# Arsen PskhuInstitute of Applied Mathematics and Automation, Russia, Nalchik · Department of Fractional Calculus

Arsen Pskhu

Doctor of Sciences (Mathematics)

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54

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Introduction

**Skills and Expertise**

## Publications

Publications (54)

We construct transmutations that intertwine the operator of fractional differentiation of distributed order and the first-order derivative. Fractional differentiation is considered in two forms, in the sense of Riemann-Liouville and in the sense of Gerasimov-Caputo. For each of them, a corresponding transmutation operator is constructed. Distribute...

In this paper we consider a boundary value problem for a loaded fractional diffusion equation. The loaded term has the form of the Riemann-Liouville fractional derivative or integral. The BVP is considered in the open right upper quadrant. The problem is reduced to an integral equation that, in some cases, belongs to the pseudo-Volterra type, and i...

The spectral energy density of the radiation of an oscillator, the motion of which is described by an equation with fractional integro-differentiation, is calculated in the dipole approximation. The fractional oscillator model can describe various types of radiation, including those with a nonexponential relaxation law. The shape of the spectral li...

A new model of damped vibrations of an oscillator based on the use of the mathematical apparatusof fractional integro-differentiation with an exponential-power function of dynamic memory is considered.Using the Wright function, the exact solution of the equation of motion of the oscillator is obtained. Theo-retical analysis shows that the model can...

A boundary value problem for a fractionally loaded heat equation is considered in the first quadrant. The loaded term has the form of the Riemann-Liouville’s fractional derivative with respect to the time variable, and the order of the derivative in the loaded term is less than the order of the differential part. The study is based on reducing the...

We consider a boundary value problem for the fractional diffusion equation in an angle domain with a curvilinear boundary. Existence and uniqueness theorems for solutions are proved. It is shown that Holder continuity of the curvilinear boundary ensures the existence of solutions. The uniqueness is proved in the class of functions that vanish at in...

In this paper, the Green functions of the first boundary-value problem for the fractional diffusion-wave equation in multidimensional (bounded and unbounded) hyper-rectangular domains are constructed.

A face shield is a type of the personal protective equipment that is used to protect a person’s face from various external hazardous influences—splashes, drops and aerosols, thermal and optical radiation. It is believed that face shields effectively prevent contamination of the mucous membranes of the human body. According to the World Health Organ...

The thermal properties of C60 and C70 fullerenes and fullerites have been studied by X-ray structural analysis and UV spectroscopy. It was shown that C70 fullerite is stabler than C60 fullerite (by ~150°C) due to the different geometry of the fullerenes and stronger intermolecular interactions. The experimental results are given a qualitative and q...

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 Riem...

We consider a model of damped vibrations of thin film membrane MEMS based on fractional differentiation. The theoretical model is based on the application of the effective velocity concept and the “fractional” principle of least action. The effective mass of the MEMS membrane was calculated. The MEMS oscillation equation is derived and its numerica...

Методами рентгеноструктурного анализа и УФ-спектроскопии исследованы термические свойства фуллеренов и фуллеритов С60 и С70. Показано, что фуллерит С70 более стабилен по сравнению с С60 (на ~150°С), что обусловлено различной геометрией фуллеренов и более сильным межмолекулярным взаимодействием. Полученным экспериментальным результатам дана качестве...

We consider a diffusion–wave equation with fractional derivative with respect to the time variable, defined on infinite interval, and with the starting point at minus infinity. For this equation, we solve an asympotic boundary value problem without initial conditions, construct a representation of its solution, find out sufficient conditions provid...

We consider the principle of least action in the context of fractional calculus. Namely, we derive the fractional Euler–Lagrange equation and the general equation of motion with the composition of the left and right fractional derivatives defined on infinite intervals. In addition, we construct an explicit representation of solutions to a model fra...

Problems on the asymptotic behavior of solutions to the Cauchy problems for the fractional-diffusion-wave equation for large values of time are examined. Sufficient conditions of stabilization in the class of rapidly growing functions and necessary and sufficient conditions of stabilization to zero in the case of asymptotically nonnegative initial...

We study a boundary value problem for a fractional differential equation modeling the damped vibrations of thin film MEMS with variable potential. The principal differential part of the equation under consideration is the composition of left- and right-sided Caputo derivatives. We find sufficient conditions for the potential which guarantee the uni...

A general expression is obtained for retarded potentials for a system of equations of macroscopic electrodynamics with fractional Caputo derivatives with respect to time. An analog of the Lienard-Wiechert potentials is obtained. All expressions contain a nonlocal (time-distributed) delay, which takes the temporal dispersion in the system into accou...

In this paper, we construct a transmutation operator for fractional multi-term differential operators. The constructed operator intertwines multi-term differential operators and the operator of first order differentiation, and allows us to find explicit representations of solutions for initial and boundary value problems for fractional multi-term e...

We construct the Green function of the first boundary-value problem for a diffusion-wave equation with fractional derivative with respect to the time variable. The Green function is sought in terms of a double-layer potential of the equation under consideration. We prove a jump relation and solve an integral equation for an unknown density. Using t...

We propose a method for describing damped vibrations of a beam with a built-in end considering the dynamic hysteresis that determines mechanical energy dissipation due to viscoelasticity. As the mathematical basis, we have used the fractional integro-differentiation apparatus. Rapidly damped vibrations of a foamed polypropylene beam have been studi...

We consider a model of damped vibrations based on fractional differentiation. The model given is completelyconsistent with the classical model of vibration with viscous damping. We find the relation between the order of fractionaldifferentiation in the equation of motion and Q-factor of an oscillator. The proposed approach seems more appropriatefor...

In this article, we construct a fundamental solution of a higher-order
equation with time-fractional derivative, give a representation for
a solution of the Cauchy problem, and prove the uniqueness theorem
in the class of functions satisfying an analogue of Tychonoff's condition.

В работе построены функции Грина первой краевой задачи для дробного диффузионно-волнового уравнения в многомерных (ограниченных и неограниченных) гиперпрямоугольных областях.

A model of forced oscillations of an oscillator based on the fractional integro-differential formalism has been considered. It is shown that this model is in good agreement with the classical model of forced oscillations of an oscillator with viscous damping. The parameters of the frequency dependence of stationary oscillations of a fractional osci...

We consider the Cauchy problem for a third-order partial differential equation with time-fractional derivative, and prove a uniqueness theorem for the problem in the class of fast-growing functions satisfying an analogue of the Tychonoff condition.

We solve the first boundary-value problem in a non-cylindrical domain for a diffusion-wave equation with the Dzhrbashyan–Nersesyan operator of fractional differentiation with respect to the time variable. We prove an existence and uniqueness theorem for this problem, and construct a representation of the solution. We show that a sufficient conditio...

We construct an explicit representation of the solution of a multidimensional Abel integral equation of the second kind with partial fractional integrals in terms of the Wright function.

We solve the first boundary-value problem in a non-cylindrical
domain for a diffusion-wave equation with the Dzhrbashyan–Nersesyan
operator of fractional differentiation with respect to the time variable.
We prove an existence and uniqueness theorem for this problem, and construct a representation of the solution. We show that a sufficient conditio...

We solve a boundary value problem for a first-order partial differential equation in a rectangular domain with a fractional discretely distributed differentiation operator. The fractional differentiation is given by Dzhrbashyan–Nersesyan operators. We construct a representation of the solution and prove existence and uniqueness theorems. The result...

We discuss an initial value problem for a fractional diffusion equation with discretely distributed fractional differentiation operator with respect to time variable. We construct a fundamental solution of the considered equation, give a solution of the problem under study, and prove a uniqueness theorem in the class of rapid growth functions. The...

We study a boundary value problem for a fractional partial differential equation of order ≤ 1 in a domain with curvilinear boundary.

We study the continuation of solutions of a fractional partial differential equation. We show that a solution can be uniquely continued into a domain that is uniquely determined by the boundary part supporting the initial conditions and outside which the continuation is no longer unique.

We construct a fundamental solution of a multi-time diffusion equation
with the Dzhrbashyan-Nersesyan fractional differentiation operator with
respect to the time variables. We give a representation for a solution
of the Cauchy problem and prove the uniqueness theorem in the class of
functions of fast growth. The corresponding results for equations...

An explicit representation for solution of the generalized Abel integral
equation , where is the Riemann-Liouville fractional integral, in terms of the Wright function, is
constructed.

An initial-value problem for a linear ordinary differential equation of noninteger order with Riemann-Liouville derivatives is stated and solved. The initial conditions of the problem ensure that (by contrast with the Cauchy problem) it is uniquely solvable for an arbitrary set of parameters specifying the orders of the derivatives involved in the...

We construct a fundamental solution of a linear fractional partial differential equation. For an equation with Dzhrbashyan-Nersesyan
fractional differentiation operators, we solve a boundary value problem and find a closed-form representation for its solution.
The corresponding results for equations with Riemann-Liouville and Caputo derivatives are...

An initial-value problem for a linear ordinary differential equation of noninteger order with Riemann-Liouville derivatives is stated and solved. The initial conditions of the problem ensure that (by contrast with the Cauchy problem) it is uniquely solvable for an arbitrary set of parameters specifying the orders of the derivatives involved in the...

We construct a fundamental solution of a diffusion-wave equation with Dzhrbashyan-Nersesyan fractional differentiation operator with respect to the time variable. We prove reduction formulae and solve the problem of sign-determinacy for the fundamental solution. A general representation for solutions is constructed. We give a solution of the Cauchy...

We construct a fundamental solution of a diffusion-wave equation with Dzhrbashyan–Nersesyan fractional differentiation operator with respect to the time variable. We prove reduction formulae and solve the problem of sign-determinacy for the fundamental solution. A general representation for solutions is constructed. We give a solution of the Cauchy...

In the present paper, we prove an assertion allowing us to extend results related to the presence or absence of real zeros of functions of Mittag-Leffler type \(E_{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \left( {z;\mu } \right) = \sum\limits_{k = 0}^\infty {\frac{{z^k }}{{\Gamma \left( {\alpha k + \mu...

In the present paper, we prove an assertion allowing us to extend results related to the presence or absence of real zeros of the Mittag-Leffler function for certain values of α and μ to more extensive ranges of these parameters. We give a geometric description of the sets of pairs (α,μ) for which this function has and does not have real zeros.

In the paper, we construct the inverse operators for a class of integro-differentiation operators of segment order, find solutions to the elementary integral and differential equations with such operators, and derive analogs of the Newton-Leibniz formulas for these operators.

In the present paper, we use the Green function method to obtain a general representation of solutions of the fractional diffusion equation and construct Green functions of the first, second, and mixed boundary value problems.

In the rectangular domain D = {(x,y): 0 <x <a,0 <y <b}, we consider the equation D_0x^α u(x,y) + λD_0y^β u(x,y) = f (x,y), where 0 < α,β≤1, αβ <1, D_0x^α, D_0y^β are partial fractional derivatives in the Riemann – Liouville sense of order α and β with respect to x and y, respectively. For this equation, a boundary value problem is formulated and so...

The Frankl problem for the Lavrentiev-Bitsadze equation is considered. The existence and uniqueness of a solution to this problem is proved under fairly general assumptions regarding the boundary of the elliptic part of the mixed domain.