Lyapunov-type inequalities for fractional differential equations

The principal aim of this paper is to discuss Lyapunov-type inequalities for fractional differential equations with fractional boundary conditions. Some new Lyapunov's inequalities are established, which almost generalize and improve some earlier results in the literature

Lyapunov-type inequalities for a higher order fractional differential equation with fractional integral boundary conditions

New Lyapunov-type inequalities are derived for the fractional boundary value problem \begin{align*} & D_a^\alpha u(t)+q(t)u(t)=0,\quad a<t<b,\\ &u(a)=u'(a)=\dots=u^{(n-2)}(a)=0,\quad u(b)=I_a^{\alpha}(hu)(b), \end{align*} where $n\in \mathbb N$, $n\geq 2$, $n-1<\alpha<n$, $D_a^\alpha$ denotes the Riemann--Liouville fractional derivative of order $\alpha$, $I_a^{\alpha}$ denotes the Riemann--Liouville fractional integral of order $\alpha$, and $q,h\in C([a,b];\mathbb R)$. As an application, we obtain numerical approximations of lower bound for the eigenvalues of corresponding equations

Hartman-Wintner-Type Inequality for a Fractional Boundary Value Problem via a Fractional Derivative with respect to Another Function

We consider a fractional boundary value problem involving a fractional derivative with respect to a certain function g. A Hartman-Wintner-type inequality is obtained for such problem. Next, several Lyapunov-type inequalities are deduced for different choices of the function g. Moreover, some applications to eigenvalue problems are presented

Computation of Generalized Averaged Gaussian Quadrature Rules

The estimation of the quadrature error of a Gauss quadrature rule when applied to the approximation of an integral determined by a real-valued integrand and a real-valued nonnegative measure with support on the real axis is an important problem in scientific computing. Laurie [2] developed anti-Gauss quadrature rules as an aid to estimate this error. Under suitable conditions the Gauss and associated anti-Gauss rules give upper and lower bounds for the value of the desired integral. It is then natural to use the average of Gauss and anti-Gauss rules as an improved approximation of the integral. Laurie also introduced these averaged rules. More recently, the author derived new averaged Gauss quadrature rules that have higher degree of exactness for the same number of nodes as the averaged rules proposed by Laurie. In [2], [5], [3] stable numerical procedures for computation of the corresponding averaged Gaussian rules are proposed. An analogous procedure can be applied also for a more general class of weighted averaged Gaussian rules introduced in [1]. Those results are presented in [4]. Here we we give a survey of the quoted results, which are obtained jointly with L. Reichel (Kent State Univ., OH (U.S.)

New Trends in Differential and Difference Equations and Applications

This is a reprint of articles from the Special Issue published online in the open-access journal Axioms (ISSN 2075-1680) from 2018 to 2019 (available at https://www.mdpi.com/journal/axioms/special issues/differential difference equations)

Dynamic output nonfragile reliable control for nonlinear fractional-order glucose–insulin system

The main intention of this paper is to scrutinize the problem of internal model-based dynamic output feedback nonfragile reliable control problem for fractional-order glucose–insulin system. Specifically, a robust control law that represents the insulin injection rate is designed in order to regulate the level of glucose in diabetes treatment in the existence of meal disturbance or external glucose infusion due to improper diet. By the construction of suitable Lyapunov functional, a novel set of sufficient conditions is derived with the aid of linear matrix inequalities for obtaining the required dynamic output feedback control law. In particular, the designed controller ensures the robust stability and disturbance attenuation performance against meal disturbance of the glucose–insulin system. Numerical simulation results are performed to verify the advantage of the developed design technique. Specifically, the irregular blood glucose level can be brought down to normal level by injecting suitable rate of insulin to the patient. The result exposes that the level of blood glucose is sustained in the identified ranges via the proposed dynamic output feedback control law.&nbsp