Helmholtz and Diffusion Equations Associated with Local Fractional Derivative Operators Involving the Cantorian and Cantor-Type Cylindrical Coordinates

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.

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. Kreß 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 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].

Our present investigation is structured as follows. In Section 2, the Helmholtz equation on the Cantor sets with local fractional derivative is investigated. The diffusion equation on the Cantor sets based upon the local fractional vector calculus is structured in Section 3. The Helmholtz and diffusion equations in the Cantor-type cylindrical coordinates are studied in Section 4. Finally, the conclusions are presented in Section 5.

2. The Helmholtz Equation on the Cantor Sets

In order to derive the Helmholtz equation on the Cantor sets, if the local fractional derivative is defined through [43–46] 𝑓^(𝛼)𝑥₀=𝑑^𝛼𝑓(𝑥)𝑑𝑥^𝛼_𝑥=𝑥0=lim_𝑥→𝑥0Δ^𝛼𝑓(𝑥)−𝑓𝑥₀𝑥−𝑥₀^𝛼 (1) with Δ^𝛼𝑓(𝑥)−𝑓𝑥₀≅Γ(1+𝛼)Δ𝑓(𝑥)−𝑓𝑥₀, (2) then the wave equation on the Cantor sets was suggested in [44] by ∇^2𝛼𝑢()=1𝑟,𝑡𝑎^2𝛼𝜕^2𝛼𝑢(𝑟,𝑡)𝜕𝑡^2𝛼, (3) where the local fractional Laplace operator is given by [43, 44, 48] ∇^2𝛼=𝜕^2𝛼𝜕𝑥^2𝛼+𝜕^2𝛼𝜕𝑦^2𝛼+𝜕^2𝛼𝜕𝑧^2𝛼, (4) where 1/𝑎^2𝛼 is a constant and 𝑢(𝑟,𝑡) is satisfied with local fractional continuous conditions (see [47]).

Using separation of variables in nondifferentiable functions, which begins by assuming that the fractal wave function 𝑢(𝑟,𝑡) may be separable, namely, 𝑢(𝑟,𝑡)=𝑀(𝑟)𝑇(𝑡), (5) we have ∇^2𝛼𝑀(𝑟)𝑀(𝑟)=1𝑎^2𝛼𝑇(𝑡)𝜕^2𝛼𝑇(𝑡)𝜕𝑡^2𝛼, (6) such that ∇^2𝛼𝑀(𝑟)+𝜔^2𝛼𝑀(𝑟)1=0,𝑎^2𝛼𝑇(𝑡)𝜕^2𝛼𝑇(𝑡)𝜕𝑡^2𝛼=−𝜔^2𝛼. (7)(8) In the three-dimensional Cantorian coordinate system, by following (7), we have 𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑥^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑦^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑧^2𝛼+𝜔^2𝛼𝑀𝑥,𝑦,𝑧=0, (9) where the operator is a local fractional derivative operator.

For the two-dimensional Cantorian coordinate system, the local fractional homogeneous Helmholtz equation is given by 𝜕^2𝛼𝑀𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦𝜕𝑦^2𝛼+𝜔^2𝛼𝑀𝑥,𝑦=0. (10) For a fractal dimension 𝛼=1, (9) becomes 𝜕²𝑀𝑥,𝑦,𝑧𝜕𝑥²+𝜕²𝑀𝑥,𝑦,𝑧𝜕𝑦²+𝜕²𝑀𝑥,𝑦,𝑧𝜕𝑧²+𝜔²𝑀𝑥,𝑦,𝑧=0, (11) which is the classical Helmholtz equation [10].

In view of (9), the inhomogeneous Helmholtz equation reads as follows: 𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑥^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑦^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑧^2𝛼+𝜔^2𝛼𝑀𝑥,𝑦,𝑧=𝑓𝑥,𝑦,𝑧, (12) where 𝑓(𝑥,𝑦,𝑧) is a local fractional continuous function.

In the two-dimensional Cantorian coordinate system, following (12), the local fractional inhomogeneous Helmholtz equation can be suggested by 𝜕^2𝛼𝑀𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦𝜕𝑦^2𝛼+𝜔^2𝛼𝑀𝑥,𝑦=𝑓𝑥,𝑦, (13) where 𝑓(𝑥,𝑦) is a local fractional continuous function.

We notice that the fractional Helmholtz equation was applied to deal with the differentiable wave equations in [14]. However, the Helmholtz equation with local fractional derivative arises in physical problems in such areas as, for example, fractal electromagnetic radiation, seismology, and acoustics, because their wave functions are the local fractional continuous functions (nondifferentiable functions). So, the Helmholtz equation on the Cantor sets can be used to describe the fractal electromagnetic radiation, the fractal seismology, the fractal acoustics, and so on.

3. Diffusion Equation on the Cantor Sets

In this section, we derive the diffusion equation on the Cantor sets with local fractional vector calculus [44].

Let us recall Fick’s law within the local fractional derivative, which was presented as 𝐉(𝑟,𝑡)=−𝐷𝜑∇^𝛼𝜑(𝑟,𝑡), (14) where 𝜑(𝑟,𝑡) and 𝐉(𝑟,𝑡) are local fractional continuous functions.

It is noticed that the flux of the diffusing material in any part of the fractal system is proportional to the local fractional density gradient. If the diffusion coefficient 𝐷(𝜑)=𝐷 is constant, the local fractional Fick law was suggested as [44] 𝐉(𝑟,𝑡)=−𝐷∇^𝛼𝜑(𝑟,𝑡), (15) which was expressed as [44] 𝐉(𝑟,𝑡)⋅𝑑𝐒^(𝛽)=−𝐷𝜑∇^𝛼𝜑(𝑟,𝑡)⋅𝑑𝐒^(𝛽), (16) where the local fractional vector integral is defined as [44] 𝐮𝑟_𝑃⋅𝑑𝐒^(𝛽)=lim^𝑁_{𝑁→∞𝑃=1}𝐮𝑟_𝑃⋅𝐧_𝑃Δ𝑆_𝑃^(𝛽), (17) with 𝑁 elements of area with a unit normal local fractional vector 𝐧_𝑃, Δ𝑆_𝑃^(𝛽)→0 as 𝑁→∞ for 𝛽=2𝛼, and 𝜑(𝑟,𝑡) is the density of the diffusing material in local fractional field.

The conservation of mass within local fractional vector operator was presented as [44] 𝑑^𝛼𝑑𝑡^𝛼𝜑(𝑟,𝑡)𝑑𝑉^(𝛾)=−𝐉(𝑟,𝑡)⋅𝑑𝐒^(𝛽), (18) where local fractional volume integral is given by [44] 𝐮𝑟_𝑃𝑑𝑉^(𝛾)=lim^𝑁_{𝑁→∞𝑃=1}𝐮𝑟_𝑃Δ𝑉_𝑃^(𝛾), (19) with 𝑁 elements of volume Δ𝑉_𝑃^(𝛾)→0 as 𝑁→∞ for 𝛾=(3/2)𝛽=3𝛼.

Following (18), and by using the divergence theorem of local fractional field [44], we have 𝑑^𝛼𝜑(𝑟,𝑡)𝑑𝑡^𝛼+∇^𝛼⋅𝐉(𝑟,𝑡)=0, (20) where 𝐉(𝑟,𝑡) is the flux of the diffusing material in local fractional field.

Submitting (14) into (20), we obtain 𝑑^𝛼𝜑(𝑟,𝑡)𝑑𝑡^𝛼+∇^𝛼−𝐷𝜑∇^𝛼𝜑(𝑟,𝑡)=0, (21) which is the so-called diffusion equation on the Cantor sets. This result differs from the fractional diffusion equation [29–36].

For the diffusion coefficient 𝐷(𝜑)=𝐷, (21) becomes 𝑑^𝛼𝜑(𝑟,𝑡)𝑑𝑡^𝛼=𝐷∇^2𝛼𝜑(𝑟,𝑡). (22) In the three-dimensional Cantorian coordinate system, following (22), we have 𝑑^𝛼𝜑𝑥,𝑦,𝑧,𝑡𝑑𝑡^𝛼𝜕=𝐷^2𝛼𝜕𝑥^2𝛼𝜑+𝜕𝑥,𝑦,𝑧,𝑡^2𝛼𝜕𝑦^2𝛼𝜑+𝜕𝑥,𝑦,𝑧,𝑡_2𝛼𝜕𝑧^2𝛼𝜑𝑥,𝑦,𝑧,𝑡. (23) In the two-dimensional Cantorian coordinate system, we get 𝑑^𝛼𝜑𝑥,𝑦,𝑡𝑑𝑡^𝛼𝜕=𝐷^2𝛼𝜕𝑥^2𝛼𝜑+𝜕𝑥,𝑦,𝑡^2𝛼𝜕𝑦^2𝛼𝜑𝑥,𝑦,𝑡. (24) In the one-dimensional Cantorian coordinate system, we obtain [48] 𝑑^𝛼𝜑(𝑥,𝑡)𝑑𝑡^𝛼𝜕=𝐷^2𝛼𝜕𝑥^2𝛼𝜑(𝑥,𝑡). (25) We notice that when fractal dimension 𝛼 is equal to 1, we get the classical diffusion equation [15, 16]. However, the diffusion equation on the Cantor sets with local fractional derivative is derived from local fractional field, whose quantities are local fractional continuous functions.

4. The Cantor-Type Cylindrical-Coordinate Method to the Helmholtz and Diffusion Equations on the Cantor Sets

Let us consider the Cantor-type cylindrical coordinates, which read as follows: 𝑥^𝛼=𝑅^𝛼cos_𝛼𝜃^𝛼,𝑦^𝛼=𝑅^𝛼sin_𝛼𝜃^𝛼,𝑧^𝛼=𝑧^𝛼, (26) with 𝑅∈(0,+∞), 𝑧∈(−∞,+∞), 𝜃∈(0,𝜋], and 𝑥^2𝛼+𝑦^2𝛼=𝑅^2𝛼.

We now have a local fractional vector given by 𝐫=𝑅^𝛼cos_𝛼𝜃^𝛼𝐞^𝛼₁+𝑅^𝛼sin_𝛼𝜃^𝛼𝐞^𝛼₂+𝑧^𝛼𝐞^𝛼₃=𝑟_𝑅𝐞^𝛼_𝑅+𝑟_𝜃𝐞^𝛼_𝜃+𝑟_z𝐞^𝛼_z, (27) such that [46] ∇^𝛼𝜙(𝑅,𝜃,𝑧)=𝐞^𝛼_𝑅𝜕^𝛼𝜕𝑅^𝛼𝜙+𝐞^𝛼_𝜃1𝑅^𝛼𝜕^𝛼𝜕𝜃^𝛼𝜙+𝐞^𝛼_𝑧𝜕^𝛼𝜕𝑧^𝛼∇𝜙,_2𝛼𝜙()=𝜕𝑅,𝜃,𝑧^2𝛼𝜕𝑅^2𝛼1𝜙+𝑅^2𝛼𝜕^2𝛼𝜕𝜃^2𝛼1𝜙+𝑅^𝛼𝜕^𝛼𝜕𝑅^𝛼𝜕𝜙+^2𝛼𝜕𝑧^2𝛼∇𝜙,_𝛼𝜕⋅𝐫=^𝛼𝑟_𝑅𝜕𝑅^𝛼+1𝑅^𝛼𝜕^𝛼𝑟_𝜃𝜕𝜃^𝛼+𝑟_𝑅𝑅^𝛼+𝜕^𝛼𝑟_𝑧𝜕𝑧^𝛼,∇^𝛼1×𝐫=𝑅^𝛼𝜕^𝛼𝑟_𝜃𝜕𝜃^𝛼−𝜕^𝛼𝑟_𝜃𝜕𝑧^𝛼𝐞^𝛼_𝑅+𝜕^𝛼𝑟_𝑅𝜕𝑧^𝛼−𝜕^𝛼𝑟_𝑧𝜕𝑅^𝛼𝐞^𝛼_𝜃+𝜕^𝛼𝑟_𝜃𝜕𝑅^𝛼+𝑟_𝑅𝑅^𝛼−1𝑅^𝛼𝜕^𝛼𝑟_𝑅𝜕𝜃^𝛼𝐞^𝛼_𝑧, (28)(29)(30)(31) where 𝐞^𝛼_𝑅=cos_𝛼𝜃^𝛼𝐞^𝛼₁+sin_𝛼𝜃^𝛼𝐞^𝛼₂,𝐞^𝛼_𝜃=−sin_𝛼𝜃^𝛼𝐞^𝛼₁+cos_𝛼𝜃^𝛼𝐞^𝛼₂,𝐞^𝛼_𝑧=𝐞^𝛼₃. (32) Submitting (29) into (9) and (12), it yields 𝜕^2𝛼𝑀(𝑅,𝜃,𝑧)𝜕𝑅^2𝛼+1𝑅^2𝛼𝜕^2𝛼𝑀(𝑅,𝜃,𝑧)𝜕𝜃^2𝛼+1𝑅^𝛼𝜕^𝛼𝑀(𝑅,𝜃,𝑧)𝜕𝑅^𝛼+𝜕^2𝛼𝑀(𝑅,𝜃,𝑧)𝜕𝑧^2𝛼+𝜔^2𝛼𝑀()𝜕𝑅,𝜃,𝑧=0,_2𝛼𝑀(𝑅,𝜃,𝑧)𝜕𝑅^2𝛼+1𝑅^2𝛼𝜕^2𝛼𝑀(𝑅,𝜃,𝑧)𝜕𝜃^2𝛼+1𝑅^𝛼𝜕^𝛼𝑀(𝑅,𝜃,𝑧)𝜕𝑅^𝛼+𝜕^2𝛼𝑀(𝑅,𝜃,𝑧)𝜕𝑧^2𝛼+𝜔^2𝛼𝑀(𝑅,𝜃,𝑧)=𝑓(𝑅,𝜃,𝑧), (33) which is the Helmholtz equation in the Cantor-type cylindrical coordinates.

In the like manner, from (23), we get 𝑑^𝛼𝜑(𝑅,𝜃,𝑧,𝑡)𝑑𝑡^𝛼𝜕=𝐷^2𝛼𝜑(𝑅,𝜃,𝑧,𝑡)𝜕𝑅^2𝛼+1𝑅^2𝛼𝜕^2𝛼𝜑(𝑅,𝜃,𝑧,𝑡)𝜕𝜃^2𝛼+1𝑅^𝛼𝜕^𝛼𝜑(𝑅,𝜃,𝑧,𝑡)𝜕𝑅^𝛼+𝜕^2𝛼𝜑(𝑅,𝜃,𝑧,𝑡)𝜕𝑧^2𝛼, (34) which is the diffusion equation in the Cantor-type cylindrical coordinates.

5. Concluding Remarks and Observations

In the present work, 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. By applying the Cantor-type cylindrical-coordinate method, we have also investigated the Helmholtz and diffusion equations on the Cantor sets in the Cantor-type cylindrical coordinates. Furthermore, we have presented two illustrative examples for the corresponding fractional Helmholtz and diffusion equations on the Cantor sets by using the Cantorian and Cantor-type cylindrical coordinates.

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 11102181) and in part by Natural Science Foundation of Hebei Province (no. A2012203117).