Abstract
The main object of this paper is to investigate the Helmholtz and diffusion equations on the Cantor sets involving local fractional derivative operators. The Cantor-type cylindrical-coordinate method is applied to handle the corresponding local fractional differential equations. Two illustrative examples for the Helmholtz and diffusion equations on the Cantor sets are shown by making use of the Cantorian and Cantor-type cylindrical coordinates.
Highlights
In the Euclidean space, we observe several interesting physical phenomena by using the differential equations in the different styles of planar, cylindrical, and spherical geometries
The main object of this paper is to investigate the Helmholtz and diffusion equations on the Cantor sets involving local fractional derivative operators
We have derived the Helmholtz and diffusion equations on the Cantor sets in the Cantorian coordinates, which are based upon the local fractional derivative operators
Summary
In the Euclidean space, we observe several interesting physical phenomena by using the differential equations in the different styles of planar, cylindrical, and spherical geometries. Fractional calculus theory (see [20,21,22,23,24,25,26,27,28]) was applied to model the diffusion problems in engineering, and fractional diffusion equation was discussed (see, e.g., [29,30,31,32,33,34,35,36]). The main aim of this paper is present in the mathematical structure of the Helmholtz and diffusion equations within local fractional derivative and to propose their forms in the Cantor-type cylindrical coordinates by using the Cantor-type cylindrical-coordinate method [46].
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