Band 24, Heft 4

We introduce three representation formulas for the fractional p -Laplace operator in the whole range of parameters 0 < s < 1 and 1 < p < ∞. Note that for p ≠ 2 this a nonlinear operator. The first representation is based on a splitting procedure that combines a renormalized nonlinearity with the linear heat semigroup. The second adapts the nonlinearity to the Caffarelli-Silvestre linear extension technique. The third one is the corresponding nonlinear version of the Balakrishnan formula. We also discuss the correct choice of the constant of the fractional p -Laplace operator in order to have continuous dependence as p → 2 and s → 0 + , 1 − . A number of consequences and proposals are derived. Thus, we propose a natural spectral-type operator in domains, different from the standard restriction of the fractional p -Laplace operator acting on the whole space. We also propose numerical schemes, a new definition of the fractional p -Laplacian on manifolds, as well as alternative characterizations of the W s , p (ℝ n ) seminorms.

This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

We prove a “hidden” regularity result for weak solutions of time fractional diffusion-wave equations where the Caputo fractional derivative is of order α ∈ (1, 2). To establish such result we analyse the regularity properties of the weak solutions in suitable interpolation spaces.

Based on the popular Caputo fractional derivative of order β in (0, 1), we define the censored fractional derivative on the positive half-line ℝ + . This derivative proves to be the Feller generator of the censored (or resurrected) decreasing β -stable process in ℝ + . We provide a series representation for the inverse of this censored fractional derivative. We are then able to prove that this censored process hits the boundary in a finite time τ ∞ , whose expectation is proportional to that of the first passage time of the β -stable subordinator. We also show that the censored relaxation equation is solved by the Laplace transform of τ ∞ . This relaxation solution proves to be a completely monotone series, with algebraic decay one order faster than its Caputo counterpart, leading, surprisingly, to a new regime of fractional relaxation models. Lastly, we discuss how this work identifies a new sub-diffusion model.

In this paper, we consider a class of nonlinear fractional impulsive differential equation involving Sturm-Liouville boundary-value conditions and p -Laplacian operator. By making use of critical point theorem and variational methods, some new criteria are given to guarantee that the considered problem has infinitely many solutions. Our results extend some recent results and the conditions of assumptions are easily verified. Finally, an example is given as an application of our fundamental results.

This paper proposes a new method for tuning the parameters of multi-input multi-output (MIMO) fractional-order PID (FOPID) controller. The aim of the proposed method is to calculate the parameters of this controller such that the rise time and steady-state errors of the feedback system are minimized without violating the predetermined stability margins. Mathematically, this problem is formulated as maximizing the spectral norm of the open-loop transfer matrix at zero frequency subject to a constraint on the H ∞ -norm of the sensitivity function. This problem is nonlinear in parameters of the MIMO FOPID, which can be solved using the iterative algorithm developed in this paper based on non-smooth H ∞ synthesis.

This paper concerns a backward problem for a nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity. Such a problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2), which is usually used to model the anomalous diffusion. We show that the problem is severely ill-posed. Using the Fourier transform and a filter function, we construct a regularized solution from the data given inexactly and explicitly derive the convergence estimate in the case of the local Lipschitz reaction term. Special cases of the regularized solution are also presented. These results extend some earlier works on the space-fractional backward diffusion problem.

On a compact connected manifold M $\mathbb{M}$, we concern the fractional power dissipative operator e−tLα $e^{-t\left\vert \mathcal{L}\right\vert ^{\alpha}}$, and obtain the almost-everywhere convergence rate (as t → 0 + ) of e−tLαf $e^{-t\left\vert \mathcal{L}\right\vert ^{\alpha}}\left( f\right)$ when f is in some Sobolev type Hardy spaces. The main result is a non-trivial extension of a recent result on ℝ n by Cao and Wang in 2.

In this paper, by applying the intertwining properties, we introduce the fractional powers of the number operator perturbed by generalized Gross Laplacians (infinite dimensional Laplacians), which are special types of the infinitesimal generators of generalized Mehler semigroups. By applying the intertwining properties and semigroup approach, we study the Langevin type equations associated with the infinite dimensional Laplacians and with white noise distributions as forcing terms. Then we investigate the unique solution of the fractional Langevin type equations associated with the Riemann-Liouville and Caputo time fractional derivatives, and the fractional power of the infinite dimensional Laplacians, for which we apply the intertwining properties again. For our purpose, we discuss the fractional integrals and fractional derivatives of white noise distribution valued functions.

We consider the following fractional reaction-diffusion equation ut(t)+∂t∫0tgα(s)Au(t−s)ds=tγf(u), $$ u_t(t) + \partial_t \int\nolimits_{0}^{t} g_{\alpha}(s) \mathcal{A} u(t-s) ds = t^{\gamma} f(u),$$ where g α ( t ) = t α −1 /Γ( α ) (0 < α < 1), f ∈ C ([0, ∞)) is a non-decreasing function, γ > −1, and A $\mathcal{A}$ is an elliptic operator whose fundamental solution of its associated parabolic equation has Gaussian lower and upper bounds. We characterize the behavior of the functions f so that the above fractional reaction-diffusion equation has a bounded local solution in L r (Ω), for non-negative initial data u 0 ∈ L r (Ω), when r > 1 and Ω ⊂ ℝ N is either a smooth bounded domain or the whole space ℝ N . The case r = 1 is also studied.

In this paper, we formulate and prove two maximum principles to nonlinear fractional differential equations. We consider a fractional derivative operator with Mittag-Leffler function of two parameters in the kernel. These maximum principles are used to establish a pre-norm estimate of solutions, and to derive certain uniqueness and positivity results to related linear and nonlinear fractional initial value problems.

In this paper, we study a loaded modified diffusion equation (the Hallaire equation with the fractional derivative with respect to time). The compact finite difference schemes of Crank-Nicolson type of higher order is developed for approximating the stated problem on uniform grids with the orders of accuracy O(h4+τ2−α) $\mathcal{O}(h^4+\tau^{2-\alpha})$ and O(h4+τ2) $\mathcal{O}(h^4+\tau^{2})$. A priori estimates are obtained for solutions of differential and difference equations. Stability of the suggested schemes and also their convergence with the rate equal to the order of the approximation error are proved. Proposed theoretical calculations are illustrated by numerical experiments on test problems.

The advanced analytical study in the field of fractional-order non-full rank inverse model control design is presented in the paper. Following the recent results in this matter it is certain, that the inverse model control-oriented perfect control law can be established for the non-full rank integer-order systems being under the discrete-time state-space reference with zero value. It is shown here, that the perfect control paradigm can be extended to cover the multivariable non-full rank plants governed by the more general Grünwald-Letnikov discrete-time state-space model. Indeed, the postulated approach significantly reduces both iterative and non-iterative computational effort, mainly derived from the approximation of the Moore-Penrose inverse of the non-full rank matrices to finally be inverted. A prevention provided by the new method excludes the detrimental matrix behavior in the form of singularity, often avoided due to the observed ill-conditioned sensitivity. Thus, the new defined robust fractional-order non-full rank instance of such control strategy, supported by the pole-free mechanism, gives rise to the introduction of the general unified non-full rank perfect control-originated theory. Numerical algorithms with simulation investigation clearly confirm the innovative peculiarities provided by the manuscript.

In this paper, we discuss a class of Caputo fractional evolution equations on Banach space with feedback control constraint whose value is non-convex closed in the control space. First, we prove the existence of solutions for the system with feedback control whose values are the extreme points of the convexified constraint that belongs to the original one. Secondly, we study the topological properties of the sets of admissible “state-control” pair for the original system with various feedback control constraints and the relations between them. Moreover, we obtain necessary and sufficient conditions for the solution set of original systems to be closed. In the end, an example is given to illustrate the applications of our main results.