Volume 24, Issue 2

In this paper, we first address the general fractional integrals and derivatives with the Sonine kernels that possess the integrable singularities of power function type at the point zero. Both particular cases and compositions of these operators are discussed. Then we proceed with a construction of an operational calculus of the Mikusiński type for the general fractional derivatives with the Sonine kernels. This operational calculus is applied for analytical treatment of some initial value problems for the fractional differential equations with the general fractional derivatives. The solutions are expressed in form of the convolution series that generalize the power series for the exponential and the Mittag-Leffler functions.

We establish intertwining relations between Riesz potentials associated with fractional powers of minus-Laplacian and orthogonal Radon transforms 𝓡 j , k of the Gonzalez-Strichartz type. The latter take functions on the Grassmannian of j -dimensional affine planes in ℝ n to functions on a similar manifold of k -dimensional planes by integration over the set of all j -planes that meet a given k -plane at a right angle. The main results include sharp existence conditions of 𝓡 j , k f on L p -functions, Fuglede type formulas connecting 𝓡 j , k with Radon-John k -plane transforms and Riesz potentials, and explicit inversion formulas for 𝓡 j , k f under the assumption that f belongs to the range of the j -plane transform. The method extends to another class of Radon transforms defined on affine Grassmannians by inclusion.

We introduce a new type of dyadic maximal operators and prove that under the log-Hölder continuity condition of the variable exponent p (⋅), it is bounded on L p (⋅) if 1 < p − ≤ p + ≤ ∞. Moreover, the space generated by the L p (⋅) -norm (resp. the L p (⋅), q -norm) of the maximal operator is equivalent to the Hardy space H p (⋅) (resp. to the Hardy-Lorentz space H p (⋅), q ). As special cases, our maximal operator contains the usual dyadic maximal operator and four other maximal operators investigated in the literature.

Obstacle avoidance is one of the main interests regarding path planning. In many situations (mostly those regarding applications in urban environments), the obstacles to be avoided are dynamical and unpredictable. This lack of certainty regarding the environment introduces the need to use local path planning techniques rather than global ones. A well-known method uses artificial potential fields introduced by Khatib. The Weyl potential definition have enabled to distinguish the dangerousness of an obstacle, however acceleration oscillations appear when the considered robot enters a danger zone close to an obstacle, thus leading to high energy consumption. In order to reduce these oscillations regarding this method, four alternative formulations for the repulsive field are proposed: corrective polynomials, tangential and radial components, Poisson potential and pseudo fractional potential. Their limitations will be explored and their performances will be compared by using criteria such as length and energy in a simulation scenario.

By observing that the fractional Caputo derivative of order α ∈ (0, 1) can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag-Leffler function. Each eigenfunction turns out to be the Laplace transform of the right-inverse of a non-decreasing self-similar Markov process associated via the so-called Lamperti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplicative convolution operators. In particular, we provide both a stochastic representation, expressed in terms of these inverse processes, and an explicit representation, given in terms of the generalized Mittag-Leffler functions, of the solution of these self-similar Cauchy problems. This work could be seen as an-in depth analysis of a specific class, the one with the self-similarity property, of the general inverse of increasing Markov processes introduced in [15].

The asymptotic behaviour of solutions in an appropriate space is discussed for a fractional problem involving Hadamard left-sided fractional derivatives of different orders. Reasonable sufficient conditions are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach a logarithmic function as time goes to infinity. This generalizes and extends earlier results on integer order differential equations to the fractional case. Our approach is based on appropriate desingularization techniques and generalized versions of Gronwall-Bellman inequality. It relies also on a kind of Hadamard fractional version of l'Hopital’s rule which we prove here.

In this paper, we investigate a nonlocal boundary value problem for an equation of special type. For y > 0 it is a fractional diffusion equation, which contains the Riemann-Liouville fractional derivative. For y < 0 it is a generalized equation of moisture transfer. A unique solvability of the considered problem is proved.

Mikusiński’s operational calculus is a formalism for understanding integral and derivative operators and solving differential equations, which has been applied to several types of fractional-calculus operators by Y. Luchko and collaborators, such as for example [26], etc. In this paper, we consider the operators of Riemann–Liouville fractional differentiation of a function with respect to another function, and discover that the approach of Luchko can be followed, with small modifications, in this more general setting too. The Mikusiński’s operational calculus approach is used to obtain exact solutions of fractional differential equations with constant coefficients and with this type of fractional derivatives. These solutions can be expressed in terms of Mittag-Leffler type functions.

We study the duality theory for fractional resolvents, extending and improving some corresponding theorems on semigroups. As applications, we develop the variational technique to analyze the finite-approximate controllability of a backward fractional control system with a right-sided Riemann-Liouville fractional derivative. Moreover, validity of our theoretical findings is given by a fractional diffusion model.

This work is devoted to examining the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal super-critical diffusion operator and a multiplicative noise. Our proof for uniqueness is based upon the analysis on double variables method and the existence is enabled by a parabolic approximation.

In this work, we consider the n -dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem. More precisely, we show that the eigenfunctions are orthogonal and the eigenvalues are real and simple. Moreover, using techniques from fractional variational calculus, we prove in the main result that the eigenvalues are separated and form an infinite sequence, where the eigenvalues can be ordered according to increasing magnitude. Finally, a connection with Clifford analysis is established.

The theory of averaging is a classical component of applied mathematics and has been applied to solve some engineering problems, such as in the filed of control engineering. In this paper, we develop a theory of averaging on both finite and infinite time intervals for fractional non-autonomous differential equations. The closeness of the solutions of fractional no-autonomous differential equations and the averaged equations has been proved. The main results of the paper are applied to the switched capacitor voltage inverter modeling problem which is described by the fractional differential equations.