arXiv subjects

Yuri Luchko

Publications and source records attributed to Yuri Luchko.

At least 19 recordsLinked to original sources

arXiv2024

Comparison principles for the time-fractional diffusion equations with the Robin boundary conditions. Part II: Semilinear equations

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.

math.AP↗

arXiv2024

Operational Calculus for the 1st Level General Fractional Derivatives and its Applications

The 1st level General Fractional Derivatives (GFDs) combine in one definition the GFDs of the Riemann-Liouville type and the regularized GFDs (or the GFDs of the Caputo type) that have been recently introduced and actively studied in the Fractional Calculus literature. In this paper, we first construct an operational calculus of Mikusiński type for the 1st level GFDs. In particular, it includes the operational calculi for the GFDs of the Riemann-Liouville type and for the regularized GFDs as its particular cases. In the second part of the paper, this calculus is applied for derivation of the closed form solution formulas to the initial-value problems for the linear fractional differential equations with the 1st level GFDs.

math.AP↗

arXiv2023

Symmetrical Sonin kernels in terms of the hypergeometric functions

In this paper, we introduce a new class of the kernels of the integral transforms of the Laplace convolution type that we call symmetrical Sonin kernels. For a symmetrical Sonin kernel given in terms of some elementary or special functions, its associated kernel has the same form with possibly different parameter values. Several known and 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. The new symmetrical Sonin kernels are provided in terms of the Wright function and some extensions of the Horn confluent hypergeometric functions in two variables.

math.CA↗

arXiv2023

Comparison principles for the time-fractional diffusion equations with the Robin boundary conditions. Part I: Linear equations

The main objective of this paper is analysis of the initial-boundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type time-fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we discuss the uniqueness and existence of solutions to these initial-boundary value problems. Under some suitable conditions on the problem data, we then prove positivity of the solutions. Based on these results, several comparison principles for the solutions to the initial-boundary value problems for the linear time-fractional diffusion equations are derived.

math.AP↗

arXiv2022

Comparison principles for the linear and semiliniar time-fractional diffusion equations with the Robin boundary condition

The main objective of this paper is analysis of the initial-boundary value problems for the linear and semilinear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we deal with the uniqueness and existence of the solutions to these initial-boundary value problems. Then we show a positivity property for the solutions and derive the corresponding comparison principles. In the case of the semilinear time-fractional diffusion equation, we also apply the monotonicity method by upper and lower solutions. As an application of our results, we present some a priori estimates for solutions to the semilinear time-fractional diffusion equations.

math.AP↗

arXiv2022

The 1st Level General Fractional Derivatives and some of their Properties

In this paper, we first provide a short summary of the main properties of the so-called general fractional derivatives with the Sonin kernels introduced so far. These are integro-differential operators defined as compositions of the first order derivative and an integral operator of convolution type. Depending on succession of these operators, the general fractional derivatives of the Riemann-Liouville and of the Caputo types were defined and studied. The main objective of this paper is a construction of the 1st level general fractional derivatives that comprise both the general fractional derivative of the Riemann-Liouville type and the general fractional derivative of the Caputo type. We also provide some of their properties including the 1st and the 2nd fundamental theorems of Fractional Calculus for these derivatives and the suitably defined general fractional integrals.

math.CA↗

arXiv2022

Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann-Liouville Sense

In this paper, we first deal with the general fractional derivatives of arbitrary order defined in the Riemann-Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of Fractional Calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann-Liouville sense. In the second part of the paper, we develop an operational calculus of the Mikusiński type for the general fractional derivatives of arbitrary order in the Riemann-Liouville sense and apply it for derivation of an explicit form of solutions to the Cauchy problems for the single- and multi-term linear fractional differential equations with these derivatives. The solutions are provided in form of the convolution series generated by the kernels of the corresponding general fractional integrals.

math.CA↗

arXiv2021

Special Functions of Fractional Calculus in Form of Convolution Series and their Applications

In this paper, we first discuss the convolution series that are generated by the Sonine kernels from a class of functions continuous on the real positive semi-axis that have an integrable singularity of power function type at the point zero. These convolution series are closely related to the general fractional integrals and derivatives with the Sonine kernels and represent a new class of the special functions of Fractional Calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel $κ(t) = t^{α-1}/Γ(α),\ 0<α<1$. The main result of the paper is derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that were not yet considered in the Fractional Calculus publications.

math.CA↗

arXiv2021

Trends, Directions for Further Research, and Some Open Problems of Fractional Calculus

The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research.

math.CA↗

arXiv2021

Convolution series and the generalized convolution Taylor formula

In this paper, we deal with the convolution series that are a far reaching generalization of the conventional power series and the power series with the fractional exponents including the Mittag-Leffler type functions. Special attention is given to the most interesting case of the convolution series generated by the Sonine kernels. In this paper, we first formulate and prove the second fundamental theorem for the general fractional integrals and the $n$-fold general sequential fractional derivatives of both the Riemann-Liouville and the Caputo types. These results are then employed for derivation of two different forms of a generalized convolution Taylor formula for representation of a function as a convolution polynomial with a remainder in form of a composition of the $n$-fold general fractional integral and the $n$-fold general sequential fractional derivative of the Riemann-Liouville and the Caputo types, respectively. We also discuss the generalized Taylor series in form of convolution series and deduce the formulas for its coefficients in terms of the $n$-fold general sequential fractional derivatives.

math.CA↗

arXiv2021

Multidimensional scaling and visualization of patterns in distribution of nontrivial zeros of the zeta-function

In this paper, we analyze the nontrivial zeros of the Riemann zeta-function using the multidimensional scaling (MDS) algorithm and computational visualization features. The nontrivial zeros of the Riemann zeta-function as well as the vectors with several neighboring zeros are interpreted as the basic elements (points or objects) of a data set. Then we employ a variety of different metrics, such as the Euclidean and Lorentzian ones, to calculate the distances between the objects. The set of the calculated distances is then processed by the MDS algorithm that produces the loci, organized according to the objects features. Then they are analyzed from the perspective of the emerging patterns. Surprisingly, in the case of the Lorentzian metric, this procedure leads to the very clear periodical structures both in the case of the objects in form of the single nontrivial zeros of the Riemann zeta-function and in the case of the vectors with a given number of neighboring zeros. The other tested metrics do not produce such periodical structures, but rather chaotic ones. In this paper, we restrict ourselves to numerical experiments and the visualization of the produced results. An analytical explanation of the obtained periodical structures is an open problem worth for investigation by the experts in the analytical number theory.

math.NT↗

arXiv2021

General fractional integrals and derivatives of arbitrary order

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives with these kernels possess the integrable singularities at the point zero, the kernels of the general fractional integrals can be - depending on their order - both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved.

math.CA↗

arXiv2021

Operational Calculus for the general fractional derivatives with the Sonine kernels

In this paper, we first address the general fractional integrals and derivatives with the Sonine kernels that possess the integrable singularities of power function type at the point zero. Both particular cases and compositions of these operators are discussed. Then we proceed with a construction of an operational calculus of the Mikusiński type for the general fractional derivatives with the Sonine kernels. This operational calculus is applied for analytical treatment of some initial value problems for the fractional differential equations with the general fractional derivatives. The solutions are expressed in form of the convolution series that generalize the power series for the exponential and the Mittag-Leffler functions.

math.CA↗

arXiv2021

On the maximum principle for the multi-term fractional transport equation

In this paper, we prove a maximum principle for the general multi-term space-time-fractional transport equation and apply it for establishing uniqueness of solution to an initial-boundary-value problem for this equation. We also derive some comparison principles for solutions to the initial-boundary-value problems with different problem data. Finally, we present a maximum principle for the Cauchy problem for a time-fractional transport equation on an unbounded domain.

math.AP↗

arXiv2021

General Fractional Integrals and Derivatives with the Sonine Kernels

In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the $n$-fold general fractional integrals and derivatives and study their properties.

math.CA↗

arXiv2020

Fractional derivatives and the fundamental theorem of Fractional Calculus

In this paper, we address the one-parameter families of the fractional integrals and derivatives defined on a finite interval. First we remind the reader of the known fact that under some reasonable conditions, there exists precisely one unique family of the fractional integrals, namely, the well-known Riemann-Liouville fractional integrals. As to the fractional derivatives, their natural definition follows from the fundamental theorem of the Fractional Calculus, i.e., they are introduced as the left-inverse operators to the Riemann-Liouville fractional integrals. Until now, three families of such derivatives were suggested in the literature: the Riemann-Liouville fractional derivatives, the Caputo fractional derivatives, and the Hilfer fractional derivatives. We clarify the interconnections between these derivatives on different spaces of functions and provide some of their properties including the formulas for their projectors and the Laplace transforms. However, it turns out that there exist infinitely many other families of the fractional derivatives that are the left-inverse operators to the Riemann-Liouville fractional integrals. In this paper, we focus on an important class of these fractional derivatives and discuss some of their properties.

math.CA↗

arXiv2020

On complete monotonicity of solution to the fractional relaxation equation with the $n$th level fractional derivative

In this paper, we first deduce the explicit formulas for the projector of the $n$th level fractional derivative and for its Laplace transform. Then the fractional relaxation equation with the $n$th level fractional derivative is discussed. It turns out that under some conditions, the solutions to the initial-value problems for this equation are completely monotone functions that can be represented in form of the linear combinations of the Mittag-Leffler functions with some power law weights. Special attention is given to the case of the relaxation equation with the 2nd level derivative.

math.CA↗

arXiv2018

Subordination principles for the multi-dimensional space-time-fractional diffusion-wave equation

This paper is devoted to an in deep investigation of the first fundamental solution to the linear multi-dimensional space-time-fractional diffusion-wave equation. This equation is obtained from the diffusion equation by replacing the first order time-deri\-va\-ti\-ve by the Caputo fractional derivative of order $β,\ 0 <β\leq 2$ and the Laplace operator by the fractional Laplacian $(-Δ)^{\fracα2}$ with $0<α\leq 2$. First, a representation of the fundamental solution in form of a Mellin-Barnes integral is deduced by employing the technique of the Mellin integral transform. This representation is then used for establishing of several subordination formulas that connect the fundamental solutions for different values of the fractional derivatives $α$ and $β$. We also discuss some new cases of completely monotone functions and probability density functions that are expressed in terms of the Mittag-Leffler function, the Wright function, and the generalized Wright function.

math.AP↗