Volume 24, Issue 3

In this editorial paper, we start by surveying of the main milestones in the organization, foundation, and development of the journal Fractional Calculus and Applied Analysis ( FCAA ). The main potential of FCAA is in its readers, authors, and editors who contribute to the scientific advance and promote the progress of the journal. Among the editors, a special role of the honorary editors who contributed significantly to the foundation of the journal should be highlighted. These are prominent scientists, lecturers, and disseminators of FC and related topics. Unfortunately, some of them have already passed away, but they remain our living and lasting memory. In the main part of this survey, we remind the readers some biographical data and achievements related to FCAA topics of these Honorary Founding Editors: Professors Eric Love, Ian Sneddon, Bogoljub Stanković, Rudolf Gorenflo, Danuta Przeworska-Rolewicz, Gary Roach, Anatoly Kilbas, and Wen Chen. In eight sections, each devoted to one of the honorary editors, their ways in science and in FC, as well as their main results and publications are shortly described. Special attention is given to their role in foundation of FCAA and to their contributions to its success story.

In this paper, we introduce a short-time coupled fractional Fourier transform ( scfrft ) using the kernel of the coupled fractional Fourier transform ( cfrft ). We then prove that it satisfies Parseval’s relation, derive its inversion and addition formulas, and characterize its range on ℒ 2 (ℝ 2 ). We also study its time delay and frequency shift properties and conclude the article by a derivation of an uncertainty principle for both the coupled fractional Fourier transform and short-time coupled fractional Fourier transform.

The formula of the all-pole low-pass frequency filter transfer function of the fractional order ( N + α ) designated for implementation by non-cascade multiple-feedback analogue structures is presented. The aim is to determine the coefficients of this transfer function and its possible variants depending on the filter order and the distribution of the fractional-order terms in the transfer function. Optimization algorithm is used to approximate the target Butterworth low-pass magnitude response, whereas the approximation errors are evaluated. The interpolated equations for computing the transfer function coefficients are provided. An example of the transformation of the fractional-order low-pass to the high-pass filter is also presented. The results are verified by simulation of multiple-feedback filter with operational transconductance amplifiers and fractional-order element.

The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different route, namely, construction of explicit asymptotic approximations for the solutions. As a study case, we consider a problem with left and right Riemann-Liouville derivatives, for which our analysis yields asymptotically sharp estimates for the sequence of eigenvalues and eigenfunctions.

In this paper we characterize the Laplace transform of functions with power growth square averages and study several multi-term Caputo and Riemann-Liouville fractional integro-differential equations in this space of functions.

This paper gathers the tools for solving Riemann-Liouville time fractional non-linear PDE’s by using a Galerkin method. This method has the advantage of not being more complicated than the one used to solve the same PDE with first order time derivative. As a model problem, existence and uniqueness is proved for semilinear heat equations with polynomial growth at infinity.

This paper presents two new classes of Müntz functions which are called Jacobi-Müntz functions of the first and second types. These newly generated functions satisfy in two self-adjoint fractional Sturm-Liouville problems and thus they have some spectral properties such as: orthogonality, completeness, three-term recurrence relations and so on. With respect to these functions two new orthogonal projections and their error bounds are derived. Also, two new Müntz type quadrature rules are introduced. As two applications of these basis functions some fractional ordinary and partial differential equations are considered and numerical results are given.

In this article, we consider the space-time fractional (nonlocal) equation characterizing the so-called “double-scale” anomalous diffusion ∂ tβ u(t,x)=− (− Δ )α /2u(t,x)− (− Δ )γ /2u(t,x),t> -->0,− 1< x< 1, $$\begin{array}{} \displaystyle \partial_t^\beta u(t, x) = -(-\Delta)^{\alpha/2}u(t,x) - (-\Delta)^{\gamma/2}u(t,x), \, \, t \gt 0, \, -1 \lt x \lt 1, \end{array}$$ where ∂ tβ $\begin{array}{} \displaystyle \partial_t^\beta \end{array}$ is the Caputo fractional derivative of order β ∈ (0, 1) and 0 < α ≤ γ < 2. We consider a nonlocal inverse problem and show that the fractional exponents β , α and γ are determined uniquely by the data u ( t , 0) = g ( t ), 0 < t ≤ T . The existence of the solution for the inverse problem is proved using the quasi-solution method which is based on minimizing an error functional between the output data and the additional data. In this context, an input-output mapping is defined and its continuity is established. The uniqueness of the solution for the inverse problem is proved by means of eigenfunction expansion of the solution of the forward problem and some basic properties of fractional Laplacian. A numerical method based on discretization of the minimization problem, namely the steepest descent method and a least squares approach, is proposed for the solution of the inverse problem. The numerical method determines the fractional exponents simultaneously. Finally, numerical examples with noise-free and noisy data illustrate applicability and high accuracy of the proposed method.

For an integro-differential equation of the convolution type defined on the half-line [0, ∞) with a power nonlinearity and variable coefficient, we use the weight metrics method to prove a global theorem on the existence and uniqueness of a solution in the cone of nonnegative functions in the space C [0, ∞). It is shown that the solution can be found by a successive approximation method of the Picard type; an estimate for the rate of convergence of the approximations is produced. We present sharp two-sided a-priori estimates for the solutions. These estimates enable us to construct a complete metric space which is invariant under the nonlinear convolution operator considered here and to prove that the equation induced by this operator has a unique solution in this space as well as in the class of all non-negative continuous functions vanishing at the origin. Such equations with operators of fractional calculus as the Riemann-Liouville, Erdélyi-Kober, Hadamard fractional integrals arise, in particular, when describing the process of propagation of shock waves in gas-filled pipes, solving the problem about heating a half-infinite body in a nonlinear heat-transfer process, considering models of population genetics, and elsewhere.

This paper introduces an efficient collocation solver, the generalized finite difference method (GFDM) combined with the recent-developed scale-dependent time stepping method (SD-TSM), to predict the anomalous diffusion behavior on surfaces governed by surface time-fractional diffusion equations. In the proposed solver, the GFDM is used in spatial discretization and SD-TSM is used in temporal discretization. Based on the moving least square theorem and Taylor series, the GFDM introduces the stencil selection algorithms to choose the stencil support of a certain node from the whole discretization nodes on the surface. It inherits the similar properties from the standard FDM and avoids the mesh generation, which is available particularly for high-dimensional irregular discretization nodes. The SD-TSM is a non-uniform temporal discretization method involving the idea of metric, which links the fractional derivative order with the non-uniform discretization strategy. Compared with the traditional time stepping methods, GFDM combined with SD-TSM deals well with the low accuracy in the early period. Numerical investigations are presented to demonstrate the efficiency and accuracy of the proposed GFDM in conjunction with SD-TSM for solving either single or coupled fractional diffusion equations on surfaces.

A specific formulation of the “classical” problem of mathematical analysis is considered. This is the problem of calculating the derivative of a function. The purpose of this work is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on the methods of control theory. The input data of the algorithm is represented by inaccurate measured function values at discrete, frequently enough, times. The proposed algorithm is based on two aspects: a local modification of the Tikhonov regularization method from the theory of ill-posed problems and the Krasovskii extremal shift method from the guaranteed control theory, both of which ensure the stability to informational noises and computational errors. Numerical experiments were carried out to illustrate the operation of the algorithm.

In this paper, we develop a sliding method for the higher order fractional Laplacians. We first obtain the key ingredients to obtain monotonicity of solutions, such as narrow region maximum principles in bounded or unbounded domains. Then we introduce a new idea of estimating the singular integrals defining the fractional Laplacian along a sequence of approximate maximum points and illustrate how this sliding method can be employed to obtain monotonicity of solutions. We believe that the narrow region maximum principles will become useful tools in analyzing higher order fractional equations.

The global stability of continuous-time fractional orders nonlinear feedback systems with positive linear parts and interval state matrices is investigated. New sufficient conditions for the global stability of this class of positive feedback nonlinear systems are established. The effectiveness of these new stability conditions is demonstrated on simple example.