Volume 20, Issue 3

We present a survey on inequalities in fractional calculus that have proven to be very useful in analyzing differential equations. We mention in particular, a “Leibniz inequality” for fractional derivatives of Riesz, Riemann-Liouville or Caputo type and its generalization to the d -dimensional case that become a key tool in differential equations; they have been used to obtain upper bounds on solutions leading to global solvability, to obtain Lyapunov stability results, and to obtain blowing-up solutions via diverging in a finite time lower bounds. We will also mention the weakly singular Gronwall inequality of Henry and its variants, principally by Medved, that plays an important role in differential equations of any kind. We will also recall some “traditional” inequalities involving fractional derivatives or fractional powers of the Laplacian.

Recent modeling of real world phenomena give rise to Caputo type fractional order differential equations with non-instantaneous impulses. The main goal of the survey is to highlight some basic points in introducing non-instantaneous impulses in Caputo fractional differential equations. In the literature there are two approaches in interpretation of the solutions. Both approaches are compared and their advantages and disadvantages are illustrated with examples. Also some existence results are derived.

A theoretical analysis of two- and three-dimensional fractional-order Hindmarsh-Rose neuronal models is presented, focusing on stability properties and occurrence of Hopf bifurcations, with respect to the fractional order of the system chosen as bifurcation parameter. With the aim of exemplifying and validating the theoretical results, numerical simulations are also undertaken, which reveal rich bursting behavior in the three-dimensional fractional-order slow-fast system.

In this paper we study the existence of solutions for an initial value problem, posed in a given Banach space, with a fractional differential equation via densifiability techniques. For our goal, we will prove a new fixed point result (not based on measures of noncompactness) which is, in forms, a generalization of the well-known Darbo’s fixed point theorem but essentially different. Some illustrative examples are given.

We discuss the existence, multiplicity and uniqueness of solutions to the periodic problem c D α u + q ( t , u ) c D β u = f ( t , u ), u (0) = u ( T ), where 0 < β < α ≤ 1. The existence results are proved by the combination of the Schauder fixed point theorem with the maximum principle for the Caputo fractional derivative and the structure of compact sets in ℝ.

In this paper, we consider a class of evolution equations with Hilfer fractional derivative. By employing the fixed point theorem and the noncompact measure method, we establish a number of new criteria to guarantee the existence and uniqueness of mild solutions when the associated semigroup is compact or not.

The existence of a unique solution for the identification problem to linear fractional differential equations in Banach spaces were proved in the cases of the nondegenerate equation and of the equation with degenerate operator at the Caputo derivative. The degenerate case was studied for the overdetermination on the phase space and on the degeneracy subspace of the corresponding homogeneous equation. Abstract results are used to the investigation of the identification problem for time-fractional order Sobolev’s system of equations in the cases of the overdetermination on the fluid velocity and on the fluid pressure gradient.

It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved due to the nonlocal properties of fractional operators. Therefore, developing some high-order numerical approximation formulas for fractional derivatives plays a more important role in numerically solving fractional differential equations. This paper focuses on constructing (generalized) high-order fractional-compact numerical approximation formulas for Riesz derivatives. Then we apply the developed formulas to the one- and two-dimension Riesz spatial fractional reaction-dispersion equations. The stability and convergence of the derived numerical algorithms are strictly studied by using the energy analysis method. Finally, numerical simulations are given to demonstrate the efficiency and convergence orders of the presented numerical algorithms.

Advances in the use of fractional order calculus in control theory increasingly make their way into control applications such as in the process industry, electrical machines, mechatronics/robotics, albeit at a slower rate into control applications in automotive and railway systems. We present work on advances in high-speed rail vehicle tilt control design enabled by use of fractional order methods. Analytical problems in rail tilt control still exist especially on simplified tilt using non-precedent sensor information (rather than use of the more complex precedence (or preview) schemes). Challenges arise due to suspension dynamic interactions (due to strong coupling between roll and lateral dynamic modes) and the sensor measurement. We explore optimized PID-based non-precedent tilt control via both direct fractional-order PID design and via fractional-order based loop shaping that reduces effect of lags in the design model. The impact of fractional order design methods on tilt performance (track curve following vs ride quality) trade off is particularly emphasized. Simulation results illustrate superior benefit by utilizing fractional order-based tilt control design.

Dedicated to the 120th anniversary of the birth of Alexey N. Gerasimov, one of the initiators of Fractional Calculus in Solid Mechanics . In the first part of this survey we provide a brief sketch of his life. In the second part, the scientific contributions of the Soviet mechanician A.N. Gerasimov are reviewed. E ditorial N ote . In a survey [28] recently published in this journal, Sect. 5, we the authors have mistaken the first name of Alexey Gerasimov with the incorrect name Andrey. This was due to unsuccessful attempts to find it unabbreviated in the available literature and by queries among colleagues. Some Internet sources (in Russian language) gave us the first name of another, also Soviet scientist in Mechanics. We do apologize for this fault.

In this paper regular fractional Sturm-Liouville boundary value problems are considered. In particular, new inner products are described in the Sobolev space and a symmetric operator is established in this space.

In this paper we discuss certain geometric properties of a function through its Caputo fractional derivative. We show that convexity and monotonicity results can be obtained provided that the fractional derivative of the function is of one sign for some value of α . Analogous result for the global extrema of a function is obtained. However, to release the condition on the Diethelm paper [3] that the fractional derivative of the function is of one sign for all values of α in certain domain, higher order fractional inequalities are required. The applicability of the new results is discussed.