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14 pages, 417 KB

Open AccessArticle

Monotonicity and Composition Analyses for Fractional Differences Involving Generalized Mittag-Leffler Kernels

by Mera Arab, Alina Alb Lupas, Christopher S. Goodrich and Pshtiwan Othman Mohammed

Axioms 2026, 15(7), 510; https://doi.org/10.3390/axioms15070510 - 6 Jul 2026

Viewed by 465

This study establishes a rigorous monotonicity and composition analysis for fractional difference operators defined via the generalized Mittag-Leffler kernel. A fundamental criterion linking the sign of the discrete Atangana–Baleanu–Riemann (ABR) fractional difference to a new generalized form of monotonicity ( [...] Read more.

w (1 - w)

−monotonicity) of the function is derived at first. Subsequently, a crucial composition rule for the ABR difference and its associated sum is proved, providing an explicit formula that elegantly incorporates initial conditions. The use of our theoretical findings is demonstrated practically by its application in solving linear discrete initial value problems, transforming them into explicit solutions by employing the composition theorem. The outcomes reveal the underlying dynamics of discrete models, significantly enhancing the theoretical frameworks and applications. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Difference and Differential Equations)

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16 pages, 791 KB

Open AccessArticle

Stability Analysis of Rössler Chaotic Attractor via the Nabla Discrete Fractional Operator: Existence, Uniqueness, Ulam–Hyers Stability, and Numerical Simulation

by B. Divya, K. Ganesan and A. Selvam

AppliedMath 2026, 6(5), 67; https://doi.org/10.3390/appliedmath6050067 - 29 Apr 2026

Cited by 1 | Viewed by 644

Abstract

This research presents a fractional-order formulation and mathematical analysis of the Rössler chaotic attractor. By utilizing the Nabla discrete Atangana–Baleanu fractional difference derivative in the Caputo sense, the classical integer-order attractor is extended into the fractional domain. The existence and uniqueness of solutions for the resulting fractional system are established via the fixed-point theorem, thereby ensuring that the recommended attractor is well-posed. Furthermore, the Ulam–Hyers stability is investigated within the Nabla discrete Atangana–Baleanu fractional difference derivative in the Caputo sense framework. For numerical investigations, an Euler numerical scheme adapted to the fractional difference derivative is developed and implemented, yielding high-quality phase portraits of a chaotic attractor. The results highlight the effectiveness of fractional-order modeling and numerical methods in capturing the dynamics and stability of the Rössler chaotic system. Full article

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22 pages, 4312 KB

Open AccessArticle

Numerical Solution to the Time-Fractional Burgers–Huxley Equation Involving the Mittag-Leffler Function

by Afzaal Mubashir Hayat, Muhammad Bilal Riaz, Muhammad Abbas, Moataz Alosaimi, Adil Jhangeer and Tahir Nazir

Mathematics 2024, 12(13), 2137; https://doi.org/10.3390/math12132137 - 7 Jul 2024

Cited by 5 | Viewed by 2387

Abstract

Fractional differential equations play a significant role in various scientific and engineering disciplines, offering a more sophisticated framework for modeling complex behaviors and phenomena that involve multiple independent variables and non-integer-order derivatives. In the current research, an effective cubic B-spline collocation method is [...] Read more.

θ

-weighted scheme to solve the proposed problem. The spatial derivative is interpolated using cubic B-spline functions, whereas the temporal derivative is discretized by the Atangana–Baleanu operator and finite difference scheme. The proposed approach is stable across each temporal direction as well as second-order convergent. The study investigates the convergence order, error norms, and graphical visualization of the solution for various values of the non-integer parameter. The efficacy of the technique is assessed by implementing it on three test examples and we find that it is more efficient than some existing methods in the literature. To our knowledge, no prior application of this approach has been made for the numerical solution of the given problem, making it a first in this regard. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations, 2nd Edition)

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22 pages, 2105 KB

Open AccessArticle

Numerical Investigation of the Fractional Diffusion Wave Equation with the Mittag–Leffler Function

by Madiha Shafiq, Muhammad Abbas, Emad K. El-Shewy, Mahmoud A. E. Abdelrahman, Noura F. Abdo and Ali A. El-Rahman

Fractal Fract. 2024, 8(1), 18; https://doi.org/10.3390/fractalfract8010018 - 26 Dec 2023

Cited by 13 | Viewed by 3447

Abstract

A spline is a sufficiently smooth piecewise curve. B-spline functions are powerful tools for obtaining computational outcomes. They have also been utilized in computer graphics and computer-aided design due to their flexibility, smoothness and accuracy. In this paper, a numerical procedure dependent on the cubic B-spline (CuBS) for the time fractional diffusion wave equation (TFDWE) is proposed. The standard finite difference (FD) approach is utilized to discretize the Atangana–Baleanu fractional derivative (ABFD), while the derivatives in space are approximated through the CuBS with a

θ

-weighted technique. The stability of the propounded algorithm is analyzed and proved to be unconditionally stable. The convergence analysis is also studied, and it is of the order

O (h^{2} + {(Δ t)}^{2})

. Numerical solutions attained by the CuBS scheme support the theoretical solutions. The B-spline technique gives us better results as compared to other numerical techniques. Full article

(This article belongs to the Section Numerical and Computational Methods)

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17 pages, 538 KB

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 2199

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 KB

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 23 | Viewed by 2626

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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13 pages, 291 KB

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 2924

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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