Volume 18, Issue 4

In this paper we investigate some properties of the n-dimensional generalized Erdélyi-Kober operator with the Bessel function in the kernel and its applications in the case n = 3 to solving a Cauchy problem for four- dimensional differential equation of hyperbolic type with singular coefficients.

We first introduce the concept of Sugeno fractional integral based on the concept of g-seminorm. Then Minkowski’s inequality for Sugeno fractional integral of the order α > 0 based on two binary operations ⋆, ∗ is given. Our results significantly generalize the previous results in this field of fuzzy measure and fuzzy integral. Some examples are given to illustrate the results.

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: tD α ∞ ( −∞ D α t u(t)) + L(t)u(t) = ∇W(t, u(t)), (0.1) u ∈ H α (ℝ,ℝ N ), where α ∈ (1/2, 1), t ∈ ℝ, u ∈ ℝ n , L ∈ C(ℝ,ℝ n2 ) is a symmetric and positive definite matrix for all t ∈ ℝ, W ∈ C 1 (ℝ × ℝ n ,ℝ), and ∇W is the gradient of W at u. The novelty of this paper is that, assuming there exists l ∈ C(ℝ,ℝ) such that (L(t)u, u) ≥ l(t)|u| 2 for all t ∈ ℝ, u ∈ ℝ n and the following conditions on l: inf t ∈ ℝ l(t) > 0 and there exists r 0 > 0 such that, for any M >0 m({t ∈ (y − r 0 , y + r 0 )/ l(t) ≤ M}) → 0 as |y| →∞ are satisfied and W is of subquadratic growth as |u| → +∞, we show that (0.1) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in Z. Zhang and R. Yuan [24] are significantly improved.

This paper is devoted to the study of synchronous phenomena of the solutions to system driven by fractional environmental noises on finite lattice. Under certain dissipative and integrability conditions, we obtain the synchronization between two solutions, and among different components of solutions when the coupling coefficient tends to infinity. This indicates that no matter how large the intensity and what kinds of the noises perturb the system, the synchronization persists

In this paper we consider a space fractional reaction-advection-diffusion equation which is actually a semi-linear Cauchy problem with a spatial fractional derivative operator of order α, 0 < α < 2. We establish and prove assertions concerning the existence and uniqueness of solution within certain Colombeau space. The proofs are given in the case when the left Liouville fractional derivative is involved, but the results are also valid in the case of the right Liouville fractional derivative as well as for the Riesz fractional derivative. The solutions are obtained by using the theory of generalized uniformly continuous semigroups of operators. We have also proved that the non-regularized and corresponding regularized operators, as well as solutions of non-regularized and corresponding regularized equations are L 2 −associated.

In this paper, we deal with existence and uniqueness of (μ, ν)-pseudo almost automorphic mild (classical) solution to semilinear fractional differential equations with the Caputo derivatives. The main results are obtained by means of the fixed point theory, Leray-Schauder alternative theorem and fractional powers of operators. Moreover, an application to fractional predator-prey system with diffusion is given.

We describe all solutions of the operator analogue of the product rule with respect to a convolution of the generalized Gel’fond-Leont’ev integration in the space of functions analytic in a domain of the complex plane. This is an important example of the Erdélyi-Kober operator of integration of fractional order, for which a series of basic results have been proposed by Dimovski and Kiryakova, and in a series of previous authors’ papers

In this paper we analyze a linear system for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides depended on time. First, we prove existence and uniqueness of the classical solution to this problem, and provide the coercive estimates of the solution. Second, based on the obtained results we establish one-to-one solvability to a linear system of a general form in the H¨older spaces.

We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck- Smoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed.

From the point of view of stochastic analysis the Caputo and Riemann- Liouville derivatives of order α ∈ (0, 2) can be viewed as (regularized) generators of stable Lévy motions interrupted on crossing a boundary. This interpretation naturally suggests fully mixed, two-sided or even multidimensional generalizations of these derivatives, as well as a probabilistic approach to the analysis of the related equations. These extensions are introduced and some well-posedness results are obtained that generalize, simplify and unify lots of known facts. This probabilistic analysis leads one to study a class of Markov processes that can be constructed from any given Markov process in R d by blocking (or interrupting) the jumps that attempt to cross certain closed set of ’check-points’.

With this short note, I would like to clarify some of the results contained in our previous paper [5]: M. Fabrizio, Fractional rheological models for thermomechanical systems. Dissipation and free energies. Fract. Calc. Appl. Anal. 17, No 1 (2014), 206-223; DOI: 10.2478/s13540-014-0163-7; http://www.degruyter.com/view/j/fca.2014.17.issue-1/issue-files/fca.2014.17.issue-1.xml, concerning the thermodynamic conditions for fractional models of heat propagation, which generalize the classical Cattaneo-Maxwell and Fourier laws.

In this work we provide a new mathematical model for the Pennes’ bioheat equation, assuming a fractional time derivative of single order. Alternative versions of the bioheat equation are studied and discussed, to take into account the temperature-dependent variability in the tissue perfusion, and both finite and infinite speed of heat propagation. The proposed bioheat model is solved numerically using an implicit finite difference scheme that we prove to be convergent and stable. The numerical method proposed can be applied to general reaction diffusion equations, with a variable diffusion coefficient. The results obtained with the single order fractional model, are compared with the original models that use classical derivatives.