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Numerical Methods
Published in Esteban Tlelo-Cuautle, Luis Gerardo De La Fraga, Omar Guillén-Fernández, Alejandro Silva-Juárez, Optimization of Integer/Fractional Order Chaotic Systems by Metaheuristics and their Electronic Realization, 2021
Esteban Tlelo-Cuautle, Luis Gerardo De La Fraga, Omar Guillén-Fernández, Alejandro Silva-Juárez
In engineering, all integer-order chaotic oscillators can be transformed to its fractional-order version. In this manner, as nowadays one can find many applications of autonomous integer-order chaotic oscillators, their fractional-order version may enhance those engineering applications. All reviews on the history of fractional calculus agree that this topic being more than 300 years ago, and the first reference may probably have been associated with Leibniz and L’Hospital in 1695, where the half-order derivative was mentioned. The majority of researchers agree that the main reason for the delay on dealing with fractional-order chaotic oscillators was the absence of solution methods for fractional-order differential equations. Therefore, and as already mentioned in [20], at the present time there are some methods that have demonstrated its suitability to find, with a good approximation, the solution of fractional-order derivatives and integrals. In this manner, fractional calculus can easily be used in wide areas of applications such as in physics, electrical engineering, control systems, robotics, signal processing, chemical mixing, bioengineering, and so on.
On Difference Operators and Their Applications
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Pinakadhar Baliarsingh, Hemen Dutta
In fact, the theory of difference sequence spaces has made a significant contribution in enveloping the theory of classical and fractional calculus. The theory of fractional calculus deals with the investigation of derivatives and integrations of a function with arbitrary orders. Fractional derivative provides an extensive knowledge for description of memory and hereditary properties of various materials and processes including certain natural and physical phenomena. The application of fractional derivatives becomes more apparent in modeling mechanical and electrical properties of real materials as well as in the description of rheological properties of rocks and in many other fields. Especially, the theory of fractional derivatives has been extensively used in the study of fractal theory, theory of control of dynamic systems, theory of viscoelasticity, electrochemistry, diffusion processes and many others (see Blutzer and Torvik (1996); Dreisigmeyer and Young (2003); Kilbas et al. (2006)).
PRELIMINARIES
Published in Marko Kostić, Abstract Volterra Integro-Differential Equations, 2015
In recent years, considerable interest in fractional calculus and fractional differential equations has been stimulated by their applications in modeling of various problems in engineering, physics, chemistry, biology and other sciences. The Mittag-Leffler and Wright functions are known to play fundamental roles in various applications of the fractional calculus. For further information about the topics mentioned above, the reader may consult the monographs by D. Baleanu, K. Diethelm, E. Scalas, J. Trujillo [33], K. Diethelm [163], A. A. Kilbas, H. M. Srivastava, J. J. Trujillo [271], J. Klafter, S. C. Lim, R. Metzler (Eds.) [278], F. Mainardi [400], K. S. Miller, B. Ross [416], K. B. Oldham, J. Spanier [447], I. Podlubny [458] and S. G. Samko, A. A. Kilbas, O. I. Marichev [478]; we also refer to the references [1], [22], [26]-[28], [34], [89], [235], [264]-[265], [281]-[282], [302]-[308], [312]-[316], [318]-[323], [329], [358], [386], [391], [410], [427], [451], [480], [495] and [513]-[514].
Accelerating flow for engine oil base fluid with graphene oxide and molybdenum disulfide nanoparticles: modified fractional simulations
Published in Waves in Random and Complex Media, 2023
Ali Raza, Sami Ullah Khan, Muhammad Yasir, Sumera Dero
Fractional calculus is a branch of mathematical analysis that deals with generalizations of differentiation and integration to non-integer orders. It extends the concepts of derivatives and integrals to fractional or non-integer orders, allowing for the analysis of functions with fractional or non-integer degrees of differentiability. Fractional calculus has found applications in various fields, including physics, engineering, finance, signal processing, and control theory. It provides powerful mathematical tools for modeling and understanding complex systems that exhibit fractional order behavior, offering insights and solutions that are not achievable with traditional integer-order calculus. In fractional calculus, various mathematical algorithms have been developed. The Caputo–Fabrizio fractional derivative is a specific type of fractional derivative operator introduced by Caputo and Fabrizio in 2013. It is used in the field of fractional calculus to generalize the concept of differentiation to non-integer orders. The Caputo–Fabrizio fractional derivative provides a way to analyze and model phenomena that exhibit fractional order dynamics, such as fractional diffusion and anomalous transport processes. It is one of several fractional derivative operators used in fractional calculus, each with its own properties and applications [31–34].
Symmetry analysis, laws of conservation, and numerical and approximate analysis of Burger’s fractional order differential equation
Published in Waves in Random and Complex Media, 2023
Shaban Mohammadi, S. Reza Hejazi
Non-standard finite difference schemes were initially introduced in an article by Mickens [1]. A small description concerning the results of this scheme up to 1994 has been mentioned in Mickens [2]. Next, Mickens’ finite difference schemes [1–3], developed for the numerical solution of fractional order ordinary/partial differential equations, have been implemented increasingly in numerous fields, including mechanics, signal processing, and control [4]. Fractional calculus is an important branch of applied mathematics that is mostly concerned with non-integer order derivatives. Theories of classic differential calculus have helped increase the focus on the matter of fractional calculus [5,6]. Nevertheless, fractional calculus has become increasingly important in different fields, namely engineering, chemistry, financial affairs, physics, etc. Many researchers believe that non-integer order derivatives can be utilized to describe numerous phenomena in the environment. It also presents a practical tool to describe different materials and processes’ persistence and hereditary properties [7]. It has been proved that fractional-order models can outperform integer-order models, which have been implemented before, in describing real/exact behaviors. Unfortunately, acquiring an analytical solution for fractional-order partial differential equations is not a simple process in most cases. Hence, suitable numerical methods can be utilized to review the qualitative behavior of fractional-order systems.
A time fractional model of hemodynamic two-phase flow with heat conduction between blood and particles: applications in health science
Published in Waves in Random and Complex Media, 2022
Farhad Ali, Fazli Haq, Naveed Khan, Anees Imtiaz, Ilyas Khan
The integer-order derivatives and integrals are discussed in classical calculus. Whereas fractional calculus is the generalization of ordinary calculus in which the non-integer order derivative and integrals have been discussed. Fractional calculus has developed three hundred years ago by the contribution of Laplace, Fourier, Abel, and Liouville [25,26]. Fractional calculus is applied in various fields of science, biosciences, mathematics, electrochemistry, physics, engineering, and technology [27]. Moreover, fractional derivatives have several applications in the field of fluid mechanics, mathematical biology, viscoelasticity, electrochemistry, and signal processing [28–32]. Therefore the significance of fractional order derivatives cannot be ignored. Imran et al. [33] Studied the differential type fluid with mass and heat transfer by considering the Caputo derivatives. Ali et al. [34] Examined the flow of blood as a Casson fluid through a horizontal tube and considered the fractional-order derivative. Bakhti et al. [35] Considered the pulsatile flow of blood through stenosis arteries because of pressure gradient using the derivatives of fractional order. For the flow of blood in the presence of a magnetic field due to periodic pressure gradient inside the oscillatory arteries, a fractional-order model is proposed by Bansi et al. [36]. Ali Shah et al. [37] Considered the fractional-order derivative to examine the flow of blood with magnetic particles and obtained the solutions by the joint use of Laplace & Hankel transformations.