Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system

Abstract New operators of differentiation have been introduced in this paper as convolution of power law, exponential decay law, and generalized Mittag-Leffler law with fractal derivative. The new operators will be referred as fractal-fractional differential and integral operators. The new operators aimed to attract more non-local natural problems that display at the same time fractal behaviors. Some new properties are presented, the numerical approximation of these new operators are also presented with some applications to real world problem.

[1]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[2]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[3]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[4]  Francesco Mainardi,et al.  The -Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey , 2010, 1004.2950.

[5]  J. Hadamard,et al.  Essai sur l'étude des fonctions données par leur développement de Taylor , 1892 .

[6]  Obaid J. Algahtani,et al.  Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model , 2016 .

[7]  H. G. Schaeffer,et al.  Book Reviews : Computer Methods for Mathematical Computations: G.E. Forsythe et al. Englewood Cliffs, NJ, Prentice-Hall, Inc., 1977 , 1979 .

[8]  J. Leveinen.  Composite model with fractional flow dimensions for well test analysis in fractured rocks , 2000 .

[9]  Michael A. Malcolm,et al.  Computer methods for mathematical computations , 1977 .

[10]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[11]  S. P. Näsholm,et al.  A causal and fractional all-frequency wave equation for lossy media. , 2011, The Journal of the Acoustical Society of America.

[12]  Ilknur Koca,et al.  Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order , 2016 .

[13]  J. H. Cushman,et al.  Anomalous diffusion as modeled by a nonstationary extension of Brownian motion. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Ryutaro Kanno,et al.  Representation of random walk in fractal space-time , 1998 .

[15]  Hongguang Sun,et al.  Anomalous diffusion modeling by fractal and fractional derivatives , 2010, Comput. Math. Appl..

[16]  A. Fröhlich.  ON GROUPS OVER A D.G. NEAR-RING (I): SUM CONSTRUCTIONS AND FREE R -GROUPS , 1960 .

[17]  W. Chen.  Time-space fabric underlying anomalous diffusion , 2005, math-ph/0505023.

[18]  Badr Saad T. Alkahtani,et al.  Chua's circuit model with Atangana–Baleanu derivative with fractional order , 2016 .

[19]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .