In the last two decades, many new fractional operators have appeared, often defined using integrals with special functions in the kernel as well as their extended or multivariable forms. Modern operators in fractional calculus have different properties which are comparable to those of classical operators.These have been intensively studied formodelling and analysing real-world phenomena. There is now a growing body of research on new methods to understand natural occurrences and tackle different problems.
This book presents ten reviews of recent fractional operators split over three sections: 1. Chaotic Systems and Control (covers the Caputo fractional derivative, and a chaotic fractional-order financial system)2. Heat Conduction (covers the Duhamel theorem for time-dependent source terms, and the Cattaneo-Hristov model for oscillatory heat transfer)3. Computational Methods and Their Illustrative Applications (covers mathematical analysis for understanding 5 real-word phenomena: HTLV-1 infection of CD4+ T-cells, traveling waves, rumor-spreading, biochemical reactions, and the computational fluid dynamics of a non-powered floating object navigating in an approach channel) This volume is a resource for researchers in physics, biology, behavioral sciences, and mathematics who are interested in new applications of fractional calculus in the study of nonlinear phenomena.
This book presents ten reviews of recent fractional operators split over three sections: 1. Chaotic Systems and Control (covers the Caputo fractional derivative, and a chaotic fractional-order financial system)2. Heat Conduction (covers the Duhamel theorem for time-dependent source terms, and the Cattaneo-Hristov model for oscillatory heat transfer)3. Computational Methods and Their Illustrative Applications (covers mathematical analysis for understanding 5 real-word phenomena: HTLV-1 infection of CD4+ T-cells, traveling waves, rumor-spreading, biochemical reactions, and the computational fluid dynamics of a non-powered floating object navigating in an approach channel) This volume is a resource for researchers in physics, biology, behavioral sciences, and mathematics who are interested in new applications of fractional calculus in the study of nonlinear phenomena.
Table of Contents
Chapter 1 Introduction- Literature Review In Chaos
- Motivations And Novelties
- Fractional Operators And Derivatives
- Fractional-Order System With Caputo Derivative
- Procedure Of The Solution Of Model
- Application Of The Numerical Scheme
- Lyapunov Exponents And Bifurcation Maps
- Stability Analysis
- Chaos And Initial Condition Impacts
- Circuit Implementations Of The Chaotic System
- Conclusion
- Consent For Publication
- Conflict Of Interest
- Acknowledgements
- References
- Introduction
- Fractional Calculus
- The Hamiltonian Optimality Conditions Formulation
- The Oham Formulation With Fractional Optimal Control Problems 35 Convergent Theorem
- Numerical Example And Results
- Conclusion
- Consent For Publication
- Conflict Of Interest
- Acknowledgements
- References
- Muhammad Farman, Parvaiz Ahmad Naik, Aqeel Ahmad, Ali Akgul And Muhammad Umer Saleem Introduction
- Materials And Method
- Circuit Implementations Of The Chaotic System
- Qualitative Analysis
- Stability Analysis
- Sensitivity Analysis
- Complex Finance System With Caputo-Fabrizio Sense
- Oluwaseun O. Okundalaye, Necati Özdemir And Wan A. M. Othman Uniqueness Of The Special Solution
- Numerical Results And Discussion
- Input And Output Stability
- Conclusion
- Consent For Publication
- Conflict Of Interest
- Acknowledgements
- References
- Jordan Hristov Introduction
- The Main Focus Of This Chapter
- Transient Heat Conduction As A Principle Problem
- Duhamel’S Method
- Duhamel’S Method And Separations Of Variables: The General Case
- Time-Dependent Internal Heat Generation
- Caputo-Fabrizio Transform Of The Classical Solution
- Caputo-Fabrizio Operator: Some Basic Properties
- Caputo-Fabrizio Derivatives Of Some Elementary Functions
- Power-Law Function
- Linear Ramp
- Quadratic Ramp
- Exponential Ramp
- Exponential Decay
- Power-Law Decay
- Time-Scales
- Relations To The Nth Fractional Order
- Fourier’S Numbers
- Solution In Terms Of Caputo-Fabrizio Operator: Basic Case
- Examples
- Example 1. Linear Ramp As Boundary Condition
- Example 2. Exponential Ramp As Boundary Condition
- Example 3. Exponential Source Term
- Exponentially Decaying Heat Generation
- Exponentially Growing Heat Generation
- Example 4. Power-Law Decay As A Source Term
- Emerging Questions On The Fourier Series Truncation
- Conclusion
- Consent For Publication
- Conflict Of Interest
- Acknowledgements
- References
- Introduction
- Mathematical Preliminaries
- Problem Formulation
- The Non-Moving Harmonic Heat Source Effects
- The Time-Moving Harmonic Heat Source Effect
- Concluding Remarks
- Appendix
- Consent For Publication
- Conflict Of Interest
- Acknowledgement
- References
- Basic Definitions
- Solution Of Htlv-1 Model By Ham
- Convergence Theorem
- Numerical Results
- An Optimal Homotopy Analysis Approach Of Solutions
- Interval Of Convergence And Optimal Value From An Appropriate Ratio
- Squared Residual Error And Different Orders Of Approximation
- Conclusion
- Appendix
- Consent For Publication
- Conflict Of Interest
- Acknowledgements
- References
- Description Of The Sub Equation Method
- Description Of The 1/G’ -Expansion Method
- Application Of Sub Equation Method
- Results And Discussions
- Comparison Of Methods
- Advantages And Disadvantages
- Conclusion
- Consent For Publication
- Conflict Of Interest
- Acknowledgements
- References
- Some Essential Theorems
- Model Formulation
- Existence And Uniqueness
- Boundedness
- Numerical Simulation
- Conclusion
- Consent For Publication
- Conflict Of Interest
- Acknowledgement
- References
- Preliminaries
- Fundamental Idea Of The Considered Scheme
- Implementation Of The 𝒒-Homotopy Analysis Transform Method
- Results And Discussion
- Conclusion
- Consent For Publication
- Conflict Of Interest
- Acknowledgement
- References
- Materials And Methods
- Comparison Of Methods
- Design Of Approach Channel
- Model Setup And Scenarios
- Results And Discussion
- Concluding Remarks
- Consent For Publication
- Conflict Of Interest
- Acknowledgement
- References
- Subject Index
Author
- Mehmet Yavuz
- Necati Özdemir
