Volume 17, Issue 4

A survey of three types of convolutions, depending on arbitrary linear functionals is made. They are convolutions for right inverse operators of the differentiation operator, the Euler operator and the square of the differentiation operator. Three lines of applications of these convolutions are outlined: characterizing their multipliers, the commutants and direct construction of operational calculi.

Let χ and ϕ be “arbitrary” linear functionals on C[0,∞) and C 1[0, a], respectively. A large class of evolution equations with one space variable and one time variable is considered. A theorem of uniqueness of the solution of this boundary value problem is proved.

An initial-boundary value problem for the velocity distribution of a viscoelastic flow with generalized fractional Oldroyd-B constitutive model is studied. The model contains two Riemann-Liouville fractional derivatives in time. The eigenfunction expansion of the solution is constructed. The behavior of the time-dependent components of the solution is studied and the results are used to establish convergence of the series under some conditions. Further, applying the convolutional calculus approach proposed by Dimovski (I.H. Dimovski, Convolutional Calculus, Kluwer, Dordrecht (1990)), a Duhamel-type representation of the solution is found, containing two convolution products of particular solutions and the given initial and source functions. A non-classical convolution with respect to spatial variable is used. The obtained representation is applied for numerical computation of the solution in the case of a generalized second grade fluid. Numerical results for several one-dimensional examples are given and the present technique is compared to a finite difference method in terms of efficiency, accuracy, and CPU time.

In 1966 Ivan Dimovski introduced and started detailed studies on the Bessel type differential operators B of arbitrary (integer) order m ≥ 1. He also suggested a variant of the Obrechkoff integral transform (arising in a paper of 1958 by another Bulgarian mathematician Nikola Obrechkoff) as a Laplace-type transform basis of a corresponding operational calculus for B and for its linear right inverse integral operator L. Later, the developments on these linear singular differential operators appearing in many problems of mathematical physics, have been continued by the author of this survey who called them hyper-Bessel differential operators, in relation to the notion of hyper-Bessel functions of Delerue (1953), shown to form a fundamental system of solutions of the IVPs for By(t) = λy(t). We have been able to extend Dimovski’s results on the hyper-Bessel operators and on the Obrechkoff transform due to the happy hint to attract the tools of the special functions as Meijer’s G-function and Fox’s H-function to handle successfully these matters. These author’s studies have lead to the introduction and development of a theory of generalized fractional calculus (GFC) in her monograph (1994) and subsequent papers, and to various applications of this GFC in other topics of analysis, differential equations, special functions and integral transforms. Here we try briefly to expose the ideas leading to this GFC, its basic facts and some of the mentioned applications.

The Delerue hyper-Bessel functions that appeared as a multi-index generalizations of the Bessel function of the first type, are closely related to the hyper-Bessel differential operators of arbitrary order, introduced by Dimovski. In this work we consider an enumerable family of hyper-Bessel functions and study the convergence of series in such a kind of functions. The obtained results are analogues to the ones in the classical theory of the widely used power series, like Cauchy-Hadamard, Abel and Fatou theorem.

We study the multiplicity of solutions for fractional differential equations subject to boundary value conditions and impulses. After introducing the notions of classical and weak solutions, we prove the existence of at least three solutions to the impulsive problem considered.

Cauchy problems for a class of linear differential equations with constant coefficients and Riemann-Liouville derivatives of real orders, are analyzed and solved in cases when some of the real orders are irrational numbers and when all real orders appearing in the derivatives are rational numbers. Our analysis is motivated by a forced linear oscillator with fractional damping. We pay special attention to the case when the leading term is an integer order derivative. A new form of solution, in terms of Wright’s function for the case of equations of rational order, is presented. An example is treated in detail.

The Stieltjes transform has recently been extended to a subspace of Boehmians. In this note, additional results are obtained which include an inversion formula plus Abelian type theorems.

A family of nonstandard Gauss-Jacobi-Lobatto quadratures for numerical calculating integrals of the form ∫-11 f′(x)(1-x)α dx, α > -1, is derived and applied to approximation of the usual fractional derivative. A software implementation of such quadratures was done by the recent Mathematica package OrthogonalPolynomials (cf. [A.S. Cvetković, G.V. Milovanović, Facta Univ. Ser. Math. Inform. 19 (2004), 17–36] and [G.V. Milovanović, A.S. Cvetković, Math. Balkanica 26 (2012), 169–184]). Several numerical examples are presented and they show the effectiveness of the proposed approach.

Fuzzy fractional differential equations with fuzzy coefficients are analyzed in the frame of Mikusiński operators. Systems of fuzzy operational algebraic equations are obtained, in view of the definition of fuzzy derivatives. Their exact and approximate solutions are constructed and their characters are analyzed, considering them as the corresponding solutions of the given problem. The described procedure of the construction of solutions is illustrated on an example and the obtained approximate solutions of the considered problems are visualized by using the GeoGebra software package.

This article deals with investigation of some important properties of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations in bounded multi-dimensional domains. In particular, we investigate the asymptotic behavior of the solutions as the time variable t → 0 and t → +∞. By the Laplace transform method, we show that the solutions decay logarithmically as t → +∞. As t → 0, the decay rate of the solutions is dominated by the term (t log(1/t))−1. Thus the asymptotic behavior of solutions to the initial-boundary-value problem for the distributed order time-fractional diffusion equations is shown to be different compared to the case of the multi-term fractional diffusion equations.

Fractional-order generalized Laguerre functions (FGLFs) are proposed depends on the definition of generalized Laguerre polynomials. In addition, we derive a new formula expressing explicitly any Caputo fractional-order derivatives of FGLFs in terms of FGLFs themselves. We also propose a fractional-order generalized Laguerre tau technique in conjunction with the derived fractional-order derivative formula of FGLFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). The fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on FGLFs and compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

In this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

In this paper, by using fixed-point theorems, and lower and upper solution method, the existence for a class of fractional initial value problem (FIVP) $\begin{gathered} D_{0 + }^\alpha u(t) = f(t,u(t)),t \in (0,h), \hfill \\ t^{2 - \alpha } u(t)|_{t = 0} = b_1 D_{0 + }^{\alpha - 1} u(t) = |_{t = 0} = b_2 , \hfill \\ \end{gathered} $ is discussed, where f ∈ C([0, h]×R,R), D 0+α u(t) is the standard Riemann-Liouville fractional derivative, 1 < α < 2. Some hidden confusion and fallacy in the literature are commented. A new condition on the nonlinear term is given to guarantee the equivalence between the solution of the FIVP and the fixed-point of the operator. Based on the new condition, some new existence results are obtained and presented as example.

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

The reflection symmetric Erdélyi-Kober type fractional integral operators are used to construct fractional quasi-particle generators. The eigenfunctions and eigenvalues of these operators are given analytically. A set of fractional creation- and annihilation-operators is defined and the properties of the corresponding free Hamiltonian are investigated. Analogously to the classical approach for interacting multi-particle systems, the results are interpreted as a fractional quantum model for a description of residual interactions of pairing type.