Computation and Modeling for Fractional Order Systems.

Format:

Book

Author/Creator:

Chakraverty, Snehashish.

Contributor:

Jena, Rajarama Mohan.

Language:

English

Subjects (All):

Fractional differential equations.

Fractional calculus.

Physical Description:

1 online resource (288 pages)

Edition:

1st ed.

Place of Publication:

San Diego : Elsevier Science & Technology, 2024.

Summary:

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. In this regard, this book brings together contemporary and computationally efficient methods for investigating real-world fractional order systems in one volume. 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. The fractional derivative has also been used in various other physical problems, such as frequency-dependent damping behavior of structures, motion of a plate in a Newtonian fluid, PID controller for the control of dynamical systems, and many others. The mathematical models in electromagnetics, rheology, viscoelasticity, electrochemistry, control theory, Brownian motion, signal and image processing, fluid dynamics, financial mathematics, and material science are well defined by fractional-order differential equations. Generally, these physical models are demonstrated either by ordinary or partial differential equations. However, modeling these problems by fractional differential equations, on the other hand, can make the physics of the systems more feasible and practical in some cases. In order to know the behavior of these systems, we need to study the solutions of the governing fractional models. The exact solution of fractional differential equations may not always be possible using known classical methods. Generally, the physical models occurring in nature comprise complex phenomena, and it is sometimes challenging to obtain the solution (both analytical and numerical) of nonlinear differential equations of fractional order. Various aspects of mathematical modeling that may include deterministic or uncertain (viz. fuzzy or interval or stochastic) scenarios along with fractional order (singular/non-singular kernels) are important to understand the dynamical systems. Computation and Modeling for Fractional Order Systems covers various types of fractional order models in deterministic and non-deterministic scenarios. 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 by the authors of the book.

Contents:

Front Cover

Computation and Modeling for Fractional Order Systems

Copyright

Contents

List of contributors

1 Response time and accuracy modeling through the lens of fractional dynamics

1.1 Introduction

1.1.1 Historical foundation and applications of sequential sampling theory

1.1.2 Lévy flight models as an extension of diffusion models

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

1.3.1 First-passage time approximation

1.3.2 Likelihood construction

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 deriv...

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

2.5.1 Numerical discussion

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

3 Fractional modeling approaches to transport phenomena

3.1 Introduction

3.2 Construction of non-local dynamic models

3.2.1 Causality

3.2.1.1 Mathematical aspects of causality

3.2.2 Constitutive modeling

3.2.2.1 Frame indifference

3.2.2.2 Principle of determinism

3.2.2.3 A thermodynamic constraint: the dissipation principle

3.2.2.4 Simple materials

3.2.3 Fading memory concept and formalism

3.2.3.1 Diffusion-flux relationship: the fading memory concept

3.2.3.2 Boltzmann's superposition

3.2.3.3 Simple heat conduction example: Cattaneo's approach.

3.2.3.4 Extended fading memory concept

3.3 Kernel effects on the constitutive equations

3.3.1 Caputo type fractional operators: the general concept

3.3.1.1 Example 1: exponential memory

3.3.1.2 Example 2: Mittag-Leffler (one-parameter) memory

3.3.1.3 Example 3: Prabhakar memory kernel

3.3.1.4 Example 4: Rabotnov kernel as a memory

3.3.2 Volterra equation approach

3.3.2.1 The concept and Riemann-Liouville operators

3.3.2.2 Example 5: exponential memory

3.3.2.3 Example 6: Mittag-Leffler (one-parameter) function as a kernel

3.3.2.4 Example 7: Prabhakar kernel as a memory

3.3.2.5 Example 8: Rabotnov kernel as a memory

3.4 Final comments and outcomes

Appendix 3.A Mittag-Leffler functions and fractional operators

3.A.1 Mittag-Leffler functions and related kernels

3.A.1.1 One-parameter Mittag-Leffler function

3.A.1.2 Two-parameter Mittag-Leffler function

3.A.1.3 Three-parameter Mittag-Leffler function

3.A.1.4 Prabhakar kernel

3.A.2 Fractional operators based on the Mittag-Leffler function

3.A.2.1 Prabhakar integral

3.A.2.2 Prabhakar derivatives

3.A.2.3 Caputo derivative

3.A.2.4 Atangana-Baleanu derivative

3.A.2.5 Caputo-Fabrizio derivative

4 Numerical solution of time-fractional nonlinear diffusion equations involving weak singularities

4.1 Introduction

4.2 Preliminaries

Regularity of the solution

DGJ method

4.3 The discretized problem

4.4 Error estimation

4.5 Numerical results

4.6 Conclusion

Acknowledgments

5 On the study of the conformal time-fractional generalized q-deformed sinh-Gordon equation

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

5.6.1 The numerical outcomes

5.7 Illustrations with graphics

5.8 Conclusion

Availability of data and materials

6 Nonlinear fractional integro-differential equations by using the homotopy perturbation method

6.1 Introduction

6.2 Preliminaries

Gamma function (GF) [22,23]

Lipschitz condition (LC) [19]

Riemann-Liouville fractional integral (RLFI) [22,23]

Caputo fractional derivative (CFD) [22,23]

Riemann-Liouville fractional derivative (RLFD) [23-25]

6.3 Model of the FVFIDE

6.4 HPM

6.5 Illustrative examples

6.6 Conclusion

7 Fractional prey-predator model with fuzzy initial conditions

7.1 Introduction

7.1.1 Fuzzy differential equations

7.1.2 Fractional differential equations

7.1.3 Novelty in this chapter

7.2 Basic concepts

7.2.1 Fuzzy number

7.2.2 Fuzzy number in parametric form

7.2.3 Triangular fuzzy number

7.2.4 Fuzzy arithmetic operations

7.2.5 Riemann-Liouville fractional derivative

7.2.6 Generalized Hukuhara derivative

7.3 Main result

7.3.1 Fuzzy Riemann-Liouville fractional derivative

7.3.2 Fuzzy fractional prey-predator model

7.4 Numerical illustration

7.5 Results and discussion

7.6 Conclusion

8 Taylor series expansion approach for solving fractional order heat-like and wave-like equations

8.1 Introduction

8.2 Taylor series expansion method

8.2.1 Fundamental approach to solve space-fractional PDEs

8.2.2 Fundamental approach to solve time-fractional PDEs

8.3 Illustrative applications

8.3.1 Solution of HLEs

8.3.1.1 One-dimensional HLE [9]

8.3.1.2 Two-dimensional HLE [9]

8.3.2 Solution of WLEs

8.3.2.1 One-dimensional WLE [6]

8.3.2.2 Two-dimensional WLE [6]

8.4 Conclusion

References.

9 A dynamical study of the fractional order King Cobra model

9.1 Introduction

9.2 Basic definitions

9.3 Mathematical model

Stability analysis of the iteration approach

9.3.1 Numerical technique for the Caputo model

9.4 King Cobra model with fractional conformable derivative

9.4.1 Numerical approach for the fractional conformable derivative

9.5 Numerical simulation

9.6 Conclusion

10 The fractional perturbed nonlinear Schrödinger equation in nanofibers: soliton solutions and dynamical behaviors

10.1 Introduction

10.2 Conformable derivative

10.3 Mathematical analysis of the model

10.4 Implementation of the exp(−Φ(ξ))-expansion method

10.4.1 Kerr law nonlinearity

10.4.2 Parabolic law nonlinearity

10.4.3 Power-law nonlinearity

10.5 Dynamical behaviors

10.6 Conclusions

11 On some recent advances in fractional order modeling in engineering and science

11.1 Introduction

11.2 Preliminaries and fundamentals

11.3 Fractional COVID-19 model

11.3.1 COVID-19 model formulation

11.3.2 Vieta-Lucas shifted polynomials

11.3.3 Simulation of the model (11.3.1.2)

11.3.4 Symmetry and invariance of COVID-19 model

11.3.5 Simulation outcomes for model (11.3.1.2)

11.4 Modeling of the nonlinear fractional Zika model

11.4.1 Zika virus model formulation

11.4.2 Basic definitions of fractional Sumudu transform

11.4.3 Homotopy Sumudu transform method

11.4.4 Symmetry, invariance, and dissipation of the nonlinear fractional Zika model

11.4.5 Optimal control for fractional order network model for Zika virus

11.4.6 Numerical simulation and outcomes

11.5 Conclusion

12 Symbolic computations for exact solutions of fractional partial differential equations with reaction term

12.1 Introduction

12.2 Methods.

12.2.1 Bernoulli approximation method

12.2.2 Trial solution method

12.3 Results

12.3.1 The generalized space-time fractional biological population model

12.3.2 The density-independent fractional diffusion-reaction equation

12.3.3 The space-time fractional generalized Duffing model

12.4 Conclusion

Conflict of interest

13 Unsupervised ANN model for solving fractional differential equations

13.1 Introduction

13.2 Preliminaries

13.2.1 Architecture of artificial neural networks

13.2.2 Riemann-Liouville fractional derivative [19]

13.2.3 Caputo fractional derivative [19]

13.2.4 Grünwald-Letnikov fractional derivative [15]

13.3 Modeling of neural networks for FDEs

13.4 Simulation results and discussion

13.5 Conclusion

Acknowledgment

14 Solitary wave solution for time-fractional SMCH equation in fuzzy environment

14.1 Introduction

14.2 Basic concept of fuzzy set theory and fractional theory

14.2.1 Definition 1 [43,44]

14.2.2 Definition 2 [43,44]

14.2.3 Definition 3 [43,44]

14.2.4 Definition 4 [43,44]

14.2.5 Definition 5 [43,44]

14.2.6 Definition 6 [43,44]

14.3 Method description

14.3.1 Local fractional Taylor theorem [28,33]

14.3.2 Definition 7 [28,33]

14.3.3 Definition 8 [28,33]

14.3.4 Some basic results related to FRDTM

14.4 Implementation of FRDTM on fuzzified SMCH equation

14.5 Results and discussion

14.6 Conclusion

15 Piecewise concept in fractional models

15.1 Introduction

15.2 Piecewise derivative and integral with global, classical, and fractional types

15.3 Piecewise derivative with fractional derivatives

15.3.1 Piecewise derivative having classical and fractional derivatives

15.3.2 Piecewise derivative with global and fractional derivatives.

15.3.3 Piecewise derivative with global derivatives with singular and non-singular kernel.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

0-443-15405-8

OCLC:

1423041563