In this paper, a time fractional diffusion equation on a finite domain is con- sidered. The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first order time derivative by a fractional derivative of order 0 < a< 1 (in the Riemann-Liovill or Caputo sence). In equation that we consider the time fractional derivative is in...

in this paper, a new numerical method for solving fractional optimal control problems (focps) is presented. the fractional derivative in the dynamic system is described in the caputo sense. the method is based upon biorthogonal cubic hermite spline multiwavelets approxima-tions. the properties of biorthogonal multiwavelets are first given. the operational matrix of fractional riemann-lioville i...

We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account “memory”. The precise model is D t u− div(u(−∆)−σu) = f, 0 < σ < 1/2. We pose the problem over {t ∈ R+, x ∈ Rn} with nonnegative initial data u(0, x) ≥ 0 as wel...

We study existence and qualitative properties of solutions for the abstract fractional relaxation equation (0.1) u′(t)−ADα t u(t) + u(t) = f(t), 0 < α < 1, t ≥ 0, u(0) = 0, on a complex Banach space X, where A is a closed linear operator, Dα t is the Caputo derivative of fractional order α ∈ (0, 1), and f is an X-valued function. We also study conditions under which the solution operator has th...

The 1D discrete fractional Laplacian operator on a cyclically closed (periodic) linear chain with finite number N of identical particles is introduced. We suggest a ”fractional elastic harmonic potential”, and obtain the N -periodic fractional Laplacian operator in the form of a power law matrix function for the finite chain (N arbitrary not necessarily large) in explicit form. In the limiting ...

A fractional order HIV model with both virus-tocell and cell-to-cell transmissions and therapy effect is investigated. The conditions for the existence of the equilibria are determined. Local stability analysis of the HIV model is studied by using the fractional Routh-Hurwitz stability conditions. We have generalized the integer LaSalle’s invariant theorem into fractional system and given some ...

In the present paper, we study Cauchy-Dirichlet problem to a nonlocal nonlinear diffusion equation with polynomial nonlinearities $$\mathcal {D}_{0|t}^{\alpha }u+(-\varDelta )^s_pu=\gamma |u|^{m-1}u+\mu |u|^{q-2}u,\,\gamma ,\mu \in \mathbb {R},\,m>0,q>1,$$ involving time-fractional Caputo derivative }$$ and space-fractional p-Laplacian operator $$(-\varDelta )^s_p$$ . We give simple proof of co...

In this paper we prove Hopf’s boundary point lemma for the fractional Laplacian. With respect to the classical formulation, in the non-local framework the normal derivative of the involved function u at z ∈ ∂Ω is replaced with the limit of the ratio u(x)/(δR(x)) , where δR(x) = dist(x, ∂BR) and BR ⊂ Ω is a ball such that z ∈ ∂BR. More precisely, we show that lim inf B∋x→z u(x) (δR(x)) > 0 . Als...