STOCHASTIC SOLUTIONS FOR FRACTIONAL CAUCHY PROBLEMS

By Boris Baeumer, Mark M. Meerschaert. 2003

229 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/a3697bb26bd6ccf792346130f669b221515e508a

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Citations in the corpus

Convolution-Type Derivatives, Hitting-Times of Subordinators and Time-Changed C0-semigroups (2013, background): “Equations of fractional order appear in a lot of physical phenomena (Meerschaert and Sikorskii [24]) and in particular for modeling anomalous diffusions (see for example Benson et al. [3]; D’Ovidio [9]).”
Continuous-time random walk and parametric subordination in fractional diffusion (2007, unknown): “1) and its variants has been investigated by several authors as pointed out in the bibliography in [27]: here we refer to some of them, [1,3,33,35], where the connection with the CTRW was also pointed out.”
Fractal mobile/immobile solute transport (2003, methodology): “…Þ ¼ sg ~C sð Þ C0 xð Þsg 1; and L H t uð Þgg t u buð Þ1=g ! buð Þ 1=g ( ) ¼ e sþbsgð Þu; where H ( ) is the Heaviside function and gg(t) is a stable density with scaling parameter 0 < g < 1 (known as a stable subordinator) so that L gg tð Þ ¼ e sg [Hille and Phillips, 1957; Baeumer et al., 2001].”
Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications (2009, methodology): “Therefore, the non-Fickian models based on the CTRW with iid jumps and waiting times, including the classical CTRW [18] and the fractional-derivative models investigated in this study, may need to be refined by considering correlation of jumps, such as the correlated CTRW developed by Meerschaert et”
Cauchy problem for fractional diffusion equations (2003, background): “There are also several papers devoted to the Cauchy problem for abstract evolution equations (1.1), in which B is a closed operator on a Banach space ([12, 8, 3, 4, 2], and others).”
Limit theorems for continuous-time random walks with infinite mean waiting times (2004, unknown): “Keywords: Operator self-similar process; continuous-time random walk 2000 Mathematics Subject Classification: Primary 60G50 Secondary 60K40”
Finite difference methods for two-dimensional fractional dispersion equation (2006, background): “Fractional derivatives are nonlocal operators of convolution type [1,9,24].”
On fractional powers of generators of fractional resolvent families (2010, background): “Moreover, Baeumer, Meerschaert and Nane [6] proved that Eq.”
Time fractional equations and probabilistic representation (2017, methodology): “The author thanks M. M. Meerschaert for the invitation to the Workshop “Future Directions in Fractional Calculus Research and Applications” held at Michigan State University, East Lansing, from October 17-21, 2016, and for helpful comments.”
Triangular array limits for continuous time random walks (2008, background): “If A(t) is a stable process with index α, then the governing equation employs fractional space derivatives of order α; see for example [3,11,20,35,36].”
Stochastic model for ultraslow diffusion (2006, unknown): “HPS was partially supported by NSF grant DMS-0417869.”
Distributed-order fractional diffusions on bounded domains (2009, background): “Meerschaert and Scheffler [33] discuss generalized Cauchy problems of the form ψW (∂/∂t)u(t, x) = Lu(t, x)+δ(x)ψW (t,∞), where ψW (s) is the Laplace exponent of a nondecreasing Lévy process (subordinator) whose Lévy measure φW has infinite total mass, and L is the generator of another Lévy proces”
A second-order accurate numerical method for the two-dimensional fractional diffusion equation (2007, background): “Fractional derivatives are non-local operators of convolution type [1,6,20].”
Brownian subordinators and fractional Cauchy problems (2007, methodology): “For example, in [10] the semigroup associated with a L´evy process X 0 ( t ) started at x = 0 is deﬁned by One physical interpretation of this formula, when f ( x ) is the probability density of a random variable W , is that W represents the location of a randomly selected particle at time t = 0,…”