Article
Mathematics, Applied
Anumanthappa Ganesh, Swaminathan Deepa, Dumitru Baleanu, Shyam Sundar Santra, Osama Moaaz, Vediyappan Govindan, Rifaqat Ali
Summary: In this paper, we discuss the standard approaches to the Hyers-Ulam Mittag Leffler problem of fractional derivatives and nonlinear fractional integrals using a fractional Fourier transform. We prove the basic properties of derivatives, provide a brief method for solving linear fractional differential equations, and derive the structure of the Hyers-Ulam Mittag Leffler problem for linear two-term equations. Additionally, we consider some physical examples.
Article
Mathematics, Interdisciplinary Applications
Zhehao Zhang
Summary: This paper proposes a new fractional Poisson process model through a recursive fractional differential equation, discussing the correlation between its probability distribution and arrival times. In addition, the underlying time randomizing process is also studied.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
X. Y. Li, X. Y. Liu
Summary: In this letter, a hybrid kernel functions collocation approach for boundary value problems with Caputo fractional derivative is proposed. The approach combines the advantages of the Sobolev and Mittag-Leffler kernel functions to improve the stability and accuracy of existing methods. The results demonstrate the effectiveness of the proposed approach.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Engineering, Mechanical
Oscar Martinez-Fuentes, Aldo Jonathan Munoz-Vazquez, Guillermo Fernandez-Anaya, Esteban Tlelo-Cuautle
Summary: In this paper, a class of dynamic observers for nonlinear fractional-order systems is studied, and the Mittag-Leffler stability is analyzed. The Riemann-Liouville integral is utilized to provide robustness against noisy measurements, and a family of high gain proportional rho-integral observers is designed for estimating unmeasured state variables.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Nasser-eddine Tatar
Summary: The stability of a fractional order Euler-Bernoulli type problem was investigated in this research. By adding lower-order fractional term and a memory term, it was shown that the system can be stabilized to rest in a Mittag-Leffler manner. The results depend heavily on established properties of fractional derivatives and newly introduced functionals.
CHAOS SOLITONS & FRACTALS
(2021)
Article
Mathematics, Applied
Nasser-eddine Tatar
Summary: The study investigates the use of non-integer derivatives in the telegraph problem, showing that both the low-order fractional derivative and the viscoelastic term can stabilize the system and are of Mittag-Leffler type. However, in the fractional case, some basic rules no longer apply, making the situation more delicate. Mittag-Leffler stability is proven under certain smallness conditions on the relaxation function.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics
Tran Thanh Binh, Bui Dinh Thang, Nguyen Duc Phuong
Summary: In this article, we study the elliptic equation under the Caputo derivative and obtain regularity results for the mild solution based on different assumptions of the input data. We analyze the mild solution using the bound of the Mittag-Leffler functions and Hilbert scales space. Additionally, we provide a regularized solution using the Fourier truncation method and estimate the error between the regularized solution and the mild solution.
DEMONSTRATIO MATHEMATICA
(2023)
Article
Mathematics, Interdisciplinary Applications
Banan Al-Homidan, Nasser-eddine Tatar
Summary: This paper investigates a non-linear fractional equation between one and two, which combines features of both the heat and wave equations. The study focuses on stabilizing the system using a lower-order fractional term or a memory term involving the Laplacian. Global and local stability results are proven under different conditions. The challenges in this case mainly stem from the memory dependence of the fractional derivatives, which invalidates the product rule.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics
Ravi Agarwal, Snezhana Hristova
Summary: This paper studies the synchronization problem for impulsive fractional-order Cohen-Grossberg neural networks with generalized proportional Caputo fractional derivatives with changeable lower limit at any point of impulse. The cases when the control input is acting continuously as well as when it is acting instantaneously at the impulsive times are considered. The global Mittag-Leffler synchronization is defined as a generalization of exponential synchronization, and some sufficient conditions for Mittag-Leffler synchronization are obtained. Our results are illustrated with examples.
Article
Mathematics, Applied
Ricardo Almeida, M. Luisa Morgado
Summary: This work studies problems in the calculus of variations with a generalized tempered fractional derivative as the differential operator. Necessary conditions for determining optimal curves are proven, and problems with additional constraints are analyzed. The numerical method presented, based on discretization of the variational problem, demonstrates efficiency through several examples.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2022)
Article
Mathematics, Interdisciplinary Applications
Alexander Iomin
Summary: This article proposes a fractional generalization of the Floquet theorem for fractional Schrodinger equations (FTSE) with time-dependent periodic Hamiltonians. The obtained result, known as the fractional Floquet theorem (fFT), is formulated using the Mittag-Leffler function as the eigenfunction of the Caputo fractional derivative. The suggested formula allows for reducing the FTSE to standard quantum mechanics with time-dependent Hamiltonians, where the standard Floquet theorem holds. Two examples related to quantum resonances are considered to provide support for the obtained result.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Physics, Multidisciplinary
A. Panneer Selvam, M. Vellappandi, V Govindaraj
Summary: The aim of this study is to investigate the controllability of dynamical systems using the psi-Caputo fractional derivative. The Grammian matrix is used to determine necessary and sufficient controllability requirements for linear systems, characterized by Mittag-Leffler functions, while the fixed point approach is used to establish adequate controllability criteria for nonlinear systems. The novelty of this research lies in exploring controllability concepts through the utilization of the psi-Caputo fractional derivative. Several numerical examples are provided to enhance the understanding of the theoretical results.
Article
Mathematics, Applied
Jean-Francois Duhe, Stephane Victor, Pierre Melchior, Youssef Abdelmounen, Francois Roubertie
Summary: Fractional derivatives are effective for modeling long-memory phenomena, but their non-local nature requires constant knowledge of the full past of the functions for differentiation. This limitation can negatively impact the speed of calculations in real-time system identification. Truncated fractional derivatives are used to mitigate this issue, but they introduce inaccuracies. This study explores the relationship between signal frequency content, the approximation of truncated fractional derivatives, and the relaxation of fractional systems to develop efficient real-time system identification algorithms.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Platon G. Surkov
Summary: This paper discusses a specific formulation of the classical problem of mathematical analysis, which is the calculation of the derivative of a function. The purpose is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on control theory methods, ensuring stability to informational noises and computational errors. Numerical experiments were conducted to demonstrate the algorithm's operation.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2021)
Article
Mathematics, Applied
B. S. Vadivoo, G. Jothilakshmi, Y. Almalki, A. Debbouche, M. Lavanya
Summary: This paper investigates the relative controllability problem for a class of fractional differential equations with multiple time delays. The solution representation and necessary and sufficient conditions are established for linear and non-linear systems. Numerical examples and diagrammatic formulations using MATLAB are provided to illustrate the theoretical results.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Mokhtar Kirane, Erkan Nane, Nguyen Huy Tuan
Article
Mathematics
Due Trong Dang, Erkan Nane, Dang Minh Nguyen, Nguyen Huy Tuan
POTENTIAL ANALYSIS
(2018)
Article
Statistics & Probability
Erkan Nane, Nguyen Hoang Tuan, Nguyen Huy Tuan
STATISTICS & PROBABILITY LETTERS
(2018)
Article
Statistics & Probability
Erkan Nane, Yinan Ni
ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS
(2018)
Article
Mathematics, Interdisciplinary Applications
Erkan Nane, Nguyen Huy Tuan
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
(2018)
Article
Mathematics
Mohammud Foondun, Wei Liu, Erkan Nane
JOURNAL OF DIFFERENTIAL EQUATIONS
(2019)
Article
Mathematics
Sunday A. Asogwa, Jebessa B. Mijena, Erkan Nane
POTENTIAL ANALYSIS
(2020)
Article
Mathematics, Applied
Xiangqian Meng, Erkan Nane
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2020)
Article
Mathematics, Applied
Erkan Nane, Yimin Xiao, Aklilu Zeleke
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2020)
Article
Mathematics, Applied
Sunday A. Asogwa, Mohananud Foondun, Jebessa B. Mijena, Erkan Nane
NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
(2020)
Article
Mathematics, Applied
Nguyen Huy Tuan, Erkan Nane, Donal O'regan, Nguyen Duc Phuong
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2020)
Article
Statistics & Probability
Ngartelbaye Guerngar, Erkan Nane
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2020)
Article
Statistics & Probability
Erkan Nane, Eze R. Nwaeze, McSylvester Ejighikeme Omaba
STATISTICS & PROBABILITY LETTERS
(2020)
Article
Mathematics, Applied
Ngartelbaye Guerngar, Erkan Nane, Ramazan Tinaztepe, Suleyman Ulusoy, Hans Werner Van Wyk
Summary: This article examines the space-time fractional equation characterizing double-scale anomalous diffusion and solves the inverse problem of uniquely determining the fractional exponents from data. The existence and uniqueness of the solution are proven using the quasi-solution method and eigenfunction expansion. Additionally, a numerical method for solving the inverse problem is proposed and demonstrated through numerical examples.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2021)
Article
Statistics & Probability
Arun Kumar, Erkan Nane
MODERN STOCHASTICS-THEORY AND APPLICATIONS
(2018)