Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

Abstract

We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

Local fractional analytical methods for solving wave equations with local fractional derivative

“…In this section, we mention the notations, definitions, and some properties of the local fractional operators and fractional calculus. ()…”

Initial boundary value problem for fractal heat equation in the semi-infinite region by Yang-Laplace transform

“…Making use of g = 0 and r a c a = K 2a , the 1-D heat conduction through a semi-infinite fractal medium is modelled by [26,27,30] …”

Mittag-Leffler function for discrete fractional modelling

“…For example, due to the beautiful memory effects, it often appears in diffusion in the porous media (Benson et al, 2000;Berkowitz et al, 2002;Bhrawya and Zaky, 2015;Liu et al, 2004;Sun et al, 2009;Yang et al, 2013), the material's properties (Bagley and Torvik, 1983;Carpinteri and Cornetti, 2002;Mainardi, 2010;Rossikhin and Shitikova, 1997), biological population (Atangana, 2014) and control systems (Baleanu et al, 2011;Li and Chen, 2004;Machado, 1997) et al In the applications of the mentioned topic, one frequently comes across the discrete Mittag-Leffler function (DMLF) (Abdeljawad, 2011;Acar and Atici, 2013;Atici and Eloe, 2007;Pillai and Jayakumar, 1995;Nagai, 2003;Liu et al, 2014). Due to the functions' infinity series' expression, the truncated form is often approximately used.…”