We establish the existence and uniqueness of an attractive fractional coupled system. Such a system has applications in biological populations of cells. We confirm that the fractional system under consideration admits a global solution in the Sobolev space. The solution is shown to be unique. The technique is founded on analytic method of the fixed point theory and the fractional differential o...

where D is the Caputo fractional derivative of order α with  < α < , A :D(A)⊂ X → X is the infinitesimal generator of a resolvent Sα(t), t ≥ , B : U → X is a bounded linear operator, u ∈ L([,b],U), X and U are two real Hilbert spaces, J–α t h denotes the  – α order fractional integral of h ∈ L([,b],X). The controllability problem has attracted a lot of mathematicians and engineers’ att...

The models that are based of fractional derivatives should be highlighted among promising new models to describe turbulent fluid flows. In the present work, a steadystate flow in a duct is considered under the condition that the turbulent diffusion is governed by a fractional power of the Laplace operator. To study numerically flows in rectangular channels, finite-difference approximations are ...

In this paper, a space fractional diffusion equation (SFDE) with nonhomogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of lin...

An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized twolevel schemes are used. The numeri...

In this article we study various convergence results for a class of nonlinear fractional heat equations of the form ⎧ ⎨ ⎩ ut(t, x)− I[u(t, ·)](x) = f(t, x), (t, x) ∈ (0, T )× Rn, u(0, x) = u0(x), x ∈ Rn, where I is a nonlocal nonlinear operator of Isaacs type. Our aim is to study the convergence of solutions when the order of the operator changes in various ways. In particular, we consider zero...

We propose a high order explicit finite difference method for fractional advection diffusion equations. These equations can be obtained from the standard advection diffusion equations by replacing the second order spatial derivative by a fractional operator of order α with 1 < α ≤ 2. This operator is defined by a combination of the left and right Riemann–Liouville fractional derivatives. We stu...