Local fractional similarity solution for the diffusion equation defined on Cantor sets

Abstract In this letter, the local fractional similarity solution is addressed for the non-differentiable diffusion equation. Structuring the similarity transformations via the rule of the local fractional partial derivative operators, we transform the diffusive operator into a similarity ordinary differential equation. The obtained result shows the non-differentiability of the solution suitable to describe the properties and behaviors of the fractal content.

[1]  Yu Wang,et al.  Stability of a Class of Fractional-Order Nonlinear Systems , 2014 .

[2]  Hari M. Srivastava,et al.  A Local Fractional Integral Inequality on Fractal Space Analogous to Anderson’s Inequality , 2014 .

[3]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

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

[5]  Jun-Sheng Duan,et al.  Similarity Solution for Fractional Diffusion Equation , 2014 .

[6]  Hari M. Srivastava,et al.  On Local Fractional Continuous Wavelet Transform , 2013 .

[7]  Hari M. Srivastava,et al.  Local Fractional Laplace Variational Iteration Method for Solving Linear Partial Differential Equations with Local Fractional Derivative , 2014 .

[8]  R. Gorenflo,et al.  Wright functions as scale-invariant solutions of the diffusion-wave equation , 2000 .

[9]  John Crank,et al.  The Mathematics Of Diffusion , 1956 .

[10]  V. E. Tarasov.  Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[11]  Ji-Huan He.  A Tutorial Review on Fractal Spacetime and Fractional Calculus , 2014 .

[12]  H. Srivastava,et al.  THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS. NORTH-HOLLAND MATHEMATICS STUDIES , 2006 .

[13]  H. Srivastava,et al.  Helmholtz and Diffusion Equations Associated with Local Fractional Derivative Operators Involving the Cantorian and Cantor-Type Cylindrical Coordinates , 2013 .

[14]  Y. Xiaojun,et al.  Advanced Local Fractional Calculus and Its Applications , 2012 .

[15]  W. Wyss.  The fractional diffusion equation , 1986 .

[16]  Ai-Min Yang,et al.  THE YANG-FOURIER TRANSFORMS TO HEAT-CONDUCTION IN A SEMI-INFINITE FRACTAL BAR , 2013 .

[17]  Hong-yan Liu,et al.  Fractional calculus for nanoscale flow and heat transfer , 2014 .

[18]  Yuan Cao,et al.  Local Fractional Functional Method for Solving Diffusion Equations on Cantor Sets , 2014 .

[19]  Yong Zhou.  Basic Theory of Fractional Differential Equations , 2014 .

[20]  Hari M. Srivastava,et al.  Local Fractional Variational Iteration Algorithms for the Parabolic Fokker-Planck Equation Defined on Cantor Sets , 2015 .

[21]  W. Jost,et al.  Diffusion in Solids, Liquids, Gases , 1952, Zeitschrift für Physikalische Chemie.

[22]  Igor M. Sokolov,et al.  Physics of Fractal Operators , 2003 .

[23]  Paul Shewmon,et al.  Diffusion in Solids , 2016 .

[24]  Yu Zhang,et al.  Solving Initial-Boundary Value Problems for Local Fractional Differential Equation by Local Fractional Fourier Series Method , 2014 .

[25]  Carlo Cattani,et al.  Application of Local Fractional Series Expansion Method to Solve Klein-Gordon Equations on Cantor Sets , 2014 .

[26]  H. Srivastava,et al.  Application of the Local Fractional Series Expansion Method and the Variational Iteration Method to the Helmholtz Equation Involving Local Fractional Derivative Operators , 2013 .