of refraction $n$ is approximately cloaked.
Let $M$ be the number of bounded and Lipschitz regular obstacles $D_j, j:=1, ..., M$ having a maximum radius $a$, $a<<1$, located in a bounded domain $Ω$ of $\mathbb{R}^3$. We are concerned with the acoustic scattering problem with a very large number of obstacles, as $M:=M(a):=O(a^{-1})$, $a\rightarrow 0$, when they are arbitrarily distributed in $Ω$ with a minimum distance between them of the order $d:=d(a):=O(a^t)$ with $t$ in an appropriate range. We show that the acoustic farfields corresponding to the scattered waves by this collection of obstacles, taken to be soft obstacles, converge uniformly in terms of the incident as well the propagation directions, to the one corresponding to an acoustic refraction index as $a\rightarrow 0$. This refraction index is given as a product of two coefficients $C$ and $K$, where the first one is related to the geometry of the obstacles (precisely their capacitance) and the second one is related to the local distribution of these obstacles. In addition, we provide explicit error estimates, in terms of $a$, in the case when the obstacles are locally the same (i.e. have the same capacitance, or the coefficient $C$ is piecewise constant) in $Ω$ and the coefficient $K$ is H$\ddot{\mbox{o}}$lder continuous. These approximations can be applied, in particular, to the theory of acoustic materials for the design of refraction indices by perforation using either the geometry of the holes, i.e. the coefficient $C$, or their local distribution in a given domain $Ω$, i.e. the coefficient $K$.
In this article, we study the local existence of solutions for a wave equation with a nonlocal in time nonlinearity. Moreover, a blow-up results are proved under some conditions on the dimensional space, the initial data and the nonlinear forcing term.