On the Solution of Two-Sided Fractional Ordinary Differential Equations of Caputo Type

Hernández-Hernández, Ma. Elena; Kolokoltsov, Vassili N.

doi:10.1515/fca-2016-0072

Cited by 12 publications

(11 citation statements)

References 30 publications

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“…The present paper is mostly based on the ideas suggested by the author in [41] and further developed in [20], [21], [22], [46] and [43].…”

Section: Additional Bibliographical Commentsmentioning

confidence: 99%

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The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

Kolokoltsov

2019

FCAA

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This paper aims at unifying and clarifying the recent advances in the analysis of the fractional and generalized fractional Partial Differential Equations of Caputo and Riemann-Liouville type arising essentially from the probabilistic point of view. This point of view leads to the path integral representation for the solutions of these equations, which is seen to be stable with respect to the initial data and key parameters and is directly amenable to numeric calculations (Monte-Carlo simulation). In many cases these solutions can be compactly presented via the wide class of operator-valued analytic functions of the Mittag-Leffler type, which are proved to be expressed as the Laplace transforms of the exit times of monotone Markov processes.

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“…The present paper is mostly based on the ideas suggested by the author in [41] and further developed in [20], [21], [22], [46] and [43].…”

Section: Additional Bibliographical Commentsmentioning

confidence: 99%

“…Remark 4.1. It is also possible to work directly with the resolvents of the CD derivatives (see [20], [21] ), but the approach via equation (4.3) seems to be simpler.…”

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confidence: 99%

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

Kolokoltsov

2019

FCAA

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“…Numerical methods for fractional equations are presented in the papers [4,14]. The problem with two-sided fractional derivatives was analyzed in [6]. Generalized fractional equations were considered in the works [7,8,12].…”

Section: Introductionmentioning

confidence: 99%

Abstract McKean-Vlasov and HJB equations, their fractional versions and related forward-backward systems on Riemannian manifolds

Kolokoltsov,

Troeva

2021

Preprint

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We introduce a class of abstract nonlinear fractional pseudo-differential equations in Banach spaces that includes both the Mc-Kean-Vlasov-type equations describing nonlinear Markov processes and the Hamilton-Jacobi-Bellman(HJB)-Isaacs equation of stochastic control and games thus allowing for a unified analysis of these equations. This leads to an effective theory of coupled forward-backward systems (forward McKean-Vlasov evolution and backward HJB-Isaacs evolution) that are central to the modern theory of mean-field games.

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“…Such problems constitute a special class of Euler-Lagrange equations, and facilitate the study of variational principles [32]. In [33], the authors applied a probabilistic approach to study equations involving both left-sided and right-sided generalized operators of Caputo type, and showed a relationship between these equations and two-sided exit problems for certain Levy processes. In [34], the author related the study of fully mixed and multidimensional extensions of the Caputo and Riemann-Liouville derivatives with Markov processes.…”

Section: Introductionmentioning

confidence: 99%

Existence Results for a Nonlocal Coupled System of Differential Equations Involving Mixed Right and Left Fractional Derivatives and Integrals

et al. 2020

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In this paper, we study the existence and uniqueness of solutions for a new kind of nonlocal four-point fractional integro-differential system involving both left Caputo and right Riemann–Liouville fractional derivatives, and Riemann–Liouville type mixed integrals. The Banach and Schaefer fixed point theorems are used to obtain the desired results. An example illustrating the existence and uniqueness result is presented.

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On the Solution of Two-Sided Fractional Ordinary Differential Equations of Caputo Type

Cited by 12 publications

References 30 publications

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

Abstract McKean-Vlasov and HJB equations, their fractional versions and related forward-backward systems on Riemannian manifolds

Existence Results for a Nonlocal Coupled System of Differential Equations Involving Mixed Right and Left Fractional Derivatives and Integrals

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