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Article
Mathematics, Applied
Amar Benkerrouche et al.
Summary: This manuscript examines the existence, uniqueness, and stability of solutions to the boundary value problem of Hadamard fractional differential equations of variable order. By converting the problem into an equivalent standard Hadamard BVP of fractional constant order, using generalized intervals and piecewise constant functions, the analysis is conducted. The established results in this study are based on the Krasnoselskii fixed-point theorem and the Banach contraction principle. Additionally, the Ulam-Hyers stability of the given problem is investigated, and an example is constructed to illustrate the validity of the observed results.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Engineering, Mechanical
Yupin Wang et al.
Summary: This paper investigates the chaotic and fractal dynamics of a fractional coupled logistic map constructed using the Caputo fractional h-difference. The chaos of the map, influenced by the memory and scale from the fractional operator, is examined using phase portrait, 0-1 test, and Lyapunov exponent. Fractal synchronization is achieved by designing a coupled controller between Julia sets generated from two different structured fractional coupled maps. Numerical simulations are provided to validate the main findings.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Applied
Hasib Khan et al.
Summary: In this article, necessary and sufficient conditions for the existence of solutions for mABC-fractional differential equations (mABC-FDEs) with initial conditions are investigated. A numerical scheme based on Lagrange's interpolation polynomial is established and applied to a dynamical system. The uniqueness and Hyers-Ulam stability of the solutions are also studied. This study extends the results for the ABC operator and provides a foundation for dynamical problems involving existence, uniqueness, and numerical simulations. The accuracy and applicability of the scheme are verified through graphical comparison with classical results.
Article
Mathematics, Applied
Benoumran Telli et al.
Summary: This paper investigates boundary-value problems for Riemann-Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is analyzed using Darbo's fixed-point theorem and the Kuratowski measure of noncompactness. Furthermore, the Ulam-Hyers stability criteria are examined. The results are established using generalized intervals and piecewise constant functions. The conversion of the Riemann-Liouville fractional variable-order problem to equivalent standard Riemann-Liouville problems of fractional-constant orders is also demonstrated. Two examples are provided to illustrate the validity of the obtained results.
Article
Mathematics, Applied
Aziz Khan et al.
Summary: In this article, the exact controllability of a fractional order differential system involving Hilfer fractional (HF) derivative, state-dependent delay function, and impulsive conditions is discussed. The semigroup theory, theory of measure of non-compactness, and fixed-point technique are used to obtain the proposed result. Finally, an example is provided to illustrate the application of the obtained results.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Interdisciplinary Applications
Mohammed Al-Refai et al.
Summary: This short paper suggests an extension of the fractional operator involving the Mittag-Leffler kernel to deal with non-singular kernels, which allows for integrable singular kernels at the origin. New solutions to related differential equations are reported, along with some perspectives from a modeling viewpoint.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2022)
Article
Materials Science, Multidisciplinary
Aziz Khan et al.
Summary: This paper investigates the existence and uniqueness of solutions, stability, and numerical simulations of a fractional order two strain epidemic model. Fixed point postulates are used to prove the existence and uniqueness of solutions, while a theoretical approach is employed to study the stability of the model. Numerical simulations using Lagrange's interpolation polynomial technique are presented for different fractional orders.
RESULTS IN PHYSICS
(2022)
Article
Engineering, Multidisciplinary
Kamal Shah et al.
Summary: This manuscript focuses on a nonlinear dynamical system with Caputo fractional order derivative, specifically addressing an infectious disease like COVID-19. The study establishes the feasibility region, boundedness, stability analysis, and reproduction number of the system using nonlinear analysis and matrix methods. The existence and uniqueness of approximate solutions are investigated using the fixed point approach. Numerical schemes and graphical presentations are implemented to analyze the system with different fractional orders.
ALEXANDRIA ENGINEERING JOURNAL
(2022)
Article
Mathematics, Applied
Hasib Khan et al.
Summary: This article explores a fractal-fractional tobacco mathematical model with Mittag-Leffler functions, focusing on qualitative and numerical studies. The numerical results are divided into four figures based on different fractal and fractional orders, showing a significant impact on the dynamical behavior of the model.
Article
Multidisciplinary Sciences
Snezhana Hristova et al.
Summary: This paper investigates a boundary value problem for Hadamard fractional differential equations of variable order, exploring the connection between the symmetry of transformations and local solvability, and examining the existence criteria and stability in the sense of Ulam-Hyers-Rassias. The results are obtained based on the Kuratowski measure of noncompactness, with an example illustrating the validity of the observed results.
Article
Mathematics
Ahmed Refice et al.
Summary: This manuscript examines the existence and stability of solutions of boundary value problems of Hadamard-type fractional differential equations of variable order. New outcomes are obtained using the Darbo's fixed point theorem combined with Kuratowski measure of noncompactness, and an example is constructed to illustrate the validity of the results.
Article
Mathematics, Applied
Shuqin Zhang et al.
Article
Mathematics
Shuqin Zhang et al.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2019)
Article
Mathematics, Applied
Hong Wang et al.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2019)
Article
Mathematics, Applied
J. Vanterler da C. Sousa et al.
COMPUTATIONAL & APPLIED MATHEMATICS
(2018)
Article
Mathematics, Applied
J. Vanterler da C. Sousa et al.
COMPUTATIONAL & APPLIED MATHEMATICS
(2018)
Article
Yunru Bai et al.
Journal of Nonlinear Sciences and Applications
(2017)
Article
Mathematics
Mouffak Benchohra et al.
STUDIA UNIVERSITATIS BABES-BOLYAI MATHEMATICA
(2017)
Review
Engineering, Mechanical
Stefan Samko
NONLINEAR DYNAMICS
(2013)
Article
Engineering, Electrical & Electronic
Duarte Valerio et al.
