Computational Fractional Dynamical Systems Fractional Differential Equations and Applications

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.

Snehashish Chakraverty, Senior Professor, Department of Mathematics (Applied Mathematics Group), National Institute of Technology Rourkela, Odisha, India.

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.

Preface

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(-φ(ξ))-expansion method

 25.1.          Introduction

 25.2.          Methodology of the exp(-φ(ξ))-expansion method

 25.3.          Numerical Examples

                   References

Index

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