Volume 18, Issue 1

To implement integration and differentiation of half-order, it is required a two-terminal element having the following impedance: Z(jω)=1Aωe-jπ4. For this purpose a chain-type RC-circuit [1] can be used. However a chaintype line comprising a large number of sections and requiring careful shielding against interferences is an expensive and massy device. In the present work, it is shown the possibility to use a small size electrolytic two-terminal element (hereinafter referred to as EEFD) to perform fractional differentiating and integrating.

This paper is devoted to study the existence of solutions of nonlinear fractional integro-differential equation, via the techniques of measure of noncompactness. The investigation is based on a Schauder's fixed point theorem. The main result is less restrictive than those given in the literature. An illustrative example is given.

In this article, we have implemented reconstruction of variational iteration method as a new approximate analytical technique for solving fractional Black-Scholes option pricing equations. Indeed, we essentially use the well-known Mittag-Leffler function to obtain explicit solutions for some examples of financial mathematics equations.

In this paper, we establish two existence theorems for multiple solutions for the following fractional Hamiltonian system {D∞α(D∞αu(t))+L(t)u(t)=∇W(t,u(t)),u∈H(ℝ,ℝN), where α∈(1/2,1),t∈ℝ,u=(u1,…,uN)T∈ℝN, and L∈C(ℝ,ℝN2) is a symmetric and positive definite matrix for all t∈ℝ,W∈C1(ℝ×ℝN×ℝ) and ∇ W is the gradient of W about u .

Some analytical properties of the Mittag-Leffler functions, eα(t)≢Eα(-tα), are established on some t-intervals. These are lower and upper bounds obtained in terms of simple rational functions (which are related to the Padé approximants of eα(t)) for t>0 and 0<α<1.. A new method, to compute such functions, solving numerically a Caputo-type fractional differential equation satisfied by them, is developed. This approach consists in an adaptive predictor-corrector method, based on the K. Diethelm's predictor-corrector algorithm, and is shown to outperform the current methods implemented by MATLAB ® and by Mathematica when t is real and even possibly large

In We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange optimal control problem is considered, and existence of a multi-integral solution obtained. Main tools include fractional calculus, semigroup theory, fractional power of operators, a singular version of Gronwall's inequality, and Leray-Schauder fixed point theorem. An example illustrating the theory is given.

Using the iterative method, this paper investigates the existence of a unique solution to systems of nonlinear fractional differential equations, which involve the right-handed Riemann-Liouville fractional derivatives $D^{q}_{T}x$ and $D^{q}_{T}y$. Systems of linear fractional differential equations are also discussed. Two examples are added to illustrate the results.

In this paper we consider a nonlinear eigenvalue problem driven by the fractional Laplacian. By applying a version of the three-critical-points theorem we obtain the existence of three solutions of the problem in Hα(Ω). In addition the existence of at least two nontrivial solutions are also been obtained when 2<μ<2α*.

We show, using invariant subspace method, how to derive exact solutions to the time fractional Korteweg-de Vries (KdV) equation, potential KdV equation with absorption term, KdV-Burgers equation and a time fractional partial differential equation with quadratic nonlinearity. Also we extend the invariant subspace method to nonlinear time fractional differential-difference equations and derive exact solutions of the time fractional discrete KdV and Toda lattice equations.

We study a class of fractional integro-differential inclusions with integral boundary conditions and establish a Filippov type existence result in the case of nonconvex set-valued maps.

In this work, we discuss the existence and Ulam's type stability concepts for a class of partial functional differential inclusions with not instantaneous impulses and a nonconvex valued right hand side in Banach spaces. An example is also provided to illustrate our results.

In this paper, we define fractional derivative of arbitrary complex order of the distributions concentrated on R+, based on convolutions of generalized functions with the supports bounded on the same side. Using distributional derivatives, which are generalizations of classical derivatives, we present a few interesting results of fractional derivatives in D'(R+), as well as the symbolic solution for the following differential equation by Babenko's method y(x)+λΓ(-α)∫0xy(σ)(x-σ)α+1dσ=δ(x), where Re α>0

Extreme catastrophic events such as earthquakes, terrorism and economic collapses are difficult to predict. We propose a tentative mathematical model for the precursors of these events based on a memory formalism and apply it to earthquakes suggesting a physical interpretation. In this case, a precursor can be the anomalous increasing rate of events (aftershocks) following a moderate earthquake, contrary to Omori's law. This trend constitute foreshocks of the main event and can be modelled with fractional time derivatives. A fractional derivative of order 0 < v < 2 replaces the first-order time derivative in the classical diffusion equation. We obtain the frequency-domain Green's function and the corresponding time-domain solution by performing an inverse Fourier transform. Alternatively, we propose a numerical algorithm, where the time derivative is computed with the Grünwald-Letnikov expansion, which is a finitedifference generalization of the standard finite-difference operator to derivatives of fractional order. The results match the analytical solution obtained from the Green function. The calculation requires to store the whole field in the computer memory since anomalous diffusion “remembers the past”.

This paper is devoted to the study of nonlinear fractional differential equation with parameter dependence and integral boundary value conditions. In the paper various existence and multiplicity results for positive solutions are derived depending of different values of the parameter. Some illustrative examples are also discussed.

In trajectory planning, flatness is used to compute inputs generating suitable trajectories, without using any integration. The flatness property of linear controllable time-invariant fractional systems is studied. The formalism of polynomial matrix of the fractional differential operator is used leading to the characterization of fractionally flat outputs. The so-called defining matrices, which are transformations that express all system variables in function of the fractionally flat outputs and a finite number of their time derivatives, are introduced and characterized in this fractional context. Flatness of fractional systems is then applied to the trajectory planning of a real thermal experiment.

In this paper, we study nonlocal boundary value problems of fractional differential equations and inclusions with slit-strips integral boundary conditions. We show the existence and uniqueness of solutions for the single valued case (equations) by means of classical contraction mapping principle while the existence result is obtained via a fixed point theorem due to D. O'Regan. The existence of solutions for the multivalued case (inclusions) is established via nonlinear alternative for contractive maps. The results are well illustrated with the aid of examples.

The goal of this short note is to let the readers of the “FCAA” journal to know that the “counterexamples” in the paper mentioned in the title of this note and referred in the sequel as [Bai et al.] are wrong in the sense that they contradict to nothing in authors' references, in contrast to what they state.