Helmholtz and Diffusion Equations Associated with Local Fractional Derivative Operators Involving the Cantorian and Cantor-Type Cylindrical Coordinates
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. 1. Introduction 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. There are many models for the anisotropic perfectly matched layers [1], the plasma source ion implantation [2], fractional paradigm and intermediate zones in electromagnetism [3, 4], fusion [5], reflectionless sponge layers [6], time-fractional heat conduction [7], singular boundary value problems [8], and so on (see also the references cited in each of these works). The Helmholtz equation was applied to deal with problems in such fields as electromagnetic radiation, seismology, transmission, and acoustics. Kre？ and Roach [9] discussed the transmission problems for the Helmholtz equation. Kleinman and Roach [10] studied the boundary integral equations for the three-dimensional Helmholtz equation. Karageorghis [11] presented the eigenvalues of the Helmholtz equation. Heikkola et al. [12] considered the parallel fictitious domain method for the three-dimensional Helmholtz equation. Fu and Mura [13] suggested the volume integrals of the inhomogeneous Helmholtz equation. Samuel and Thomas [14] proposed the fractional Helmholtz equation. Diffusion theory has become increasingly interesting and potentially useful in solids [15, 16]. Some applications of physics, such as superconducting alloys [17], lattice theory [18], and light diffusion in turbid material [19], were considered. Fractional calculus theory (see [20–28]) was applied to model the diffusion problems in engineering, and fractional diffusion equation was discussed (see, e.g., [29–36]). Recently, the local fractional calculus theory was applied to process the nondifferentiable phenomena in fractal domain (see [37–48] and the references cited therein). There are some local fractional models, such as the local fractional Fokker-Planck equation [37], the local fractional stress-strain relations [38], the local fractional heat conduction equation [45], wave equations on the Cantor sets [47], and the local fractional Laplace equation [48]. The main aim of this paper
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