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From Sturm-Liouville problems to fractional and anomalous diffusions
TLDR
In this paper, the explicit representation of solutions to fractional and anomalous diffusions related to Sturm-Liouville problems of fractional order associated with fractional power function spaces is discussed.Abstract:
Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those diffusions, the explicit laws of certain stable processes turn out to be fundamental. This paper directs one's efforts towards the explicit representation of solutions to fractional and anomalous diffusions related to Sturm-Liouville problems of fractional order associated to fractional power function spaces. Furthermore, we study a new version of the Bochner's subordination rule and we establish some connections between subordination and space-fractional operatorread more
Citations
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Journal ArticleDOI
Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)
Journal ArticleDOI
Fractional Sturm-Liouville problem
Malgorzata Klimek,Om P. Agrawal +1 more
TL;DR: It is shown that the Legendre Polynomials resulting from an FLE are the same as those obtained from the integer order Legendre equation; however, the eigenvalues of the two equations differ.
Journal ArticleDOI
Convolution-Type Derivatives, Hitting-Times of Subordinators and Time-Changed C 0 -semigroups
TL;DR: In this article, the authors take under consideration subordinators and their inverse processes (hitting-times) and present the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives.
Journal ArticleDOI
Fractional Sturm-Liouville boundary value problems in unbounded domains
TL;DR: It is proved that these fractional Sturm-Liouville operators are self-adjoint and the obtained eigenvalues are all real, the corresponding eigenfunctions are orthogonal with respect to the weight function associated to F SLOs-1 and FSLOs-2 and form two sets of non-polynomial bases.
Journal ArticleDOI
A fractional approach to the Sturm-Liouville problem
TL;DR: In this paper, an approach to the fractional version of the Sturm-Liouville problem, by using different fractional operators that return to the ordinary operator for integer order, is presented.
References
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Book
Theory and Applications of Fractional Differential Equations
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
A table of integrals
TL;DR: Basic Forms x n dx = 1 n + 1 x n+1 (1) 1 x dx = ln |x| (2) udv = uv − vdu (3) 1 ax + bdx = 1 a ln|ax + b| (4) Integrals of Rational Functions
Journal ArticleDOI
The random walk's guide to anomalous diffusion: a fractional dynamics approach
Ralf Metzler,Joseph Klafter +1 more
TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.
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Fractional Integrals and Derivatives: Theory and Applications
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.