Computational Fractional Dynamical Systems

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.

• PrefaceAcknowledgmentsAbout the AuthorsIntroduction to Fractional Calculus1.1.          Introduction1.2.          Birth of fractional calculus1.3.          Useful mathematical functions1.3.1.       The gamma function1.3.2.       The beta function1.3.3.       The Mittag-Leffler function     1.3.4.       The Mellin-Ross function1.3.5.       The Wright function1.3.6.       The error function1.3.7.       The hypergeometric function1.3.8.       The H-function1.4.          Riemann–Liouville fractional integral and derivative1.5.          Caputo fractional derivative1.6.          Grünwald-Letnikov fractional derivative and integral1.7.          Riesz fractional derivative and integral1.8.          Modified Riemann-Liouville derivative1.9.          Local fractional derivative1.9.1.       Local fractional continuity of a function1.9.2.       Local fractional derivativeReferences  Recent Trends in Fractional Dynamical Models and Mathematical Methods2.1.          Introduction2.2.          Fractional calculus: A generalization of integer-order calculus2.3.          Fractional derivatives of some functions and their graphical illustrations2.4.          Applications of fractional calculus2.4.1.       N.H. Abel and Tautochronous problem2.4.2.       Ultrasonic wave propagation in human cancellous bone2.4.3.       Modeling of speech signals using fractional calculus2.4.4.       Modeling the cardiac tissue electrode interface using fractional calculus2.4.5.     Application of fractional calculus to the sound waves propagation in rigid porous                      Materials2.4.6.        Fractional calculus for lateral and longitudinal control of autonomous vehicles2.4.7.        Application of fractional calculus in the theory of viscoelasticity2.4.8.        Fractional differentiation for edge detection2.4.9.        Wave propagation in viscoelastic horns using a fractional calculus rheology model2.4.10.      Application of fractional calculus to fluid mechanics2.4.11.      Radioactivity, exponential decay and population growth2.4.12.      The Harmonic oscillator2.5.           Overview of some analytical/numerical methods2.5.1.        Fractional Adams–Bashforth/Moulton methods2.5.2.        Fractional Euler method2.5.3.          Finite difference method2.5.4.          Finite element method2.5.5.        Finite volume method2.5.6.        Meshless method2.5.7.        Reproducing kernel Hilbert space method2.5.8.        Wavelet method2.5.9.        The Sine-Gordon expansion method2.5.10.      The Jacobi elliptic equation method2.5.11.      The generalized Kudryashov methodReferences  Adomian Decomposition Method (ADM)3.1.           Introduction3.2.           Basic Idea of  ADM3.3.           Numerical ExamplesReferences  Adomian Decomposition Transform Method4.1.            Introduction4.2.            Transform methods for the Caputo sense derivatives4.3.            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 Examples4.7.1.         Implementation of ADLTM4.7.2.         Implementation of ADSTM4.7.3.         Implementation of ADETM4.7.4.         Implementation of ADATM    References  Homotopy Perturbation Method (HPM)5.1.            Introduction5.2.            Procedure of HPM5.3.            Numerical examplesReferences  Homotopy Perturbation Transform Method6.1.            Introduction6.2.            Transform methods for the Caputo sense derivatives6.3.            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 Examples6.7.1.         Implementation of HPLTM6.7.2.         Implementation of HPSTM6.7.3.         Implementation of HPETM6.7.4.         Implementation of HPATMReferences  Fractional Differential Transform Method7.1.            Introduction7.2.            Fractional differential transform method7.3.            Illustrative ExamplesReferences  Fractional Reduced Differential Transform Method8.1.            Introduction8.2.            Description of FRDTM8.3.            Numerical ExamplesReferences  Variational Iterative Method9.1.            Introduction9.2.            Procedure for VIM9.3.            ExamplesReferences      Method of Weighted Residuals10.1.         Introduction10.2.         Collocation method10.3.         Least-square method10.4.         Galerkin method10.5.         Numerical ExamplesReferences  Boundary Characteristics Orthogonal Polynomials11.1.         Introduction11.2.         Gram–Schmidt orthogonalization procedure11.3.         Generation of BCOPs11.4.         Galerkin method with BCOPs11.5.         Least-Square method with BCOPs11.6.         Application ProblemsReferences  Residual Power Series Method12.1.           Introduction12.2.           Theorems and lemma related to RPSM12.3.           Basic idea of RPSM12.4.           Convergence Analysis12.5.           ExamplesReferences  Homotopy Analysis Method13.1.           Introduction13.2.           Theory of homotopy analysis method13.3.           Convergence theorem of HAM13.4.           Test ExamplesReferences  Homotopy Analysis Transform Method14.1.           Introduction14.2.           Transform methods for the Caputo sense derivative14.3.           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 Examples14.7.1.         Implementation of HALTM14.7.2.         Implementation of HASTM14.7.3.         Implementation of HAETM14.7.4.         Implementation of HAATMReferences  q-Homotopy Analysis Method15.1.         Introduction15.2.         Theory of q-HAM15.3.         Illustrative ExamplesReferences  q-Homotopy Analysis transform Method16.1.         Introduction16.2.         Transform methods for the Caputo sense derivative16.3.         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 Problems16.7.1.        Implementation of q-HALTM16.7.2.        Implementation of q-HASTM16.7.3.        Implementation of q-HAETM16.7.4.        Implementation of q-HAATMReferences  (G'/G)-Expansion Method17.1.          Introduction17.2.          Description of the (G'/G)-expansion method17.3.          Application ProblemsReferences  (G’/G^2)-Expansion Method18.1.          Introduction18.2.            Description of the (G’/G^2)-expansion method18.3.            Numerical ExamplesReferences  (G’/G,1/G)-Expansion Method19.1.           Introduction19.2.           Algorithm of the (G’/G,1/G)-expansion method19.3.           Illustrative ExamplesReferences  The modified simple equation method20.1.           Introduction20.2.           Procedure of the modified simple equation method20.3.           Application ProblemsReferences  Sine-Cosine Method21.1.           Introduction21.2.           Details of Sine-Cosine method21.3.           Numerical ExamplesReferences  Tanh Method22.1.            Introduction22.2.            Description of the Tanh method22.3.            Numerical ExamplesReferences  Fractional sub-equation method23.1.            Introduction23.2.            Implementation of the fractional sub-equation method23.3.            Numerical ExamplesReferences  Exp-function Method24.1.           Introduction24.2.           Procedure of the Exp-function method24.3.           Numerical ExamplesReferences  Exp(-φ(ξ))-expansion method25.1.          Introduction25.2.          Methodology of the exp(-φ(ξ))-expansion method25.3.          Numerical ExamplesReferencesIndex