Observing diffusion problems defined on cantor sets in different coordinate systems

Abstract

In this paper, the 2-D and 3-D diffusions defined on Cantor sets with local fractional differential operator were discussed in different co-ordinate systems. The 2-D diffusion in Cantorian co-ordinate system can be converted into the symmetric diffusion defined on Cantor sets. The 3-D diffusions in Cantorian co-ordinate system can be observed in the Cantor-type cylindrical and spherical co-ordinate methods.

Maximum principles for time-fractional Caputo-Katugampola diffusion equations

“…Fractional order integral and differential equations have been demonstrated to be an efficient tool in numerous applications, for example, fractal phenomena [3,[23][24][25][26][27][28], contact mechanics problems [15], control problems [20], fluid mechanics problems [16] and so on [4,21,22,30].…”

About local fractional three-dimensional compressible Navier-Stokes equations in Cantor-type cylindrical co-ordinate system

“…Local fractional partial differential equations were observed in different co-ordinate systems, such as heat-conduction equation [1], Helmholtz equation [2], Maxwell's equation [3], wave equation [4], and diffusion equation [5]. The Cantor-type spherical co-ordinate [1], Cantor-type circle co-ordinate [2], and Cantor-type cylindrical co-ordinate [6] methods were proposed and developed to describe the heat transfer problems.…”

On three-dimensional incompressible Navier-Stokes fluid on cantor sets in spherical Cantor type co-ordinate system

“…Local fractional calculus was used to solve the non-differentiable problems in heat [1][2][3][4][5][6][7] and fluid [8][9][10][11][12] flows. The fractal Cauchy stress tensor of the incompressible Navier--Stokes fluid on Cantor sets was written in the form [11] The systems of the incompressible Navier-Stokes equations on Cantor sets was expressed as [11,12]: 0…”

“…The Cantor type spherical co-ordinate method was reported in [1, 5,8]. In this paper, the aim is to investigate the systems of the incompressible Navier-Stokes equations on Cantor sets in spherical Cantor type co-ordinate system.…”