Volume 14, Issue 2

Shifts and periodicity for functional-differential equations and their generalizations have been studied by the author in various aspects (cf. for instance, [17]–[20] and following papers). Here we would like to give a comprehensive survey of some of these results (without proofs) in order to recall the most important properties of the considered shifts. In particular, it is shown that the so-called true shifts in complete linear metric spaces are hypercyclic and that a necessary and sufficient condition for true shifts in commutative algebras to be multiplicative is that the generating operator D satisfies the Leibniz condition. A consequence of this fact is that in commutative Leibniz algebras with logarithms the operator D is uniquely determined by an isomorphism acting on $$ \frac{d} {{dt}} $$. There are also studied generalized periodic and exponential-periodic solutions of linear and some nonlinear equations with shifts and generalizations of the classical Birkhoff theorem and Floquet theorem. These results are obtained by means of tools given by Algebraic Analysis (cf. the author [17]). A generalization of binomial formula of Umbral Calculus is shown in Section 7 (cf. Roman and Rota [36]). Section 11 contains a perturbation theorem for linear differential-difference equations with non-commensurable deviations and some its consequences.

A time-fractional diffusion process defined in a discrete probability setting is studied. Working in continuous time, the infinitesimal generators of random processes are discretized and the diffusion equation generalized by allowing the time derivative to be fractional, i.e. of non-integer order. The properties of the resulting distributions are studied in terms of the Mittag-Leffler function. We discuss the computation of these distribution functions by deriving new global rational approximations for the Mittag-Leffler function that account for both its initial Taylor series and asymptotic power-law tail behaviours. Furthermore, we derive integral representations for both the continuous and the discrete time-fractional distributions and use these to prove a convergence theorem.

We define and study the d-plane Radon transform, namely R, on the n-dimensional (flat) torus. The transformation R is obtained by integrating a suitable function f over all d-dimensional geodesics (d-planes in the torus). We specially establish an explicit inversion formula of R and we give a characterization of the image, under the d-plane Radon transform, of the space of smooth functions on the torus.

The current in a simple electric circuit consisting of a resistor and a power supply is studied under the assumption that the current starts from zero and reaches its maximum value. It suggests that an evolution process of a current occurs approximating the circuit temporarily to an R-L circuit. The current equation is solved analytically using Mittag-Leffler function. The equation was also solved numerically. Results supported the assumed behavior of current to evolve from zero to its saturation value.

Two approaches for defining fractional derivatives of periodic distributions are presented. The first is a distributional version of the Weyl fractional derivative in which a derivative of arbitrary order of a periodic distribution is defined via Fourier series. The second is based on the Grünwald-Letnikov formula for defining a fractional derivative as a limit of a fractional difference quotient. The equivalence of the two approaches is established and an application to a fractional diffusion equation, posed in a space of periodic distributions, is also discussed.

We study the use of the wavelet expansion in terms of Meyer’s wavelet to the problem of mollification in the numerical calculation of fractional derivative. It is shown that, when the simplest of Meyer’s wavelet is used, the expansion is equivalent to the transform by the de la Vallée Poussin kernel, that was proposed by Hào et al. It is expected that better results are attained by using the rapidly decaying harmonic wavelet, which is another of Meyer’s wavelet. We examine this. It is also shown that there exists an estimation of the error of approximation in the mollification by using only simple inequalities, without sophisticated inequalities which are invoked in the paper by Hào et al. Examples of numerical calculation are given.

We contribute to the existence theory of abstract time-fractional equations by stating the sufficient conditions for generation of not exponentially bounded α-times C-regularized resolvent families (α > 1) in sequentially complete locally convex spaces. We also consider the growth order of constructed solutions.

The Lamb-Bateman integral equation was introduced to study the solitary wave diffraction and its solution was written in terms of an integral transform. We prove that it is essentially the Abel integral equation and its solution can be obtained using the formalism of fractional calculus.