Volume 23, Issue 6

Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to the Karhunen-Loéve expansion theorem are presented as a model for random perturbances. Results show that displacement disturbances propagate with an infinite speed. Some corrections of already published results for a non-stochastic model are also provided.

The survey is devoted to numerical solution of the equation Aαu=f$ {\mathcal A}^\alpha u=f $, 0 <  α <1, where A$ {\mathcal A} $ is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in ℝ d . The fractional power Aα$ {\mathcal A}^\alpha $ is a non-local operator and is defined though the spectrum of A$ {\mathcal A} $. Due to growing interest and demand in applications of sub-diffusion models to physics and engineering, in the last decade, several numerical approaches have been proposed, studied, and tested. We consider discretizations of the elliptic operator A$ {\mathcal A} $ by using an N -dimensional finite element space V h or finite differences over a uniform mesh with N points. In the case of finite element approximation we get a symmetric and positive definite operator Ah:Vh→Vh$ {\mathcal A}_h: V_h \to V_h $, which results in an operator equation Ahαuh=fh$ {\mathcal A}_h^{\alpha} u_h = f_h $ for u h  ∈  V h . The numerical solution of this equation is based on the following three equivalent representations of the solution: (1) Dunford-Taylor integral formula (or its equivalent Balakrishnan formula, (2.5), (2) extension of the a second order elliptic problem in Ω  ×  (0, ∞)⊂ ℝ d +1 [17,55] (with a local operator) or as a pseudo-parabolic equation in the cylinder ( x , t ) ∈ Ω  ×  (0, 1), [70, 29], (3) spectral representation (2.6) and the best uniform rational approximation (BURA) of z α on [0, 1], [37,40]. Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of Ah−α$ {\mathcal A}_h^{-\alpha} $. In this paper we present the main ideas of these methods and the corresponding algorithms, discuss their accuracy, computational complexity and compare their efficiency and robustness.

The identification of the right order of the equation in applied fractional modeling plays an important role. In this paper we consider an inverse problem for determining the order of time fractional derivative in a subdiffusion equation with an arbitrary second order elliptic differential operator. We prove that the additional information about the solution at a fixed time instant at a monitoring location, as “the observation data”, identifies uniquely the order of the fractional derivative.

The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions E α , β ( i λ ϕ ( x )),  x  ∈ ℝ N and E α , β ( i α λ ϕ ( x )),  x  ∈ ℝ N for the various range of α and β . Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

Two inverse problems with final overdetermination for diffusion and wave equations containing the Caputo fractional time derivative and a fractional Laplacian of distributed order are considered. They are: 1) the problem to reconstruct a time-dependent source term; 2) the problem to recover simultaneously the source term, the order of the time derivative and the fractional Laplacian. Uniqueness of solutions of these problems is proved. Sufficient conditions for the uniqueness are stricter for the 2nd problem than for the 1st problem.

We study uniqueness of a solution for an inverse source problem arising in linear time-fractional diffusion equations with time-dependent coefficients. We consider source term in a separated form h ( t ) f ( x ). The unknown source f ( x ) is recovered from the final time measurement u ( x , T ). A new uniqueness result is formulated in Theorem 3.1 under the assumption that h ∈ C ([0, T ]) and 0 ≢ h ≥ 0. No monotonicity in time for h ( t ) and for coefficients of the differential operator is required.

We investigate extreme properties of a class of integro-differential operators. We prove an assertion that extends the Nakhushev extremum principle, known for fractional Riemann-Liouville derivatives, to integro-differential operators with kernels of a general form. We establish the weighted extremum principle for convolution operators and the Riemann-Liouville fractional derivative. In addition, as an application, we prove a uniqueness theorem for a boundary value problem in a non-cylindrical domain for the fractional diffusion equation with the Riemann-Lioville fractional derivative.

The present paper deals with initial value problem (IVP) for semilinear fractional Schrödinger integro-differential equation idudt+Au=∫ 0tfs,Dsα u(s)ds,0< t< T,u0=0$$\begin{array}{} \displaystyle i\!\frac{du}{dt}+Au = \int\limits_{0}^{t}f\left( s,D_{s}^{\alpha }u(s)\right) ds,\, \, \, 0 \lt t \lt T,\, u\left( 0\right) = 0 \end{array} $$ in a Hilbert space H with a self-adjoint positive definite (SAPD) operator A . Stable difference schemes (DSs) have significant interest in investigations of fractional partial differential equations. The main theorem concerns the existence and uniqueness of the uniformly bounded solutions (UBSs) with respect to step time of second order of accuracy DSs for this semilinear fractional Schrödinger differential problem. In practice, existence and uniqueness theorems for a UBS of the one-dimensional initial boundary value problem (BVP) with nonlocal condition and multi-dimensional problem with local condition on the boundary are proved. Numerical results and explanatory illustrations are presented to show the validation of the theoretical results.

We consider a nonlinear stochastic heat equation with Riesz space-fractional derivative and variable thermal conductivity, on infinite domain. First we approximate the original problem by regularizing the Riesz space-fractional derivative. Then we prove that the approximate problem has almost surely a unique solution within a Colombeau generalized stochastic process space. In our solving procedure we use the theory of Colombeau generalized uniformly continuous semigroups of operators. At the end, we study the relation of the original and the approximate problem and prove that, under certain conditions, the derivative operators appearing in these two problems are associated. Even more, we prove that under some additional conditions, solutions of the original and the approximate problem are almost certainly associated as well (assuming that the first one almost surely exists).

In this article, we aim to analyze a mathematical model of tumor growth as a problem of fractional optimal control. The considered fractional-order model describes the interaction of effector-immune cells and tumor cells, including combined chemo-immunotherapy. We deduce the necessary optimality conditions together with implementing the Adomian decomposition method on the suggested fractional-order optimal control problem. The key motive is to perform numerical simulations that shall facilitate us in understanding the behavior of state and control variables. Further, the graphical interpretation of solutions effectively validates the applicability of the present analysis to investigate the growth of cancer cells in the presence of medical treatment.

This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

Editorial Note: This is a paper by M.M. Djrbashian and A.B. Nersesian of 1968, that was published in Russian. There is a constant interest to Djrbashian’s contributions to the topic of fractional calculus and theory of Mittag-Leffler function. Unfortunately, his works were published in Russian and thus, are not easy accessible and not enough popular. Therefore, we invited hS. Rogosin and M. Dubatovskaya to prepare the survey paper in this same issue of “FCAA” and also to translate and edit the present paper in English. On behalf of Editorial Board and fractional calculus’ community, we express to them our thanks for this hard work, including also retyping, mentioning some typos, etc. Authors’ Summary: The concept of fractional integro-differentiation has found a number of applications in earlier papers of the present authors. With this paper we begin the publication of our results in the field of boundary problems for differential operators of fractional order.