Initial-value problem for a linear ordinary differential equation of noninteger order

Pskhu, A. V.

doi:10.1070/sm2011v202n04abeh004156

Cited by 45 publications

(34 citation statements)

References 3 publications

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“…Pskhu [11], and references in these works). In multidimensional case (x ∈ R N ) instead of the differential expression u xx it has been considered either the Laplace operator [6,22], or the elliptic differential or pseudo-differential operator in the whole space R N with constant coefficients [10]. In both cases the authors investigated the Cauchy type problems applying either the Laplace transform or the Fourier transform.…”

Section: Resultsmentioning

confidence: 99%

Initial-boundary Value Problem for a Time-fractional Subdiffusion Equation with an Arbitrary Elliptic Differential Operator

Ashurov

Muhiddinova

2021

Lobachevskii J Math

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An initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary order elliptic differential operator is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. In the case of an initial-boundary value problem on N -dimensional torus, one can easily see that these conditions are not only sufficient, but also necessary.

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Section: Resultsmentioning

confidence: 99%

Initial-boundary Value Problem for a Time-fractional Subdiffusion Equation with an Arbitrary Elliptic Differential Operator

Ashurov

Muhiddinova

2021

Lobachevskii J Math

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“…In the second approach we stick to the idea of interruption on crossing the boundary, so that D and its complement are treated symmetrically. To present this approach in a proper generality assume a finite set Then the modification of the process on R generated by (30), with jumps interrupted on crossing B (think of B as a set of road blocks or check points placed to control the free motion given by A) can be specified by the generator…”

Section: Preliminaries: Classical Fractional Derivativesmentioning

confidence: 99%

On Fully Mixed and Multidimensional Extensions of the Caputo and Riemann-Liouville Derivatives, Related Markov Processes and Fractional Differential Equations

Kolokoltsov

2015

FCAA

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From the point of view of stochastic analysis the Caputo and Riemann-Liouville derivatives of order α ∈ (0, 2) can be viewed as (regularized) generators of stable Lévy motions interrupted on crossing a boundary. This interpretation naturally suggests fully mixed, two-sided or even multidimensional generalizations of these derivatives, as well as a probabilistic approach to the analysis of the related equations. These extensions are introduced and some well-posedness results are obtained that generalize, simplify and unify lots of known facts. This probabilistic analysis leads one to study a class of Markov processes that can be constructed from any given Markov process in R d by blocking (or interrupting) the jumps that attempt to cross certain closed set of 'check-points'. (2010): 34A08, 35S15, 60J50, 60J75 Mathematics Subject Classification

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“…Explicit representations for solutions of initial and boundary value problems were found in terms of the generalized Mittag-Leffler functions and Wright functions. A detailed exposition of these results and the references can be found in works [14,15]. To avoid technically complicated theory of special functions, in our work we solve equation (9) by reducing it to the second kind integral Volterra equation with a power kernel and solving it by the method of successive approximations.…”

Section: Solution For Homogeneous Aller-lykov Equation With Fractionamentioning

confidence: 99%

Dirichlet boundary value problem for Aller - Lykov moisture transfer equation with fractional derivative in time

Gekkieva¹,

Kerefov²

2019

Ufimskii Matematicheskii Zhurnal

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The heat-moisture transfer in soils is a fundamental base in addressing many problems of hydrology, agrophysics, building physics and other fields of science. The researchers focus on possibility of reflecting specific features of the studied arrays in the equations as well as their structure, physical properties, the processes going on in them, etc. In view of this, there arises a new class of fractional differential equations of state and transport being the base for most mathematical models describing a wide class of physical and chemical processes in media with a fractal structure and memory.This paper studies the Dirichlet boundary value problem for the Aller-Lykov moisture transfer equation with the Riemann-Liouville fractional derivative in time. The considered equation is a generalization of the Aller-Lykov equation obtained by means of introducing the concept of the fractal rate of humidity change, which accounts the presence of flows moving against the moisture potential.The existence of the solution to the Dirichlet boundary value problem is proved by the Fourier method. By means of energy inequalities method, for the solution we obtain an apriori estimate in terms of fractional Riemann-Liouville derivative, which implies the uniqueness of the solution.

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Initial-value problem for a linear ordinary differential equation of noninteger order

Cited by 45 publications

References 3 publications

Initial-boundary Value Problem for a Time-fractional Subdiffusion Equation with an Arbitrary Elliptic Differential Operator

Initial-boundary Value Problem for a Time-fractional Subdiffusion Equation with an Arbitrary Elliptic Differential Operator

On Fully Mixed and Multidimensional Extensions of the Caputo and Riemann-Liouville Derivatives, Related Markov Processes and Fractional Differential Equations

Dirichlet boundary value problem for Aller - Lykov moisture transfer equation with fractional derivative in time

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