Explicit solutions for linear variable–coefficient fractional differential equations with respect to functions

Restrepo, Joel E.; Ruzhansky, Michael; Сураган, Дурвудхан

doi:10.1016/j.amc.2021.126177

Cited by 26 publications

(26 citation statements)

References 32 publications

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“…-Develop the distribution theory for the estimators of H(t); -Investigate changes of the Hölder exponents depending on evolutions of random fields driven by SPDEs on the sphere, see [27,46,50,51];…”

Section: Discussionmentioning

confidence: 99%

On Multifractionality of Spherical Random Fields with Cosmological Applications

Broadbridge¹,

Nanayakkara²,

Olenko³

2021

Preprint

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This paper studies random fields on the unit sphere. Traditionally, isotropic Gaussian random fields are considered as the underlying statistical model of the cosmic microwave background (CMB) data. This paper discusses the generalized multifractional Brownian motion and its pointwise Hölder exponent on the sphere.The multifractional approach is used to investigate the CMB data from the Planck mission. These data consist of CMB radiation measurements at narrow angles of the sky sphere. The obtained results suggest that the estimated Hölder exponents for different CMB regions do change from location to location. Therefore, CMB data are multifractional. Then the developed methodology is used to suggest two approaches for detecting regions with anomalies in cleaned CMB maps.

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Section: Discussionmentioning

confidence: 99%

On Multifractionality of Spherical Random Fields with Cosmological Applications

Broadbridge¹,

Nanayakkara²,

Olenko³

2021

Preprint

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“…Then, more details are given in [7][8][9]. Some propositions were given in the function space is C n [a, b] (see also [10,11]). However, it is not enough or convenient in numerical analysis.…”

Section: Introductionmentioning

confidence: 99%

Continuous time random walk to a general fractional Fokker–Planck equation on fractal media

Yang

et al. 2021

Eur. Phys. J. Spec. Top.

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A general fractional calculus is described using fractional operators with respect to another function, and some often used propositions are presented in this framework. Together with the continuous time random walk (CTRW), a general time-fractional Fokker-Planck equation is derived and the governing equation meets the general fractional derivative. Finally, various new probability density functions are proposed in this paper.

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“…where is a positive real number, g(t) is strictly increasing function on [a, b] and g(a) ≥ 0 . Due to the new features in comparison with the standard fractional derivatives, much attention has been paid to theoretical research and applications, for example, fractional calculus of variations [4], the Laplace transform [5,6], exact solution [7,8] and numerical methods [9]. Motivated by the continuous time random walk understood by means of the standard fractional calculus [10,11], suppose a long-tailed waiting time probability density function [12] which is more general than the power law function Then, one comes across a general time-fractional Fokker-Planck equation with the general fractional integral where T is the temperature, P(x, t) is the probability density function, K is the diffusion coefficient, F(x) represents external force field and k b represents the Boltzmann constant.…”

Section: Introductionmentioning

confidence: 99%

A Note on Function Space and Boundedness of the General Fractional Integral in Continuous Time Random Walk

Fan

2021

J Nonlinear Math Phys

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The general fractional calculus becomes popular in continuous time random walk recently. However, the boundedness condition of the general fractional integral is one of the fundamental problems. It wasn’t given yet. In this short communication, the classical norm space is used, and a general boundedness theorem is presented. Finally, various long–tailed waiting time probability density functions are suggested in continuous time random walk since the general fractional integral is well defined.

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Explicit solutions for linear variable–coefficient fractional differential equations with respect to functions

Cited by 26 publications

References 32 publications

On Multifractionality of Spherical Random Fields with Cosmological Applications

On Multifractionality of Spherical Random Fields with Cosmological Applications

Continuous time random walk to a general fractional Fokker–Planck equation on fractal media

A Note on Function Space and Boundedness of the General Fractional Integral in Continuous Time Random Walk

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