Fractional Cauchy problems on bounded domains

By Mark M. Meerschaert, Erkan Nane, Palaniappan Vellaisamy. 2008

223 citations recorded by the source corpus.

Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain DR d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordi- nator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brow- nian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time. 1. Introduction. In this paper, we extend the approach of Meerschaert and Scheffler ( 23) and Meerschaert et al. (24) to fractional Cauchy problems on bounded domains. Our methods involve eigenfunction expansions, killed Markov processes and inverse stable subordinators. In a recent related paper (7), we establish a connection between fractional Cauchy problems with index β = 1/2 on an unbounded domain, and iterated Brownian motion (IBM), defined as Zt = B(|Yt|), where B is a Brownian motion with values in R d and Y is an independent one-dimensional Brownian motion. Since IBM is also the stochastic solution to a Cauchy problem involving a fourth-order derivative in space (2, 14), that paper also establishes a connection between certain higher-order Cauchy problems and their time-fractional analogues. More generally, Baeumer, Meerschaert and Nane (7) shows a connection between fractional Cauchy problems with β = 1/2 and higher-order Cauchy problems that involve the square of the generator. In the present paper, we

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Citations in the corpus

Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation (2011, unknown): “In the paper [28], the stochastic analogues for the fractional Cauchy problems in a bounded domain with the Dirichlet boundary conditions were introduced and investigated.”
Distributed-order fractional diffusions on bounded domains (2009, background): “379 (2011) 216–228 217 suitable technical conditions and assuming Dirichlet boundary conditions, the diffusion equation ∂u/∂t = Lu governs a Markov process X(t) killed at the boundary, and the corresponding fractional diffusion equation ∂u/∂t = Lu governs the time-changed process X(Et) [28].”
The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation (2015, methodology): “In [24], [17], the existence and properties of solutions for a time fractional equation in a bounded domain were considered by applying eigenfunction expansions.”
Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation (2011, background): “to [1], [6], [8], [21], [23], [25] to mention only few of many recent publications.”
Fractional Skellam processes with applications to finance (2014, unknown): “Since the one-parameter Mittag-Leffeler function is the eigenfunction for the Caputo derivative [20, 24], Dα t Eα(−λt) = −λEα(−λt), and equation (3.”
Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives ✩ (2012, background): “[11] developed classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain D ⊂Rd with Dirichlet boundary conditions, stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional time derivat”
Time fractional equations and probabilistic representation (2017, background): “(iv) Cauchy problems with distributed order time fractional derivatives (where κ = 0) were also studied in [16] for uniformly elliptic generators of divergence form in bounded C1,γ domains”
Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations (2009, background): “With regards to Example 3.6, it is possible to discuss SDEs and associated time-fractional PDEs with smooth boundary conditions. Time-fractional PDEs with Dirichlet boundary conditions are treated in [10], but without specifying the connection to SDEs of the form (3.15). 10 4 SDEs Including Terms Dr”
Asymptotic properties of Brownian motion delayed by inverse subordinators (2013, background): “In this case, the operator Φ t is the Riemann-Liouville fractional derivative 0 D 1 − α t (cf. [52]), Then, (5) becomes the fractional diﬀusion equation [2, 35, 36, 43, 50].”
Convolution-Type Derivatives, Hitting-Times of Subordinators and Time-Changed C0-semigroups (2013, unknown): “[21]; Saichev and Zaslavsky [31]; Zaslavsky [38].”