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Patnaik S, Hollkamp JP, Semperlotti F. Applications of variable-order fractional operators: a review. Proc Math Phys Eng Sci 2020; 476:20190498. [PMID: 32201475 PMCID: PMC7069480 DOI: 10.1098/rspa.2019.0498] [Citation(s) in RCA: 74] [Impact Index Per Article: 18.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2019] [Accepted: 01/08/2020] [Indexed: 11/12/2022] Open
Abstract
Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.
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Affiliation(s)
- Sansit Patnaik
- School of Mechanical Engineering, Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN 47907, USA
| | - John P Hollkamp
- School of Mechanical Engineering, Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN 47907, USA
| | - Fabio Semperlotti
- School of Mechanical Engineering, Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN 47907, USA
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Ullah MZ, Alzahrani AK, Baleanu D. An efficient numerical technique for a new fractional tuberculosis model with nonsingular derivative operator. JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 2019. [DOI: 10.1080/16583655.2019.1688543] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
Affiliation(s)
- Malik Zaka Ullah
- Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
| | - Abdullah K. Alzahrani
- Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
| | - Dumitru Baleanu
- Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara, Turkey
- Institute of Space Sciences, Magurele-Bucharest, Romania
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Dynamics Analysis of a New Fractional-Order Hopfield Neural Network with Delay and Its Generalized Projective Synchronization. ENTROPY 2018; 21:e21010001. [PMID: 33266717 PMCID: PMC7514113 DOI: 10.3390/e21010001] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/27/2018] [Revised: 12/13/2018] [Accepted: 12/18/2018] [Indexed: 11/17/2022]
Abstract
In this paper, a new three-dimensional fractional-order Hopfield-type neural network with delay is proposed. The system has a unique equilibrium point at the origin, which is a saddle point with index two, hence unstable. Intermittent chaos is found in this system. The complex dynamics are analyzed both theoretically and numerically, including intermittent chaos, periodicity, and stability. Those phenomena are confirmed by phase portraits, bifurcation diagrams, and the Largest Lyapunov exponent. Furthermore, a synchronization method based on the state observer is proposed to synchronize a class of time-delayed fractional-order Hopfield-type neural networks.
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Vats M, Mishra SK, Baghini MS, Chauhan DS, Srivastava R, De A. Near Infrared Fluorescence Imaging in Nano-Therapeutics and Photo-Thermal Evaluation. Int J Mol Sci 2017; 18:E924. [PMID: 28452928 PMCID: PMC5454837 DOI: 10.3390/ijms18050924] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2017] [Revised: 04/11/2017] [Accepted: 04/17/2017] [Indexed: 01/03/2023] Open
Abstract
The unresolved and paramount challenge in bio-imaging and targeted therapy is to clearly define and demarcate the physical margins of tumor tissue. The ability to outline the healthy vital tissues to be carefully navigated with transection while an intraoperative surgery procedure is performed sets up a necessary and under-researched goal. To achieve the aforementioned objectives, there is a need to optimize design considerations in order to not only obtain an effective imaging agent but to also achieve attributes like favorable water solubility, biocompatibility, high molecular brightness, and a tissue specific targeting approach. The emergence of near infra-red fluorescence (NIRF) light for tissue scale imaging owes to the provision of highly specific images of the target organ. The special characteristics of near infra-red window such as minimal auto-fluorescence, low light scattering, and absorption of biomolecules in tissue converge to form an attractive modality for cancer imaging. Imparting molecular fluorescence as an exogenous contrast agent is the most beneficial attribute of NIRF light as a clinical imaging technology. Additionally, many such agents also display therapeutic potentials as photo-thermal agents, thus meeting the dual purpose of imaging and therapy. Here, we primarily discuss molecular imaging and therapeutic potentials of two such classes of materials, i.e., inorganic NIR dyes and metallic gold nanoparticle based materials.
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Affiliation(s)
- Mukti Vats
- Department of Biosciences and Bioengineering, Indian Institute of Technology Bombay, Powai, Mumbai 410210, India.
| | - Sumit Kumar Mishra
- Molecular Functional Imaging Laboratory, Advanced Centre for Treatment, Research and Education in Cancer (ACTREC), Tata Memorial Centre, Kharghar, Mumbai 410210, India.
| | - Mahdieh Shojaei Baghini
- Molecular Functional Imaging Laboratory, Advanced Centre for Treatment, Research and Education in Cancer (ACTREC), Tata Memorial Centre, Kharghar, Mumbai 410210, India.
| | - Deepak S Chauhan
- Department of Biosciences and Bioengineering, Indian Institute of Technology Bombay, Powai, Mumbai 410210, India.
| | - Rohit Srivastava
- Department of Biosciences and Bioengineering, Indian Institute of Technology Bombay, Powai, Mumbai 410210, India.
| | - Abhijit De
- Molecular Functional Imaging Laboratory, Advanced Centre for Treatment, Research and Education in Cancer (ACTREC), Tata Memorial Centre, Kharghar, Mumbai 410210, India.
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Wu H, Wang L, Niu P, Wang Y. Global projective synchronization in finite time of nonidentical fractional-order neural networks based on sliding mode control strategy. Neurocomputing 2017. [DOI: 10.1016/j.neucom.2017.01.022] [Citation(s) in RCA: 42] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Ding Z, Shen Y. Projective synchronization of nonidentical fractional-order neural networks based on sliding mode controller. Neural Netw 2016; 76:97-105. [DOI: 10.1016/j.neunet.2016.01.006] [Citation(s) in RCA: 83] [Impact Index Per Article: 10.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2015] [Revised: 12/15/2015] [Accepted: 01/13/2016] [Indexed: 11/25/2022]
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Chen B, Chen J. Global O(t−α) stability and global asymptotical periodicity for a non-autonomous fractional-order neural networks with time-varying delays. Neural Netw 2016; 73:47-57. [DOI: 10.1016/j.neunet.2015.09.007] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2015] [Revised: 09/04/2015] [Accepted: 09/07/2015] [Indexed: 10/22/2022]
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Global Mittag-Leffler synchronization of fractional-order neural networks with discontinuous activations. Neural Netw 2016; 73:77-85. [PMID: 26562442 DOI: 10.1016/j.neunet.2015.10.010] [Citation(s) in RCA: 74] [Impact Index Per Article: 9.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2015] [Revised: 10/08/2015] [Accepted: 10/20/2015] [Indexed: 11/23/2022]
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Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw 2014; 51:1-8. [DOI: 10.1016/j.neunet.2013.11.016] [Citation(s) in RCA: 416] [Impact Index Per Article: 41.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2013] [Revised: 11/19/2013] [Accepted: 11/22/2013] [Indexed: 11/18/2022]
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Das S, Maharatna K. Fractional dynamical model for the generation of ECG like signals from filtered coupled Van-der Pol oscillators. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2013; 112:490-507. [PMID: 24028797 DOI: 10.1016/j.cmpb.2013.08.012] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/23/2012] [Revised: 08/07/2013] [Accepted: 08/20/2013] [Indexed: 06/02/2023]
Abstract
In this paper, an incommensurate fractional order (FO) model has been proposed to generate ECG like waveforms. Earlier investigation of ECG like waveform generation is based on two identical Van-der Pol (VdP) family of oscillators, which are coupled by time delays and gains. In this paper, we suitably modify the three state equations corresponding to the nonlinear cross-product of states, time delay coupling of the two oscillators and low-pass filtering, using the concept of fractional derivatives. Our results show that a wide variety of ECG like waveforms can be simulated from the proposed generalized models, characterizing heart conditions under different physiological conditions. Such generalization of the modelling of ECG waveforms may be useful to understand the physiological process behind ECG signal generation in normal and abnormal heart conditions. Along with the proposed FO models, an optimization based approach is also presented to estimate the VdP oscillator parameters for representing a realistic ECG like signal.
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Affiliation(s)
- Saptarshi Das
- School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, United Kingdom.
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Fractional dynamics pharmacokinetics-pharmacodynamic models. J Pharmacokinet Pharmacodyn 2010; 37:257-76. [PMID: 20455076 PMCID: PMC2889283 DOI: 10.1007/s10928-010-9159-z] [Citation(s) in RCA: 36] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2010] [Accepted: 04/21/2010] [Indexed: 10/28/2022]
Abstract
While an increasing number of fractional order integrals and differential equations applications have been reported in the physics, signal processing, engineering and bioengineering literatures, little attention has been paid to this class of models in the pharmacokinetics-pharmacodynamic (PKPD) literature. One of the reasons is computational: while the analytical solution of fractional differential equations is available in special cases, it this turns out that even the simplest PKPD models that can be constructed using fractional calculus do not allow an analytical solution. In this paper, we first introduce new families of PKPD models incorporating fractional order integrals and differential equations, and, second, exemplify and investigate their qualitative behavior. The families represent extensions of frequently used PK link and PD direct and indirect action models, using the tools of fractional calculus. In addition the PD models can be a function of a variable, the active drug, which can smoothly transition from concentration to exposure, to hyper-exposure, according to a fractional integral transformation. To investigate the behavior of the models we propose, we implement numerical algorithms for fractional integration and for the numerical solution of a system of fractional differential equations. For simplicity, in our investigation we concentrate on the pharmacodynamic side of the models, assuming standard (integer order) pharmacokinetics.
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Dokoumetzidis A, Macheras P. Fractional kinetics in drug absorption and disposition processes. J Pharmacokinet Pharmacodyn 2009; 36:165-78. [PMID: 19340400 DOI: 10.1007/s10928-009-9116-x] [Citation(s) in RCA: 73] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2008] [Accepted: 03/23/2009] [Indexed: 10/20/2022]
Abstract
We explore the use of fractional order differential equations for the analysis of datasets of various drug processes that present anomalous kinetics, i.e. kinetics that are non-exponential and are typically described by power-laws. A fractional differential equation corresponds to a differential equation with a derivative of fractional order. The fractional equivalents of the "zero-" and "first-order" processes are derived. The fractional zero-order process is a power-law while the fractional first-order process is a Mittag-Leffler function. The latter behaves as a stretched exponential for early times and as a power-law for later times. Applications of these two basic results for drug dissolution/release and drug disposition are presented. The fractional model of dissolution is fitted successfully to datasets taken from literature of in vivo dissolution curves. Also, the proposed pharmacokinetic model is fitted to a dataset which exhibits power-law terminal phase. The Mittag-Leffler function describes well the data for small and large time scales and presents an advantage over empirical power-laws which go to infinity as time approaches zero. The proposed approach is compared conceptually with fractal kinetics, an alternative approach to describe datasets with non exponential kinetics. Fractional kinetics offers an elegant description of anomalous kinetics, with a valid scientific basis, since it has already been applied in problems of diffusion in other fields, and describes well the data.
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Abstract
Control of complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. I have attempted to emphasize the difference between homeostatic control and allometric control mechanisms. Homeostatic control is familiar and has as its basis a negative feedback character, which is both local and relatively fast. Allometric control, on the other hand, is a new concept that can take into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the system variable. Allometric control introduces the fractal character into otherwise featureless random time series to enhance the robustness of physiological networks by introducing the fractional calculus into the control of the networks.
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