Computation and Modeling for Fractional Order Systems

1st Edition - February 20, 2024
Imprint: Academic Press
Editors: Snehashish Chakraverty, Rajarama Mohan Jena
Language: English
Paperback ISBN:
9 7 8 - 0 - 4 4 3 - 1 5 4 0 4 - 1
eBook ISBN:
9 7 8 - 0 - 4 4 3 - 1 5 4 0 5 - 8

Computation and Modeling for Fractional Order Systems provides readers with problem-solving techniques for obtaining exact and/or approximate solutions of governing equations… Read more

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Computation and Modeling for Fractional Order Systems provides readers with problem-solving techniques for obtaining exact and/or approximate solutions of governing equations arising in fractional dynamical systems presented using various analytical, semi-analytical, and numerical methods. Various analytical/semi-analytical/numerical methods are applied for solving real-life fractional order problems. The comprehensive descriptions of different recently developed fractional singular, non-singular, fractal-fractional, and discrete fractional operators, along with computationally efficient methods, are included for the reader to understand how these may be applied to real-world systems, and a wide variety of dynamical systems such as deterministic, stochastic, continuous, and discrete are addressed.

Fractional calculus has gained increasing popularity and relevance over the last few decades, due to its well-established applications in various fields of science and engineering. It deals with the differential and integral operators with non-integral powers. Fractional differential equations are the pillar of various systems occurring in a wide range of science and engineering disciplines, namely physics, chemical engineering, mathematical biology, financial mathematics, structural mechanics, control theory, circuit analysis, and biomechanics, among others.

Cover image
Title page
Table of Contents
Copyright
List of contributors
1: Response time and accuracy modeling through the lens of fractional dynamics
Abstract
1.1. Introduction
1.2. Lévy–Brownian model as a model with both Lévy and diffusion properties
1.3. A tutorial on how to fit the Lévy–Brownian model
1.4. Fitting to experimental data
1.5. Discussion
1.6. Conclusion
References
2: An efficient analytical method for the fractional order Sharma–Tasso–Olever equation by means of the Caputo–Fabrizio derivative
Abstract
2.1. Introduction
2.2. Progress of fractional derivatives in the absence of singular kernel
2.3. Fundamental scheme of the modified form of HATM with new derivative
2.4. Analysis of MHATM with Caputo–Fabrizio derivative
2.5. Numerical solution of the time-fractional STO equation
2.6. Comparison of MHATM with VIM, ADM, HPM, RPSM, and the exact solution when μ=1
2.7. L2 and L∞ error norms for the STO equation
2.8. Conclusion
References
3: Fractional modeling approaches to transport phenomena
Abstract
3.1. Introduction
3.2. Construction of non-local dynamic models
3.3. Kernel effects on the constitutive equations
3.4. Final comments and outcomes
Appendix 3.A. Mittag-Leffler functions and fractional operators
References
4: Numerical solution of time-fractional nonlinear diffusion equations involving weak singularities
Abstract
Acknowledgements
4.1. Introduction
4.2. Preliminaries
4.3. The discretized problem
4.4. Error estimation
4.5. Numerical results
4.6. Conclusion
References
5: On the study of the conformal time-fractional generalized q-deformed sinh-Gordon equation
Abstract
5.1. Introduction
5.2. q-Calculus concepts and conformal time-fractional definition
5.3. Analyzing the model mathematically
5.4. The strategy of the analytical technique
5.5. The model's mathematical solution
5.6. The numerical solution to the model
5.7. Illustrations with graphics
5.8. Conclusion
Availability of data and materials
References
6: Nonlinear fractional integro-differential equations by using the homotopy perturbation method
Abstract
6.1. Introduction
6.2. Preliminaries
6.3. Model of the FVFIDE
6.4. HPM
6.5. Illustrative examples
6.6. Conclusion
References
7: Fractional prey–predator model with fuzzy initial conditions
Abstract
7.1. Introduction
7.2. Basic concepts
7.3. Main result
7.4. Numerical illustration
7.5. Results and discussion
7.6. Conclusion
References
8: Taylor series expansion approach for solving fractional order heat-like and wave-like equations
Abstract
8.1. Introduction
8.2. Taylor series expansion method
8.3. Illustrative applications
8.4. Conclusion
References
9: A dynamical study of the fractional order King Cobra model
Abstract
9.1. Introduction
9.2. Basic definitions
9.3. Mathematical model
Stability analysis of the iteration approach
9.4. King Cobra model with fractional conformable derivative
9.5. Numerical simulation
9.6. Conclusion
References
10: The fractional perturbed nonlinear Schrödinger equation in nanofibers: soliton solutions and dynamical behaviors
Abstract
10.1. Introduction
10.2. Conformable derivative
10.3. Mathematical analysis of the model
10.4. Implementation of the exp(−Φ(ξ))-expansion method
10.5. Dynamical behaviors
10.6. Conclusions
References
11: On some recent advances in fractional order modeling in engineering and science
Abstract
11.1. Introduction
11.2. Preliminaries and fundamentals
11.3. Fractional COVID-19 model
11.4. Modeling of the nonlinear fractional Zika model
11.5. Conclusion
References
12: Symbolic computations for exact solutions of fractional partial differential equations with reaction term
Abstract
12.1. Introduction
12.2. Methods
12.3. Results
12.4. Conclusion
Conflict of interest
References
13: Unsupervised ANN model for solving fractional differential equations
Abstract
Acknowledgement
13.1. Introduction
13.2. Preliminaries
13.3. Modeling of neural networks for FDEs
13.4. Simulation results and discussion
13.5. Conclusion
References
14: Solitary wave solution for time-fractional SMCH equation in fuzzy environment
Abstract
14.1. Introduction
14.2. Basic concept of fuzzy set theory and fractional theory
14.3. Method description
14.4. Implementation of FRDTM on fuzzified SMCH equation
14.5. Results and discussion
14.6. Conclusion
References
15: Piecewise concept in fractional models
Abstract
15.1. Introduction
15.2. Piecewise derivative and integral with global, classical, and fractional types
15.3. Piecewise derivative with fractional derivatives
15.4. Numerical algorithm with piecewise derivative
15.5. Illustrative examples
15.6. Conclusion
References
Index

Snehashish Chakraverty

Snehashish Chakraverty has thirty-one years of experience as a researcher and teacher. Presently, he is working in the Department of Mathematics (Applied Mathematics Group), National Institute of Technology Rourkela, Odisha, as a senior (Higher Administrative Grade) professor. Dr Chakraverty received his PhD in Mathematics from IIT-Roorkee in 1993. Thereafter, he did his post-doctoral research at the Institute of Sound and Vibration Research (ISVR), University of Southampton, UK, and at the Faculty of Engineering and Computer Science, Concordia University, Canada. He was also a visiting professor at Concordia and McGill Universities, Canada, during 1997–1999 and visiting professor at the University of Johannesburg, Johannesburg, South Africa, during 2011–2014. He has authored/co-authored/edited 33 books, published 482 research papers (till date) in journals and conferences. He was the president of the section of mathematical sciences of Indian Science Congress (2015–2016) and was the vice president of Orissa Mathematical Society (2011–2013). Prof. Chakraverty is a recipient of prestigious awards, viz. “Careers360 2nd Faculty Research Award” for the Most Outstanding Researcher in the country in the field of Mathematics, Indian National Science Academy (INSA) nomination under International Collaboration/Bilateral Exchange Program (with the Czech Republic), Platinum Jubilee ISCA Lecture Award (2014), CSIR Young Scientist Award (1997), BOYSCAST Fellow. (DST), UCOST Young Scientist Award (2007, 2008), Golden Jubilee Director’s (CBRI) Award (2001), INSA International Bilateral Exchange Award (2015), Roorkee University Gold Medals (1987, 1988) for first positions in MSc and MPhil (Computer Application). He is in the list of 2% world scientists (2020 to 2024) in the Artificial Intelligence and Image Processing category based on an independent study done by Stanford University scientists.

Affiliations and expertise

HAG Professor, Department of Mathematics, Applied Mathematics Group, National Institute of Technology Rourkela, Rourkela, Odisha, India. Differential Equations (ordinary, partial, and fractional), Numerical Analysis, Computational Methods, Structural Dynamics (FGM, Nano), Fluid Dynamics, Mathematical and Uncertainty Modelling, Soft Computing and Machine Intelligence (Artificial Neural Network, Fuzzy, Interval, and Affine Computations).

Rajarama Mohan Jena

Rajarama Mohan Jena is Senior Research Fellow in the Department of Mathematics (Applied Mathematics Group) at the National Institute of Technology Rourkela, Odisha, India. He has an M.Sc. in Applied Mathematics and Computing from the Indian Institute of Technology, Dhanbad, India. Rajarama’s area of research interest includes Fractional Calculus, Partial Differential Equations, Numerical Analysis, Mathematical Modelling, and Uncertainty Modelling, and he has been assisting Dr. Chakraverty in various research projects relating to this book.

Affiliations and expertise

Senior Research Fellow, Department of Mathematics (MA), National Institute of Technology Rourkela, India

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