Yuri Luchko

Fermat’s theorem, often referred to as the interior extrema theorem, is one of the fundamental results in calculus and optimization theory. Recently, the statement of this theorem has been generalized to the case of fractional derivatives. Unlike the classical case, these fractional analogs are typically expressed as inequalities rather than equalities. Moreover, the exact form of these inequalities depends on the particular definition of the fractional derivative being used. In this paper, for the first time, we present Fermat-type results for the first-level general fractional derivative that includes the Caputo, Riemann–Liouville, and Hilfer derivatives, as well as the general fractional derivatives and the regularized general fractional derivatives with Sonin kernels, among its particular cases. We also discuss some applications of this fractional analogy of Fermat’s theorem, including derivation of a comparison principle for the fractional differential inequalities involving the first-level general fractional derivatives, as well as a priori estimates for solutions of the initial-value problems for the fractional differential equations with the first-level general fractional derivatives.

The Fermat theorem that is often referred to as the interior extrema theorem is one of the fundamental results in calculus and optimization theory. Recently, the statement of this theorem has been generalized to the case of fractional derivatives. Unlike in the classical case, these fractional analogs typically appear in the form of inequalities rather than equalities. Moreover, the exact form of these inequalities depends on the particular definition of the fractional derivative being used. In this paper, for the first time, we present Fermat-type results for the 1st level general fractional derivative that has the Caputo, Riemann–Liouville, and Hilfer derivatives, as well as the general fractional derivatives and the regularized general fractional derivatives with Sonin kernels among its particular cases. We also discuss some applications of this fractional analogy of Fermat’s theorem including derivation of a comparison principle for the fractional differential inequalities involving the 1st level general fractional derivatives as well as a priori estimates for solutions of the initial-value problems for the fractional differential equations with the 1st level general fractional derivatives.

A diusion-type process with a non-linear relationship between the meansquared displacement (MSD) of the diusing particles and time is nowadaysreferred to as anomalous diusion. Recently, the authors of this paper suggested a very general framework of the anomalous diusion models in formof fractional diusion equation involving the regularized general fractionalderivative (RGFD) with Sonin kernels. The MSD of the diusing particles governed by this fractional diusion equation is expressed in terms ofthe Sonin kernel associated with the kernel of the RGFD. Variation of theSonin kernels in the fractional diusion equation leads to a great diversityof possible forms of the MSD that can be tted to the measurements datacollected for a concrete anomalous diusion process. In this paper, we treata special case of the fractional diusion equation involving the RGFD withSonin kernel associated with the kernel in form of the power law temperedby an exponential function that will be referred to as the tempered fractionalderivative with Sonin kernel. Even if the kernel of this RGFD belongs to theclass of the Sonin kernels, the theory developed in the previous publicationsdoes not apply to this case because some assumptions needed for validityof this theory are not satised. However, we could show that the fractionaldiusion equation with the tempered time-fractional derivative with Soninkernel can serve as an anomalous diusion model. In particular, its fundamental solution can be interpreted as a spatial probability density functionevolving in time. One of the most surprising characteristics of this model isa transition of the anomalous diusion with the power law MSD for shorttimes to a stationary process with a constant MSD for long times.

In this paper, we introduce a general fractional master equation involving regularized general fractional derivatives with Sonin kernels, and we discuss its physical characteristics and mathematical properties. First, we show that this master equation can be embedded into the framework of continuous time random walks, and we derive an explicit formula for the waiting time probability density function of the continuous time random walk model in form of a convolution series generated by the Sonin kernel associated with the kernel of the regularized general fractional derivative. Next, we derive a fractional diffusion equation involving regularized general fractional derivatives with Sonin kernels from the continuous time random walk model in the asymptotical sense of long times and large distances. Another important result presented in this paper is a concise formula for the mean squared displacement of the particles governed by this fractional diffusion equation. Finally, we discuss several mathematical aspects of the fractional diffusion equation involving regularized general fractional derivatives with Sonin kernels, including the non-negativity of its fundamental solution and the validity of an appropriately formulated maximum principle for its solutions on the bounded domains.

In this paper, a new class of kernels of integral transforms of the Laplace convolution type that we named symmetrical Sonin kernels is introduced and investigated. For a symmetrical Sonin kernel given in terms of elementary or special functions, its associated kernel has the same form with possibly different parameter values. In the paper, several new kernels of this type are derived by means of the Sonin method in the time domain and using the Laplace integral transform in the frequency domain. Moreover, for the first time in the literature, a class of Sonin kernels in terms of the convolution series, which are a far-reaching generalization of the power series, is constructed. The new symmetrical Sonin kernels derived in the paper are represented in terms of the Wright function and the new special functions of the hypergeometric type that are extensions of the corresponding Horn functions in two variables.

In this paper, we deal with analysis of the initial-boundary value problems for the semilinear time-fractional diffusion equations, while the case of the linear equations was considered in the first part of the present work. These equations contain uniformly elliptic spatial differential operators of the second order and the Caputo type fractional derivative acting in the fractional Sobolev spaces as well as a semilinear term that depends on the spatial variable, the unknown function and its gradient. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. For these problems, we first prove uniqueness and existence of their solutions. Under some suitable conditions, we then show the non-negativity of the solutions and derive several comparison principles. We also apply the monotonicity method by upper and lower solutions to deduce some a priori estimates for solutions to the initial-boundary value problems for the semilinear time-fractional diffusion equations. Finally, we consider some initial-boundary value problems for systems of the linear and semilinear time-fractional diffusion equations and prove non-negativity of their solutions under the suitable conditions.

In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its domain, null-space, and projector operator; establish the interrelations between its different realizations; and present a generalized fractional Taylor formula involving the generic fractional derivative. Then, we consider the fractional relaxation equation containing the generic fractional derivative, derive a closed-form formula for its unique solution, and study its complete monotonicity.