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In his first paper on the generalization of the tautochrone problem, that was published in 1823, Niels Henrik Abel presented a complete framework for fractional-order calculus, and used the clear and appropriate notation for fractional-order integration and differentiation.

In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich’s fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

The aim of the paper is twofold. First, we present a new general approach to the definition of a class of mixed norm spaces of analytic functions 𝓐 q ; X (𝔻), 1 ⩽ q < ∞ on the unit disc 𝔻. We study a problem of boundedness of Bergman projection in this general setting. Second, we apply this general approach for the new concrete cases when X is either Orlicz space or generalized Morrey space, or generalized complementary Morrey space. In general, such introduced spaces are the spaces of functions which are in a sense the generalized Hadamard type derivatives of analytic functions having l q summable Taylor coefficients.

In this paper, we discuss the maximum principle for a time-fractional diffusion equation ∂ tα u(x,t)=∑ i,j=1n∂ i(aij(x)∂ ju(x,t))+c(x)u(x,t)+F(x,t),t> -->0,x∈ Ω ⊂ Rn,$$\begin{array}{} \displaystyle \partial_t^{\alpha} u(x,t) = \sum\limits_{i,j=1}^n \partial_i(a_{ij}(x)\partial_j u(x,t))+ c(x)u(x,t) + F(x,t),\,\, t\gt 0,\,\,x \in \Omega \subset {\mathbb R}^n, \end{array} $$ with the Caputo time-derivative of the order α ∈ (0, 1) in the case of the homogeneous Dirichlet boundary condition. Compared to the already published results, our findings have two important special features. First, we derive a maximum principle for a suitably defined weak solution in the fractional Sobolev spaces, not for the strong solution. Second, for the non-negative source functions F = F ( x , t ) we prove the non-negativity of the weak solution to the problem under consideration without any restrictions on the sign of the coefficient c = c ( x ) by the derivative of order zero in the spatial differential operator. Moreover, we prove the monotonicity of the solution with respect to the coefficient c = c ( x ).

In this paper, we consider the following Brézis-Nirenberg problem involving the fractional Laplacian operator: (− Δ )su=λ u+|u|2s∗ − 2uinΩ ,u=0on∂ Ω ,$$\begin{array}{} \displaystyle\left\{\begin{array}{ll} (-\Delta)^s u=\lambda u+|u|^{2_s^{*}-2}u & \textrm{in}\ \, \Omega, \\ u=0 & \textrm{on}\ \, \partial\Omega, \end{array} \right. \end{array} $$ where s ∈ (0, 1), Ω is a bounded smooth domain of ℝ N ( N > 6 s ) and 2s∗ =2NN− 2s$\begin{array}{} \displaystyle 2_s^{*}=\frac{2N}{N-2s} \end{array}$ is the critical fractional Sobolev exponent. We show that, for each λ > 0, this problem has infinitely many sign-changing solutions by using a compactness result obtained in [34] and a combination of invariant sets method and Ljusternik-Schnirelman type minimax method.

In this paper, we investigate some aspects of the qualitative theory for multi-order fractional differential equation systems. First, we obtain a fundamental result on the existence and uniqueness for multi-order fractional differential equation systems. Next, a representation of solutions of homogeneous linear multi-order fractional differential equation systems in series form is provided. Finally, we give characteristics regarding the asymptotic behavior of solutions to some classes of linear multi-order fractional differential equation systems.

Recently S. Gerhold and R. Garra – F. Polito independently introduced a new function related to the special functions of the Mittag-Leffler family. This function is a generalization of the function studied by É. Le Roy in the period 1895-1905 in connection with the problem of analytic continuation of power series with a finite radius of convergence. In our note we obtain two integral representations of this special function, calculate its Laplace transform, determine an asymptotic expansion of this function on the negative semi-axis (in the case of an integer third parameter γ ) and provide its continuation to the case of a negative first parameter α . An asymptotic result is illustrated by numerical calculations. Discussion on possible further studies and open questions are also presented.

We characterize the function space whose elements have a Mellin transform with exponential decay at infinity. This result can be seen as a generalization of the Paley–Wiener theorem for Mellin transforms. As a byproduct in a similar spirit, we also characterize spaces of functions whose distances from Mellin–Paley–Wiener spaces have a prescribed asymptotic behavior. This leads to Mellin–Sobolev type spaces of fractional order.

For a class of tempered fractional terminal value problems of the Caputo type, we study the existence and uniqueness of the solution, analyze the continuous dependence on the given data, and using a shooting method we present and discuss three numerical schemes for the numerical approximation of such problems. Some numerical examples are considered in order to illustrate the theoretical results and evidence the efficiency of the numerical methods.

In this paper the accurate relationships between the averaging procedure of a smooth function over 1D- fractal sets and the fractional integral of the RL-type are established. The numerical verifications are realized for confirmation of the analytical results and the physical meaning of these obtained formulas is discussed. Besides, the generalizations of the results for a combination of fractal circuits having a discrete set of fractal dimensions were obtained. We suppose that these new results help understand deeper the intimate links between fractals and fractional integrals of different types. These results can be used in different branches of the interdisciplinary physics, where the different equations describing the different physical phenomena and containing the fractional derivatives and integrals are used.

This paper is devoted to the fractional generalization of the Fokker-Planck equation associated with a nonlinear stochastic differential equation on a bounded domain. The driving process of the stochastic differential equation is a Lévy process subordinated to the inverse of Lévy’s mixed stable subordinators. The Fokker-Planck equation is given through the general Waldenfels operator, while the boundary condition is given through the general Wentcel’s boundary condition. As a fractional operator a distributed order differential operator with a Borel mixing measure is considered. In the paper fractional generalizations of the Fokker-Planck equation are derived and the existence of a unique solution of the corresponding initial-boundary value problems is proved.

There are many numerical algorithms for solving the fractional-order ordinary differential equations (FODEs). They are usually very different in nature, and it is difficult to compare their performances. To solve this problem, a set of five benchmark problems of different categories of FODEs with known analytical solution are designed and proposed, they can be used as benchmark problems for testing the numerical algorithms. A Simulink block diagram scheme is used for solving these benchmark problems, with computing errors and the running times reported.