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25 pages, 1117 KiB

Open AccessArticle

Instantaneously Impulsive Stabilization of Mittag–Leffler Numerical Chua’s Oscillator

by Huizhen Qu, Tianwei Zhang and Jianwen Zhou

Fractal Fract. 2025, 9(6), 332; https://doi.org/10.3390/fractalfract9060332 - 23 May 2025

Viewed by 382

The Euler difference approach has become a prevalent tool in the research of integral order differential equations. Nevertheless, a review of the literature reveals a dearth of studies examining fractional order models using the exponential Euler difference approach. The present study employs an exponential Euler difference approach to examine the properties of nonlocal discrete-time oscillators with Mittag–Leffler kernels and piecewise features, with the aim of providing insights into a continuous-time nonlocal nonlinear system. By employing impulsive equations of variations in constants with different forms in conjunction with the Gronwall inequality, a controller that is capable of instantaneously responding and stabilizing the nonlocal discrete-time oscillator is devised. This controller is realized through an associated algorithm. As a case study, the primary outcome is applied to a problem of impulsive stabilization in nonlocal discrete-time Chua’s oscillator. This article presents a stabilizing algorithm for piecewise nonlocal discrete-time oscillators developed using a novel impulsive approach. Full article

(This article belongs to the Special Issue Fractional-Order Systems: Modeling and Control Applications in Engineering)

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12 pages, 297 KiB

Open AccessArticle

Improved Fractional Differences with Kernels of Delta Mittag–Leffler and Exponential Functions

by Miguel Vivas-Cortez, Pshtiwan Othman Mohammed, Juan L. G. Guirao, Majeed A. Yousif, Ibrahim S. Ibrahim and Nejmeddine Chorfi

Symmetry 2024, 16(12), 1562; https://doi.org/10.3390/sym16121562 - 21 Nov 2024

Cited by 4 | Viewed by 1021

Abstract

Special functions have been widely used in fractional calculus, particularly for addressing the symmetric behavior of the function. This paper provides improved delta Mittag–Leffler and exponential functions to establish new types of fractional difference operators in the setting of Riemann–Liouville and Liouville–Caputo. We give some properties of these discrete functions and use them as the kernel of the new fractional operators. In detail, we propose the construction of the new fractional sums and differences. We also find the Laplace transform of them. Finally, the relationship between the Riemann–Liouville and Liouville–Caputo operators are examined to verify the feasibility and effectiveness of the new fractional operators. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Partial Differential Equations)

17 pages, 538 KiB

Open AccessArticle

Piecewise Business Bubble System under Classical and Nonsingular Kernel of Mittag–Leffler Law

by Chao Zhang and Bo Li

Entropy 2023, 25(3), 459; https://doi.org/10.3390/e25030459 - 6 Mar 2023

Viewed by 1758

Abstract

This study aims to investigate the dynamics of three agents in the emerging business bubble model based on the Mittag–Leffler law pertaining to the piecewise classical derivative and non-singular kernel. By generalizing the business bubble dynamics in terms of fractional operators and the piecewise concept, this study presents a new perspective to the field. The entire set of intervals is partitioned into two piecewise intervals to analyse the classical order and conformable order derivatives of an Atangana–Baleanu operator. The subinterval analysis is critical for removing discontinuities in each sub-partition. The existence and uniqueness of the solution based on a piecewise global derivative are tested for the considered model. The approximate root of the system is determined using the piecewise numerically iterative technique of the Newton polynomial. Under the classical order and non-singular law, the approximate root scheme is applied to the piecewise derivative. The curve representation for the piece-wise globalised system is tested by applying the data for the classical and different conformable orders. This establishes the entire density of each compartment and shows a continuous spectrum instead of discrete dynamics. The concept of this study can also be applied to investigate crossover behaviours or abrupt changes in the dynamics of the values of each market. Full article

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12 pages, 303 KiB

Open AccessArticle

Modified Fractional Difference Operators Defined Using Mittag-Leffler Kernels

by Pshtiwan Othman Mohammed, Hari Mohan Srivastava, Dumitru Baleanu and Khadijah M. Abualnaja

Symmetry 2022, 14(8), 1519; https://doi.org/10.3390/sym14081519 - 25 Jul 2022

Cited by 20 | Viewed by 2144

Abstract

The discrete fractional operators of Riemann–Liouville and Liouville–Caputo are omnipresent due to the singularity of the kernels. Therefore, convexity analysis of discrete fractional differences of these types plays a vital role in maintaining the safe operation of kernels and symmetry of discrete delta and nabla distribution. In their discrete version, the generalized or modified forms of various operators of fractional calculus are becoming increasingly important from the viewpoints of both pure and applied mathematical sciences. In this paper, we present the discrete version of the recently modified fractional calculus operator with the Mittag-Leffler-type kernel. Here, in this article, the expressions of both the discrete nabla derivative and its counterpart nabla integral are obtained. Some applications and illustrative examples are given to support the theoretical results. Full article

(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications III)

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28 pages, 608 KiB

Open AccessArticle

Lévy Interest Rate Models with a Long Memory

by Donatien Hainaut

Risks 2022, 10(1), 2; https://doi.org/10.3390/risks10010002 - 23 Dec 2021

Cited by 2 | Viewed by 3013

Abstract

This article proposes an interest rate model ruled by mean reverting Lévy processes with a sub-exponential memory of their sample path. This feature is achieved by considering an Ornstein–Uhlenbeck process in which the exponential decaying kernel is replaced by a Mittag–Leffler function. Based on a representation in term of an infinite dimensional Markov processes, we present the main characteristics of bonds and short-term rates in this setting. Their dynamics under risk neutral and forward measures are studied. Finally, bond options are valued with a discretization scheme and a discrete Fourier’s transform. Full article

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29 pages, 554 KiB

Open AccessArticle

Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

by Thomas M. Michelitsch, Federico Polito and Alejandro P. Riascos

Fractal Fract. 2020, 4(4), 51; https://doi.org/10.3390/fractalfract4040051 - 31 Oct 2020

Cited by 13 | Viewed by 2765

Abstract

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs. Full article

(This article belongs to the Special Issue Fractional Behavior in Nature 2019)

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13 pages, 291 KiB

Open AccessFeature PaperArticle

On a New Class of Fractional Difference-Sum Operators with Discrete Mittag-Leffler Kernels

by Thabet Abdeljawad and Arran Fernandez

Mathematics 2019, 7(9), 772; https://doi.org/10.3390/math7090772 - 22 Aug 2019

Cited by 10 | Viewed by 2415

Abstract

We formulate a new class of fractional difference and sum operators, study their fundamental properties, and find their discrete Laplace transforms. The method depends on iterating the fractional sum operators corresponding to fractional differences with discrete Mittag–Leffler kernels. The iteration process depends on the binomial theorem. We note in particular the fact that the iterated fractional sums have a certain semigroup property, and hence, the new introduced iterated fractional difference-sum operators have this semigroup property as well. Full article

(This article belongs to the Special Issue Modelling Problems Arising in Science and Engineering with Fractional Differential Operators: Beyond the Power-Law Limit)

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