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13 pages, 299 KiB

Open AccessArticle

Fractional Integrals Associated with the One-Dimensional Dunkl Operator in Generalized Lizorkin Space

by Fethi Bouzeffour

Symmetry 2023, 15(9), 1725; https://doi.org/10.3390/sym15091725 - 8 Sep 2023

Viewed by 1389

This paper explores the realm of fractional integral calculus in connection with the one-dimensional Dunkl operator on the space of tempered functions and Lizorkin type space. The primary objective is to construct fractional integral operators within this framework. By establishing the analogous counterparts of well-known operators, including the Riesz fractional integral, Feller fractional integral, and Riemann–Liouville fractional integral operators, we demonstrate their applicability in this setting. Moreover, we show that familiar properties of fractional integrals can be derived from the obtained results, further reinforcing their significance. This investigation sheds light on the utilization of Dunkl operators in fractional calculus and provides valuable insights into the connections between different types of fractional integrals. The findings presented in this paper contribute to the broader field of fractional calculus and advance our understanding of the study of Dunkl operators in this context. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)

24 pages, 422 KiB

Open AccessArticle

On the Quantitative Properties of Some Market Models Involving Fractional Derivatives

by Jean-Philippe Aguilar, Jan Korbel and Nicolas Pesci

Mathematics 2021, 9(24), 3198; https://doi.org/10.3390/math9243198 - 11 Dec 2021

Cited by 3 | Viewed by 2733

Abstract

We review and discuss the properties of various models that are used to describe the behavior of stock returns and are related in a way or another to fractional pseudo-differential operators in the space variable; we compare their main features and discuss what behaviors they are able to capture. Then, we extend the discussion by showing how the pricing of contingent claims can be integrated into the framework of a model featuring a fractional derivative in both time and space, recall some recently obtained formulas in this context, and derive new ones for some commonly traded instruments and a model involving a Riesz temporal derivative and a particular case of Riesz–Feller space derivative. Finally, we provide formulas for implied volatility and first- and second-order market sensitivities in this model, discuss hedging and profit and loss policies, and compare with other fractional (Caputo) or non-fractional models. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False” II)

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21 pages, 1181 KiB

Open AccessArticle

The Approximate and Analytic Solutions of the Time-Fractional Intermediate Diffusion Wave Equation Associated with the Fokker–Planck Operator and Applications

by Entsar A. Abdel-Rehim

Axioms 2021, 10(3), 230; https://doi.org/10.3390/axioms10030230 - 17 Sep 2021

Cited by 5 | Viewed by 2406

Abstract

In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the [...] Read more.

G_{3} (t)

of the three-term time-fractional wave equation with constant coefficients is also studied for two physical and biological models. The explicit analytic solutions, for the two studied models, are expressed in terms of the Weber, hypergeometric, exponential, and Mittag–Leffler functions. The relation to the diffusion equation is given. The asymptotic behaviors of the Mittag–Leffler function, the hypergeometric function

1 F 1

, and the exponential functions are compared numerically. The Grünwald–Letnikov scheme is used to derive the approximate difference schemes of the Caputo time-fractional operator and the Feller–Riesz space-fractional operator. The explicit difference scheme is numerically studied, and the simulations of the approximate solutions are plotted for different values of the fractional orders. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

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19 pages, 6132 KiB

Open AccessArticle

Fractional Fokker-Planck Equation

by Gerd Baumann and Frank Stenger

Mathematics 2017, 5(1), 12; https://doi.org/10.3390/math5010012 - 11 Feb 2017

Cited by 14 | Viewed by 7106

Abstract

We shall discuss the numerical solution of the Cauchy problem for the fully fractional Fokker-Planck (fFP) equation in connection with Sinc convolution methods. The numerical approximation is based on Caputo and Riesz-Feller fractional derivatives. The use of the transfer function in Laplace and Fourier spaces in connection with Sinc convolutions allow to find exponentially converging computing schemes. Examples using different initial conditions demonstrate the effective computations with a small number of grid points on an infinite spatial domain. Full article

(This article belongs to the Special Issue Fractional Differential and Difference Equations)

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22 pages, 416 KiB

Open AccessArticle

Non-Local Meta-Conformal Invariance, Diffusion-Limited Erosion and the XXZ Chain

by Malte Henkel

Symmetry 2017, 9(1), 2; https://doi.org/10.3390/sym9010002 - 24 Dec 2016

Cited by 7 | Viewed by 4954

Abstract

Diffusion-limited erosion is a distinct universality class of fluctuating interfaces. Although its dynamical exponent

z = 1

, none of the known variants of conformal invariance can act as its dynamical symmetry. In

d = 1

spatial dimensions, its infinite-dimensional dynamic symmetry is [...] Read more.

Diffusion-limited erosion is a distinct universality class of fluctuating interfaces. Although its dynamical exponent

z = 1

, none of the known variants of conformal invariance can act as its dynamical symmetry. In

d = 1

spatial dimensions, its infinite-dimensional dynamic symmetry is constructed and shown to be isomorphic to the direct sum of three loop-Virasoro algebras. The infinitesimal generators are spatially non-local and use the Riesz-Feller fractional derivative. Co-variant two-time response functions are derived and reproduce the exact solution of diffusion-limited erosion. The relationship with the terrace-step-kind model of vicinal surfaces and the integrable XXZ chain are discussed. Full article

(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models)

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14 pages, 228 KiB

Open AccessArticle

Computational Solutions of Distributed Order Reaction-Diffusion Systems Associated with Riemann-Liouville Derivatives

by Ram K. Saxena, Arak M. Mathai and Hans J. Haubold

Axioms 2015, 4(2), 120-133; https://doi.org/10.3390/axioms4020120 - 2 Apr 2015

Cited by 9 | Viewed by 4478

Abstract

This article is in continuation of the authors research attempts to derive computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative as space derivative. This article presents computational solutions of distributed order fractional reaction-diffusion equations associated with Riemann-Liouville derivatives of fractional orders as the time-derivatives and Riesz-Feller fractional derivatives as the space derivatives. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computational form in terms of the familiar Mittag-Leffler function. It provides an elegant extension of results available in the literature. The results obtained are presented in the form of two theorems. Some results associated specifically with fractional Riesz derivatives are also derived as special cases of the most general result. It will be seen that in case of distributed order fractional reaction-diffusion, the solution comes in a compact and closed form in terms of a generalization of the Kampé de Fériet hypergeometric series in two variables. The convergence of the double series occurring in the solution is also given. Full article

15 pages, 249 KiB

Open AccessArticle

Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative

by Ram K. Saxena, Arak M. Mathai and Hans J. Haubold

Axioms 2014, 3(3), 320-334; https://doi.org/10.3390/axioms3030320 - 4 Aug 2014

Cited by 17 | Viewed by 5712

Abstract

This paper deals with the investigation of the computational solutions of a unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann–Liouville fractional derivative defined by others and the space derivative of second order by the Riesz–Feller fractional derivative and adding a function ɸ(x, t). The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag–Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained by others and the result very recently given by others. At the end, extensions of the derived results, associated with a finite number of Riesz–Feller space fractional derivatives, are also investigated. Full article

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