Local Fractional Calculus and Its Applications

In this paper we point out the interpretations of local fractional derivative and local fractional integration from the fractal geometry point of view. From Cantor set to fractional set, local fractional derivative and local fractional integration are investigated in detail, and some applications are given to elaborate the local fractional Fourier series, the Yang-Fourier transform, the Yang-Laplace transform, the local fractional short time transform, the local fractional wavelet transform in fractal space.

[1]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[2]  Ji-Huan He.  A NEW FRACTAL DERIVATION , 2011 .

[3]  Jacky Cresson,et al.  About Non-differentiable Functions , 2001 .

[4]  G. Jumarie,et al.  Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results , 2006, Comput. Math. Appl..

[5]  Y. Xiaojun.  Research on Fractal Mathematics and Some Application in Mechanics , 2009 .

[6]  B. Guo,et al.  Global Well-Posedness for the Fractional Nonlinear Schrödinger Equation , 2010 .

[7]  Heinz-Otto Peitgen,et al.  The science of fractal images , 2011 .

[8]  Guy Jumarie,et al.  On the representation of fractional Brownian motion as an integral with respect to (dt)alpha , 2005, Appl. Math. Lett..

[9]  Kewei Zhang,et al.  On the local fractional derivative , 2010 .

[10]  Kiran M. Kolwankar,et al.  Local Fractional Fokker-Planck Equation , 1998 .

[11]  T. Vicsek,et al.  Dynamics of fractal surfaces , 1991 .

[12]  R. Koeller.  Applications of Fractional Calculus to the Theory of Viscoelasticity , 1984 .

[13]  Xiao-Jun Yang.  The Discrete Yang-Fourier Transforms in Fractal Space , 2012 .

[14]  Xiao‐Jun Yang,et al.  A novel approach to processing fractal dynamical systems using the Yang-Fourier transforms , 2012 .

[15]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[16]  Xiao-Jun Yang,et al.  Local Fractional Integral Transforms , 2011 .

[17]  Heping Xie,et al.  Fractals in Rock Mechanics , 2020 .

[18]  J. A. Tenreiro Machado,et al.  Entropy Analysis of Integer and Fractional Dynamical Systems , 2010 .

[19]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[20]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[21]  C. Sparrow.  The Fractal Geometry of Nature , 1984 .

[22]  Varsha Daftardar-Gejji,et al.  On calculus of local fractional derivatives , 2002 .

[23]  F. Tatom.  THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS , 1995 .

[24]  Xiao-Jun Yang.  A short introduction to Yang-Laplace Transforms in fractal space , 2012, IAIT 2012.

[25]  I. Podlubny.  Fractional differential equations , 1998 .

[26]  Arcady Dyskin,et al.  Effective characteristics and stress concentrations in materials with self-similar microstructure , 2005 .

[27]  Pietro Cornetti,et al.  Static–kinematic duality and the principle of virtual work in the mechanics of fractal media , 2001 .

[28]  Guang-Sheng Chen,et al.  A generalized Young inequality and some new results on fractal space , 2011, 1107.5222.

[29]  F. Mainardi.  Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

[30]  Pietro Cornetti,et al.  Calculation of the tensile and flexural strength of disordered materials using fractional calculus , 2004 .

[31]  Xiao‐Jun Yang.  Fast Yang-Fourier Transforms in Fractal Space , 2012 .

[32]  Xiao-jun Yang.  The Zero-mass Renormalization Group Differential Equations and Limit Cycles in Non-smooth Initial Value Problems , 2012 .

[33]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[34]  Yang Xiao-Jun.  Local Fractional Functional Analysis and Its Applications , 2011 .

[35]  Guy Jumarie,et al.  Lagrange characteristic method for solving a class of nonlinear partial differential equations of fractional order , 2006, Appl. Math. Lett..

[36]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[37]  J. Machado.  Optimal tuning of fractional controllers using genetic algorithms , 2010 .

[38]  Kiran M. Kolwankar,et al.  Hölder exponents of irregular signals and local fractional derivatives , 1997, chao-dyn/9711010.

[39]  Kiran M. Kolwankar,et al.  Fractional differentiability of nowhere differentiable functions and dimensions. , 1996, Chaos.

[40]  Xiao‐Jun Yang.  A generalized model for Yang-Fourier transforms in fractal space , 2011, 1106.3615.

[41]  J. Machado.  Root locus of fractional linear systems , 2011 .

[42]  W. Zhong,et al.  Applications of Yang-Fourier Transform to Local Fractional Equations with Local Fractional Derivative and Local Fractional Integral , 2012 .

[43]  Wen Chen,et al.  Investigation on Fractional and Fractal Derivative Relaxation- Oscillation Models , 2010 .

[44]  A. D. Gangal,et al.  Calculus on fractal subsets of real line - I: formulation , 2003 .

[45]  Guy Jumarie,et al.  The Minkowski's space-time is consistent with differential geometry of fractional order , 2007 .

[46]  Liquan Mei,et al.  The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics ✩ , 2012 .

[47]  A. Tofighi,et al.  Probability Structure of Time Fractional Schrödinger Equation , 2009 .

[48]  Xiao-Jun Yang.  Local fractional Fourier analysis , 2012 .

[49]  Xiao-Jun Yang.  Local Fractional Laplace’s Transform Based on the Local Fractional Calculus , 2011, CSIE 2011.

[50]  N. Laskin,et al.  Fractional quantum mechanics , 2008, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[51]  A New Viewpoint to Fourier Analysis in Fractal Space , 2013 .

[52]  Xiao-Jun Yang,et al.  A Novel Approach to Processing Fractal Signals Using the Yang-Fourier Transforms☆ , 2012 .