Volume 17, Issue 1

We consider the zeros of the solution ℧α(t) = E α(−t α), 1 < α < 2, of the fractional oscillation equation in terms of the Mittag-Leffler function, and give a wholly and clarified description for these zeros. We find that the number of zeros can be any finite number: 1, 2, 3, 4, ..., not necessarily an odd number. When the number of zeros of ℧α(t) is an even number, ℧α(t) has a critical zero. All of the values of α for which ℧α(t) has an even number of zeros constitute a countable set S. For each α ∈ (1, 2) \ S, ℧α(t) has an odd number of zeros. These results are a supplement and a perfecting for the existed related documents. We also show that the eigenvalue problems are related with the zeros of the Mittag-Leffler functions.

Because of the complexity of the practical environments, many distributed multi-agent systems can not be illustrated with the integer-order dynamics and can only be described with the fractional-order dynamics. Under the connected network with directed weighted topologies, the dynamical characteristics of agents with fractional-order derivative operator is analyzed in this paper. Applying the Laplace transform and frequency domain theory of the fractional-order operator, the distributed coordination of fractional-order multi-agent systems (FOMAS) with communication delays is studied, and a critical value of time delay is obtained to ensure the consensus of FOMAS. Since the integer-order model is a special case of fractional-order model, the extended results in this paper are in accordance with that of the integer-order model. Finally, numerical examples are provided to verify our results.

In this paper we study explicit solutions of fractional integro-differential equations with variable coefficients involving Prabhakar-type operators. Analytic solutions to equations involving Prabhakar operators and Laguerre derivatives are obtained by means of operational methods. Cauchy type problems for fractional integro-differential equations of Volterra type with generalized Riemann-Liouville derivative operator, which contain generalized Mittag-Leffler function in the kernel are also considered. Using the Laplace transform method, explicit solutions of the fractional integro-differential equations of Volterra type with variable coefficients, proposed by Srivastava and Tomovski in [28] are established in terms of the multinomial Wright function.

In this paper we investigate the existence and uniqueness of positive continuous solutions for a q-analogue of Volterra and Fredholm integral equations of first and second kinds. We derive the results by using three fixed point theorems introduced by Bushell in [7, 8]. Bushell derived his theorems by using the Cayley-Hilbert projective metric and Banach fixed point theorem. We also include some uniqueness criteria for the solutions of certain nonlinear q-integral equations provided that the solution exists in certain function spaces.

This paper provides results on the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski’s type. We prove that the initial value problem has the solution if and only if some initial values are zero.

The aim of this paper is to establish the existence of decay integral solutions to a class of retarded fractional differential equations involving impulsive effects. The results are obtained by using the fixed point approach and fractional calculus tools in Banach spaces. Applications to both ordinary and partial differential equations are presented.

The diffusion-wave equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in polar coordinates in a domain 0 ≤ r < ∞, 0 < φ < φ 0 under Dirichlet and Neumann boundary conditions. The Laplace integral transform with respect to time, the finite sin- and cos-Fourier transforms with respect to the angular coordinate, and the Hankel transform with respect to the radial coordinate are used. The numerical results are illustrated graphically.

This paper considers the robust stability bound problem of uncertain fractional-order systems. The system considered is subject either to a two-norm bounded uncertainty or to a infinity-norm bounded uncertainty. The robust stability bounds on the uncertainties are derived. The fact that these bounds are not exceeded guarantees that the asymptotical stability of the uncertain fractional-order systems is preserved when the nominal fractional-order systems are already asymptotically stable. Simulation examples are given to demonstrate the effectiveness of the proposed theoretical results.

A pseudo-differential operator (p.d.o.) associated with the fractional Hankel transform involving the symbol a(x, ξ) is defined. An integral representation of p.d.o. and boundedness result of the composition of operators Δμr and A μ,α are obtained. A generalized integral operator A μ,aα corresponding to p.d.o. is also defined and the properties of the product of two generalized integral operators corresponding to p.d.o. are studied.

The present work provides a critical assessment of numerical solutions of the space-fractional diffusion-advection equation, which is of high significance for applications in various natural sciences. In view of the fact that, in contrast to the case of normal (Gaussian) diffusion, no standard methods and corresponding numerical codes for anomalous diffusion problems have been established yet, it is of importance to critically assess the accuracy and practicability of existing approaches. Three numerical methods, namely a finite-difference method, the so-called matrix transfer technique, and a Monte-Carlo method based on the solution of stochastic differential equations, are analyzed and compared by applying them to three selected test problems for which analytical or semi-analytical solutions were known or are newly derived. The differences in accuracy and practicability are critically discussed with the result that the use of stochastic differential equations appears to be advantageous.

A fractional heat equation is used to model thermal diffusion in a one-dimensional bar that exhibits subdiffusive behavior. The left end of the bar is subjected to a nonlinear influx of heat. For the boundary constraint at the right end of the bar, two cases are considered, namely a homogeneous Neumann condition and a homogeneous Dirichlet condition. By reducing both cases to a nonlinear Volterra equation, it is shown that a blow-up always occurs. The asymptotic behavior near the blow-up is determined for both cases. It is also shown that the solution for the Neumann case dominates that of the Dirichlet case.

Within the fractional derivative framework, we study thermomechanical models with memory and compare them with the classical Volterra theory. The fractional models involve significant differences in the type of kernels and predicts important changes in the behavior of fluids and solids. Moreover, an analysis of the thermodynamic restrictions provides compatibility conditions on the kernels and allows us to determine certain free energies, which in turn enables the definition of a topology on the history space. Finally, an analogous analysis is carried out for the phenomenon of heat propagation with memory.

We expose some recent and less recent results related to the existence and the basic properties of the solution to the linear stochastic heat equation with additive Gaussian noise. We will make a comparative study of the behavior of the solution in function of the covariance structure of the driving noise.

Using the memory conception developed in the framework of the Mori-Zwanzig formalism, the kinetic equations for relaxation functions that correspond to the previously suggested empirical functions (Cole-Davidson and Havriliak-Negami) are derived. The obtained kinetic equations contain differential operators of non-integer order and have clear physical meaning and interpretation. The derivation of the memory function corresponding to the Havriliak-Negami relaxation law in the frame of Mori-Zwanzig formalism is given. A physical interpretation of the power-law exponents involved in the Havriliak-Negami empirical expression is provided too.

In this article, we examine L p-solutions of fractional integral equations in Banach spaces involving the Riemann-Liouville integral operator. Using a compactness type condition, we obtain local and global existence of solutions. Also other types of existence and uniqueness results are established. At the end, an application is given to illustrate the main result.

We improve the formulation of the main statement in the paper “Note on Riesz fractional integrals in the limiting case α(x)p(x) ≡ n”, published in this journal, Fract. Calc. Appl. Anal., Vol. 16, No 2 (2013), DOI: 10.2478/s13540-013-0023-x; http://link.springer.com/article/10.2478/s13540-013-0023-x.