# Computational Fractional Dynamical Systems

## Fractional Differential Equations and Applications

1. Edition November 2022

272 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-119-69695-7**

Computational Fractional Dynamical Systems

A rigorous presentation of different expansion and semi-analytical methods for fractional differential equations

Fractional differential equations, differential and integral operators with non-integral powers, are used in various science and engineering applications. Over the past several decades, the popularity of the fractional derivative has increased significantly in diverse areas such as electromagnetics, financial mathematics, image processing, and materials science. Obtaining analytical and numerical solutions of nonlinear partial differential equations of fractional order can be challenging and involve the development and use of different methods of solution.

Computational Fractional Dynamical Systems: Fractional Differential Equations and Applications presents a variety of computationally efficient semi-analytical and expansion methods to solve different types of fractional models. Rather than focusing on a single computational method, this comprehensive volume brings together more than 25 methods for solving an array of fractional-order models. The authors employ a rigorous and systematic approach for addressing various physical problems in science and engineering.

* Covers various aspects of efficient methods regarding fractional-order systems

* Presents different numerical methods with detailed steps to handle basic and advanced equations in science and engineering

* Provides a systematic approach for handling fractional-order models arising in science and engineering

* Incorporates a wide range of methods with corresponding results and validation

Computational Fractional Dynamical Systems: Fractional Differential Equations and Applications is an invaluable resource for advanced undergraduate students, graduate students, postdoctoral researchers, university faculty, and other researchers and practitioners working with fractional and integer order differential equations.

Acknowledgments

About the Authors

Introduction to Fractional Calculus

1.1. Introduction

1.2. Birth of fractional calculus

1.3. Useful mathematical functions

1.3.1. The gamma function

1.3.2. The beta function

1.3.3. The Mittag-Leffler function

1.3.4. The Mellin-Ross function

1.3.5. The Wright function

1.3.6. The error function

1.3.7. The hypergeometric function

1.3.8. The H-function

1.4. Riemann-Liouville fractional integral and derivative

1.5. Caputo fractional derivative

1.6. Grünwald-Letnikov fractional derivative and integral

1.7. Riesz fractional derivative and integral

1.8. Modified Riemann-Liouville derivative

1.9. Local fractional derivative

1.9.1. Local fractional continuity of a function

1.9.2. Local fractional derivative

References

Recent Trends in Fractional Dynamical Models and Mathematical Methods

2.1. Introduction

2.2. Fractional calculus: A generalization of integer-order calculus

2.3. Fractional derivatives of some functions and their graphical illustrations

2.4. Applications of fractional calculus

2.4.1. N.H. Abel and Tautochronous problem

2.4.2. Ultrasonic wave propagation in human cancellous bone

2.4.3. Modeling of speech signals using fractional calculus

2.4.4. Modeling the cardiac tissue electrode interface using fractional calculus

2.4.5. Application of fractional calculus to the sound waves propagation in rigid porous Materials

2.4.6. Fractional calculus for lateral and longitudinal control of autonomous vehicles

2.4.7. Application of fractional calculus in the theory of viscoelasticity

2.4.8. Fractional differentiation for edge detection

2.4.9. Wave propagation in viscoelastic horns using a fractional calculus rheology model

2.4.10. Application of fractional calculus to fluid mechanics

2.4.11. Radioactivity, exponential decay and population growth

2.4.12. The Harmonic oscillator

2.5. Overview of some analytical/numerical methods

2.5.1. Fractional Adams-Bashforth/Moulton methods

2.5.2. Fractional Euler method

2.5.3. Finite difference method

2.5.4. Finite element method

2.5.5. Finite volume method

2.5.6. Meshless method

2.5.7. Reproducing kernel Hilbert space method

2.5.8. Wavelet method

2.5.9. The Sine-Gordon expansion method

2.5.10. The Jacobi elliptic equation method

2.5.11. The generalized Kudryashov method

References

Adomian Decomposition Method (ADM)

3.1. Introduction

3.2. Basic Idea of ADM

3.3. Numerical Examples

References

Adomian Decomposition Transform Method

4.1. Introduction

4.2. Transform methods for the Caputo sense derivatives

4.3. Adomian decomposition Laplace transform method (ADLTM)

4.4. Adomian decomposition Sumudu transform method (ADSTM)

4.5. Adomian decomposition Elzaki transform method (ADETM)

4.6. Adomian decomposition Aboodh transform method (ADATM)

4.7. Numerical Examples

4.7.1. Implementation of ADLTM

4.7.2. Implementation of ADSTM

4.7.3. Implementation of ADETM

4.7.4. Implementation of ADATM

References

Homotopy Perturbation Method (HPM)

5.1. Introduction

5.2. Procedure of HPM

5.3. Numerical examples

References

Homotopy Perturbation Transform Method

6.1. Introduction

6.2. Transform methods for the Caputo sense derivatives

6.3. Homotopy perturbation Laplace transform method (HPLTM)

6.4. Homotopy perturbation Sumudu transform method (HPSTM)

6.5. Homotopy perturbation Elzaki transform method (HPETM)

6.6. Homotopy perturbation Aboodh transform method (HPATM)

6.7. Numerical Examples

6.7.1. Implementation of HPLTM

6.7.2. Implementation of HPSTM

6.7.3. Implementation of HPETM

6.7.4. Implementation of HPATM

References

Fractional Differential Transform Method

7.1. Introduction

7.2. Fractional differential transform method

7.3. Illustrative Examples

References

Fractional Reduced Differential Transform Method

8.1. Introduction

8.2. Description of FRDTM

8.3. Numerical Examples

References

Variational Iterative Method

9.1. Introduction

9.2. Procedure for VIM

9.3. Examples

References

Method of Weighted Residuals

10.1. Introduction

10.2. Collocation method

10.3. Least-square method

10.4. Galerkin method

10.5. Numerical Examples

References

Boundary Characteristics Orthogonal Polynomials

11.1. Introduction

11.2. Gram-Schmidt orthogonalization procedure

11.3. Generation of BCOPs

11.4. Galerkin method with BCOPs

11.5. Least-Square method with BCOPs

11.6. Application Problems

References

Residual Power Series Method

12.1. Introduction

12.2. Theorems and lemma related to RPSM

12.3. Basic idea of RPSM

12.4. Convergence Analysis

12.5. Examples

References

Homotopy Analysis Method

13.1. Introduction

13.2. Theory of homotopy analysis method

13.3. Convergence theorem of HAM

13.4. Test Examples

References

Homotopy Analysis Transform Method

14.1. Introduction

14.2. Transform methods for the Caputo sense derivative

14.3. Homotopy analysis Laplace transform method (HALTM)

14.4. Homotopy analysis Sumudu transform method (HASTM)

14.5. Homotopy analysis Elzaki transform method (HAETM)

14.6. Homotopy analysis Aboodh transform method (HAATM)

14.7. Numerical Examples

14.7.1. Implementation of HALTM

14.7.2. Implementation of HASTM

14.7.3. Implementation of HAETM

14.7.4. Implementation of HAATM

References

q-Homotopy Analysis Method

15.1. Introduction

15.2. Theory of q-HAM

15.3. Illustrative Examples

References

q-Homotopy Analysis transform Method

16.1. Introduction

16.2. Transform methods for the Caputo sense derivative

16.3. q-homotopy analysis Laplace transform method (q-HALTM)

16.4. q-homotopy analysis Sumudu transform method (q-HASTM)

16.5. q-homotopy analysis Elzaki transform method (q-HAETM)

16.6. q-homotopy analysis Aboodh transform method (q-HAATM)

16.7. Test Problems

16.7.1. Implementation of q-HALTM

16.7.2. Implementation of q-HASTM

16.7.3. Implementation of q-HAETM

16.7.4. Implementation of q-HAATM

References

(G'/G)-Expansion Method

17.1. Introduction

17.2. Description of the (G'/G)-expansion method

17.3. Application Problems

References

(G'/G^2)-Expansion Method

18.1. Introduction

18.2. Description of the (G'/G^2)-expansion method

18.3. Numerical Examples

References

(G'/G,1/G)-Expansion Method

19.1. Introduction

19.2. Algorithm of the (G'/G,1/G)-expansion method

19.3. Illustrative Examples

References

The modified simple equation method

20.1. Introduction

20.2. Procedure of the modified simple equation method

20.3. Application Problems

References

Sine-Cosine Method

21.1. Introduction

21.2. Details of Sine-Cosine method

21.3. Numerical Examples

References

Tanh Method

22.1. Introduction

22.2. Description of the Tanh method

22.3. Numerical Examples

References

Fractional sub-equation method

23.1. Introduction

23.2. Implementation of the fractional sub-equation method

23.3. Numerical Examples

References

Exp-function Method

24.1. Introduction

24.2. Procedure of the Exp-function method

24.3. Numerical Examples

References

Exp(-phi(xi))-expansion method

25.1. Introduction

25.2. Methodology of the exp(-phi(xi))-expansion method

25.3. Numerical Examples

References

Index

Rajarama Mohan Jena, Senior Research Fellow, Department of Mathematics, National Institute of Technology Rourkela, Odisha, India.

Subrat Kumar Jena, Senior Research Fellow, Department of Mathematics, National Institute of Technology Rourkela, Odisha, India.