Volume 19, Issue 2

We present an overview of the literature on solutions to impulsive Caputo fractional differential equations. Lyapunov direct method is used to obtain sufficient conditions for stability properties of the zero solution of nonlinear impulsive fractional differential equations. One of the main problems in the application of Lyapunov functions to fractional differential equations is an appropriate definition of its derivative among the differential equation of fractional order. A brief overview of those used in the literature is given, and we discuss their advantages and disadvantages. One type of derivative, the so called Caputo fractional Dini derivative, is generalized to impulsive fractional differential equations. We apply it to study stability and uniform stability. Some examples are given to illustrate the results.

This paper contains a survey on one of the mathematical approaches used to solve a fractional differential equation whose solution gives the free dynamic response of viscoelastic single degree of freedom systems (viscosity is actually modelled by a fractional displacement derivative instead of first order one). The paper shall deal with Caputo’s fractional derivative since its Laplace Transform (on which the resolution method is based) only depends by lower integer order (and therefore measurable and physically meaningful) derivatives given as initial conditions. The paper provides a deep mathematical analysis of the properties of the solution expressed in terms of the mechanical parameters meaning. Additionally, important physical implications are reported exhibiting a richer dynamic behavior if compared to the standard damping case (velocity linear dependence). Some important consequences in the use of Caputo’s fractional derivative are reported, and some limitations to possible viscous parameters values are obtained. Finally, it is shown that free response of fractional derivative equation solution is mathematically equivalent to a suitably forced solution of the integer model.

In this paper, we revisit a Legendre-tau method for two-point boundary value problems with a Caputo fractional derivative in the leading term, and establish an L 2 error estimate for smooth solutions. Further, we apply the method to the Sturm-Liouville problem. Numerical experiments indicatethat for the source problem, it converges steadily at an algebraic rate even for nonsmooth data, and the convergence rate enhances with problem data regularity, whereas for the Sturm-Liouville problem, it always yields excellent convergence for eigenvalue approximations.

The purpose of this paper is to study the existence of solutions for equations driven by a non-local regional operator with homogeneous Dirichlet boundary conditions. More precisely, we consider the problem (−Δ)ραu(x)=f(x,u),x∈Ω,u(x)=0,x∈∂Ω$$\begin{array}{c}{(-\Delta )_{\rho }^{\alpha }u(x)=f(x, u),\quad x\in \Omega,}\cr {u(x)=0, \quad x\in \partial \Omega }\end{array}$$ where the nonlinear term f satisfies superlinear and subcritical growth conditions at zero and at infinity. These equations have a variational structure, and so its solutions can be found as critical points of the energy functional I ρ associated to the problem. Here we get such critical pointsusing the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary.

In this paper, we define and consider the fractional Gross Laplacian which is characterized by the Laplace transform. As application, we study the generalized Riemann-Liouville time fractional diffusion equation in infinite dimensions. We show that the explicit solution is given as the convolution between the initial condition and a generalized function related to the Mittag-Leffler function.

The stochastic solution with Gaussian stationary increments is established for the symmetric space-time fractional diffusion equation when 0 < β < α ≤ 2, where 0 < β ≤ 1 and 0 < α ≤ 2 are the fractional derivation orders in time and space, respectively. This solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with respect to previous approaches such as the scaling limit of the continuous time random walk, the parametric subordination and the subordinated Langevin equation. A new integral representation formula for the fundamental solution of the space-time fractional diffusion equation is firstly derived. It is then shown that, in the symmetric case, a stochastic solution can be obtained by a Gaussian process with stationary increments and with a random wideness scale variable distributed according to an arrangement of two extremal Lévy stable densities. This stochastic solution is self-similar with stationary increments and uniquely defined in a statistical sense by the mean and the covariance structure. Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental solution of the symmetric space-time fractional diffusion equation.

We extend the semigroup {Φ λ } λ∈ℂ of Gel’fand and Shilov to the semigroup {Φ λ } λ∈ℂ n and the one-dimensional fractional calculus of distributions to the case of distributions on Rn with supports in the nonnegative cone ℝ¯+n$\overline{\mathbb{R}}_{+}^{n}$. We use distributional mixed derivatives of an arbitrary complex order to solve Abel’s multidimensional integral equation and a distributional fractional differential equation of wave type.

In this paper, we introduce and study a new class of boundary value problems of one-dimensional higher-order semi-linear Caputo type fractional differential equations and nonlocal multi-point discrete and integral boundary conditions. Our existence results are new in the given setting and rest on some standard tools of fixed point theory. We also discuss Riemann-Liouville and Stieltjes variants of the proposed problem. The obtained results are well illustrated with the aid of examples.

In this paper we study a new class of Riemann-Liouville fractional differential equations subject to nonlocal Erdélyi-Kober fractional integral boundary conditions. Existence and uniqueness results are obtained by using a variety of fixed point theorems, such as Banach fixed point theorem, Nonlinear Contractions, Krasnoselskii fixed point theorem, Leray-Schauder Nonlinear Alternative and Leray-Schauder degree theory. Examples illustrating the obtained results are also presented.

We apply the framework of tempered fractional calculus to investigate the spatial dispersion of elastic waves in a one-dimensional elastic bar characterized by range-dependent nonlocal interactions. The measure of the interaction is given by the attenuation kernel present in the constitutive stress-strain relation of the bar, which follows from the Kröner-Eringen’s model of nonlocal elasticity. We employ a fractional power-law attenuation kernel and spatially temper it, to make the model physically valid and mathematically consistent. The spatial dispersion relation is derived, but it turns out to be difficult to solve, both analytically and numerically. Consequently, we use numerical techniques to extract the real and imaginary parts of the complex wavenumber for a wide range of frequency values. From the dispersion plots, it is found that the phase velocity dispersion of elastic waves in the tempered nonlocal elastic bar is similar to that from the time-fractional Zener model. Further, we also examine the unusual attenuation pattern obtained for the elastic wave propagation in the bar.

The space–time fractional diffusion equations on finite domain model anomalous diffusion behavior with large particle jumps combined with long waiting times. In this work we prove existence of strong solutions for such equations. Our proofs strongly depend on the fractional Sturm–Liouville theory, precisely on the problem of finding eigenvalues and corresponding eigenfunctions to the certain fractional differential equation. Using the method of separating variables and applying theorem ensuring existence of solutions to the fractional Sturm–Liouville problem we solve several types of fractional diffusion equations.

The new fractional calculus modeling based on Mittag-Leffler function has been employed to generate a better fit model to analyze the ZLC desorption curves for para-xylene diffusion in ZSM-5 zeolites. The diffusivity values generated herewith at 100, 125 and 150°C are reported as 4.4×10 −13 , 4.98 ×10 −13 and 5.2 ×10 −13 m 2 / s , respectively. The activation energy for this diffusion process is found 4.2 kJ/mol and diffusion proportional constant ( D 0 ) is 1.85 × 10 −12 m 2 / s . The simplified model for ZLC response can be a better way to treat desorption data in ZLC experiments.

It is well known that a continuously differentiable function is monotone in an interval [ a, b ] if and only if its first derivative does not change its sign there. We prove that this is equivalent to requiring that the Caputo derivatives of all orders α ∈ (0, 1) with starting point a of this function do not have a change of sign there. In contrast to what is occasionally conjectured, it is not sufficient if the Caputo derivatives have a constant sign for a few values of α ∈ (0, 1) only.

In this note we consider some classes of integral equations with the M-Wright functions as kernels. We show that the Volterra μ −functions are eigenfunctions of the introduced integral operators. We derive also some new integral identities related to the Weyl fractional integrals and derivatives of the M-Wright (Mainardi) functions.