Volume 20, Issue 2

Since the 60s of last century Fractional Calculus exhibited a remarkable progress and presently it is recognized to be an important topic in the scientific arena. This survey analyzes and measures the evolution that occurred during the last five decades in the light of books, journals and conferences dedicated to the theory and applications of this mathematical tool, dealing with operations of integration and differentiation of arbitrary (fractional) order and their generalizations.

We consider the homogeneous equation Au = 0, where A is a symmetric and coercive elliptic operator in H 1 (Ω) with Ω bounded domain in ℝ d . The boundary conditions involve fractional power α , 0 < α < 1, of the Steklov spectral operator arising in Dirichlet to Neumann map. For such problems we discuss two different numerical methods: (1) a computational algorithm based on an approximation of the integral representation of the fractional power of the operator and (2) numerical technique involving an auxiliary Cauchy problem for an ultra-parabolic equation and its subsequent approximation by a time stepping technique. For both methods we present numerical experiment for a model two-dimensional problem that demonstrate the accuracy, efficiency, and stability of the algorithms.

Most existing research on applying the finite element method to discretize space fractional operators is studied on regular domains using either uniform structured triangular meshes, or quadrilateral meshes. Since many practical problems involve irregular convex domains, such as the human brain or heart, which are difficult to partition well with a structured mesh, the existing finite element method using the structured mesh is less efficient. Research on the finite element method using a completely unstructured mesh on an irregular domain is of great significance. In this paper, a novel unstructured mesh finite element method is developed for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. The novel unstructured mesh Galerkin finite element method is used to discretize in space and the Crank-Nicholson scheme is used to discretize the Caputo time fractional derivative. The implementation of the unstructured mesh Crank-Nicholson Galerkin method (CNGM) is detailed and the stability and convergence of the numerical scheme are analyzed. Numerical examples are presented to verify the theoretical analysis. To highlight the ability of the proposed unstructured mesh Galerkin finite element method, a comparison of the unstructured mesh with the structured mesh in the implementation of the numerical scheme is conducted. The proposed numerical method using an unstructured mesh is shown to be more effective and feasible for practical applications involving irregular convex domains.

Considerable attention has been recently given to the existence of solutions of initial or boundary value problems for fractional differential equations and inclusions with Hilfer fractional derivative. Motivated by these results, in this paper we will present existence, data dependence and Ulam stability results for some differential inclusions with Hilfer fractional derivative. The results follow as applications of the multi-valued weakly Picard operator theory. An example illustrates the main result of the paper.

We consider a semi-infinite one-dimensional phase-change material with two unknown constant thermal coefficients among the latent heat per unit mass, the specific heat, the mass density and the thermal conductivity. Aiming at the determination of them, we consider an inverse one-phase Stefan problem with an over-specified condition at the fixed boundary and a known evolution for the moving boundary. We assume that it is given by a sharp front and we consider a time fractional derivative of order α (0 < α < 1) in the Caputo sense to represent the temporal evolution of the temperature as well as the moving boundary. This might be interpreted as the consideration of latent-heat memory effects in the development of the phase-change process. According to the choice of the unknown thermal coefficients, six inverse fractional Stefan problems arise. For each of them, we determine necessary and sufficient conditions on data to obtain the existence and uniqueness of a solution of similarity type. Moreover, we present explicit expressions for the temperature and the unknown thermal coefficients. Finally, we show that the results for the classical statement of this problem, associated with α = 1, are obtained through the fractional model when α → 1 — .

The purpose of this paper is to study generalized fractional integral operators whose kernels involve a very special class of generalized hypergeometric functions r +2 F r +1 . The importance of these operators stems from the summability of this special class of generalized hypergeometric functions at unit argument which is realized in the present paper by making use of the Karlsson-Minton type summation theorems. We obtain for these operators several results including the generalized fractional integrals of the H -function, certain mapping properties and their Mellin transforms. Applications to inequalities are also pointed out.

An accurate and efficient new class of predictor-corrector schemes are proposed for solving nonlinear differential equations of fractional order. By introducing a new prediction method which is explicit and of the same accuracy order as that of the correction stage, the new schemes achieve a uniform accuracy order regardless of the values of fractional order α . In cases of 0 < α ≤ 1, the new schemes significantly improve the numerical accuracy comparing with other predictor-corrector methods whose accuracy depends on a. Furthermore, by computing the memory term just once for both the prediction and correction stages, the new schemes reduce the computational cost of the so-called memory effect, which make numerical schemes for fractional differential equations expensive in general. Both 2nd-order scheme with linear interpolation and the high-order 3rd-order one with quadratic interpolation are developed and show their advantages over other comparing schemes via various numerical tests.

In this paper invariant subspace method has been employed for solving linear and non-linear time and space fractional partial differential equations involving Caputo derivative. A variety of illustrative examples are solved to demonstrate the effectiveness and applicability of the method.

In the paper, existence, uniqueness and a Cauchy formula for the solution to a time-varying linear system containing fractional Caputo derivative is obtained. This formula shows that nonnegativity of the data of the system implies nonnegativity of the solution. In the context of a strenghthening of this result, an example illustrating the absence (in the case of Caputo derivative) of the standard relation “monotonicity of function - sign of derivative”.

Series defined by means of the three-parametric Mittag-Leffler functions, called also the Prabhakar functions, are considered in this paper. Their behaviour is investigated on the boundaries of the convergence domains. Necessary and sufficient conditions for their overconvergence are proposed. The corresponding results for series in Mittag-Leffler functions are discussed as a particular case. Such kind of results are motivated by the fact that the solutions of some fractional order differential and integral equations can be written in terms of series (or series of integrals) of Mittag-Leffler type functions.

We study a viscoelastic body involving a constitutive equation with distributed order fractional derivatives of complex order. Using a dissipation inequality in a weak form, we derive a sufficient conditions on coefficients of a model that guarantee that the Second law of thermodynamics under isothermal conditions is satisfied. Several known constitutive equations follow from our model as special cases. As an application, a new constitutive equation is related to an equation of motion of a generalized linear oscillator.

In this paper we establish an observability inequality for partial differential equations with time derivatives of non-integer order. The observation region is a subset of positive measure in Ω × (0, T ). This inequality is equivalent to the null controllable property for the linear controlled fractional diffusion equation.

This paper concerns the null controllability of fractional dynamical systems with constrained control. We assume that the linear system is controllable with square integrable control and provide sufficient conditions for the null controllability with constrained control. Further, sufficient conditions for the null controllability of nonlinear system also are established. Examples are provided to illustrate the theory.