Local Fractional Functional Method for Solving Diffusion Equations on Cantor Sets

Abstract

The analytical solutions for the diffusion equations on Cantor sets with the nondifferentiable terms are discussed by using the local fractional functional method, which is a coupling method for local fractional Fourier series and Laplace transform.

1. Introduction

The local fractional calculus [1, 2], as a new branch of fractional calculus, was successfully applied to describe the fractal problems from science and engineering. For example, the local fractional Fokker-Planck equation [3], the local fractional diffusion equations defined on Cantor sets [4, 5], the local fractional wave equation defined on Cantor sets [6, 7], the local fractional Korteweg-de Vries equation [8], the local fractional Schrödinger equation [9], local fractional Navier-Stokes equations on cantor sets [10], the local fractional Laplace equation [11], the local fractional heat-conduction equation [12–16], the local fractional differential equations arising in the fractal forest gap [17], and others [18–21] were discussed.

In this paper, we consider the local fractional diffusion equations defined on Cantor sets [5] given by 𝑢^2𝛼_𝑥𝑥=1𝑎^2𝛼𝑢^𝛼_𝑡, (1) subject to the initial-boundary conditions 𝜕^𝛼𝜕𝑥^𝛼𝑢()(𝑡)()(𝑡),𝑢()()0,𝑡=𝑔,𝑢0,𝑡=𝑓𝑥,0=𝑢𝑥,𝑙=0, (2) where the local fractional partial derivatives denote 𝑢^𝛼_𝑡=𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼,𝑢^2𝛼_𝑦𝑥=𝜕^2𝛼𝑢(𝑥,𝑡)𝜕𝑥^𝛼𝜕𝑦^𝛼, (3) and 𝑢(𝑥,𝑡),𝑔(𝑡), and 𝑓(𝑡) are the local fractional continuous functions. In the high-speed railway healthy monitor system, the problems of diffusion equations with the nondifferentiable terms always exist in fault diagnosing of high-speed trains and their control systems, so we solve this by the local fractional diffusion equations defined on Cantor sets. The local fractional function decomposition method structured in [11, 22], which is a coupling method of the local fractional Fourier series [21, 22] and the Yang-Laplace transform [14, 16, 18, 22], was used to solve the inhomogeneous local fractional wave equations defined on Cantor sets. The main aim of this paper is to discuss the local fractional diffusion equations defined on Cantor sets by the local fractional functional method.

The paper is organized as follows. In Section 2 the basic theory of the local fractional calculus and the Yang-Laplace transform were given. In Section 3, the local fractional functional method is analyzed. Section 4 presents the applications for the local fractional diffusion equations defined on Cantor sets. Finally, the conclusions are given in Section 5.

2. Preliminaries

In this section, we present the basic theory of the local fractional calculus and the local fractional Laplace transform.

Definition 1 (see [1, 5–7]). The local fractional derivative of 𝑓(𝑥) at 𝑥=𝑥₀ is given as follows: 𝐷^𝛼_𝑥𝑓𝑥₀=𝑑^𝛼𝑑𝑥^𝛼𝑓(𝑥)_𝑥=𝑥0=𝑓^(𝛼)(𝑥)=lim_𝑥→𝑥0Δ^𝛼𝑓(𝑥)−𝑓𝑥₀𝑥−𝑥₀^𝛼, (4) where Δ^𝛼(𝑓(𝑥)−𝑓(𝑥₀))≅Γ(𝛼+1)Δ(𝑓(𝑥)−𝑓(𝑥₀)).
The local fractional partial derivative of order 𝛼 is defined as follows [1]: 𝜕^𝛼𝜕𝑥^𝛼𝑓𝑥,𝑦_𝑥=𝑥0=𝑓^(𝛼)𝑥,𝑦=lim_𝑥→𝑥0Δ^𝛼𝑓𝑥,𝑦−𝑓𝑥₀,𝑦𝑥−𝑥₀^𝛼, (5) and the local fractional partial derivative of high order [1] is 𝜕^𝑘𝛼𝑓𝑥,𝑦𝑥^𝑘𝛼=^𝑘times𝜕^𝛼𝜕𝑥^𝛼𝜕^𝛼𝜕𝑥^𝛼⋯𝜕^𝛼𝜕𝑥^𝛼𝑓𝑥,𝑦, (6) where Δ^𝛼(𝑓(𝑥)−𝑓(𝑥₀))≅Γ(𝛼+1)Δ(𝑓(𝑥)−𝑓(𝑥₀)).

Definition 2 (see [1, 8–12]). Let us consider a partition of the interval [𝑎,𝑏], which is denoted as (𝑡_𝑗,𝑡_𝑗+1),𝑗=0,…,𝑁−1,𝑡₀=𝑎 and 𝑡_𝑁=𝑏 with Δ𝑡_𝑗=𝑡_𝑗+1−𝑡_𝑗 and Δ𝑡=max{Δ𝑡₀,Δ𝑡₁,…}. Local fractional integral of 𝑓(𝑥) in the interval [𝑎,𝑏] is defined as follows: _𝑎𝐼_𝑏^(𝛼)𝑓(𝑥)=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑡)(𝑑𝑡)^𝛼=1Γ(1+𝛼)lim_Δ𝑡→0^𝑁−1_𝑗=0𝑓𝑡_𝑗Δ𝑡_𝑗^𝛼. (7)

Definition 3 (see [1, 5, 11, 16, 21]). The Mittag Leffler, sine and cosine functions defined on Cantor sets are given as follows: 𝐸_𝛼𝑥^𝛼=^∞_𝑘=0𝑥^𝛼𝑘Γ(),1+𝑘𝛼sin_𝛼𝑥^𝛼=^∞_𝑘=0(−1)^𝑘𝑥^{𝛼(2𝑘+1)}Γ[(,1+𝛼2𝑘+1)]cos_𝛼𝑥^𝛼=^∞_𝑘=0(−1)^𝑘𝑥^2𝛼𝑘Γ(),1+2𝛼𝑘 (8) for 𝑥∈𝑅,0<𝛼<1.

Definition 4 (see [11, 20–22]). Let 𝑓(𝑥) be 2𝑙-periodic. For 𝑘∈𝑍, local fraction Fourier series of 𝑓(𝑥) is given as 𝑓(𝑥)=𝑎₀2+^∞_𝑘=1𝑎_𝑛cos_𝛼𝜋^𝛼(𝑘𝑥)^𝛼𝑙^𝛼+𝑏_𝑛sin_𝛼𝜋^𝛼(𝑘𝑥)^𝛼𝑙^𝛼, (9) where the local fraction Fourier coefficients are as follows: 𝑎_𝑛=2𝑙^𝛼𝑙₀𝑓(𝑥)cos_𝛼𝜋^𝛼(𝑘𝑥)^𝛼𝑙^𝛼(𝑑𝑥)^𝛼,𝑏_𝑛=2𝑙^𝛼𝑙₀𝑓(𝑥)sin_𝛼𝜋^𝛼(𝑘𝑥)^𝛼𝑙^𝛼(𝑑𝑥)^𝛼. (10)

Definition 5 (see [14, 16, 18, 22]). Let (1/Γ(1+𝛼))∫^∞₀|𝑓(𝑥)|(𝑑𝑥)^𝛼<𝑘<∞. The local fractional Laplace transform of 𝑓(𝑥) is given as 𝐿_𝛼𝑓(𝑥)=𝑓_𝑠^𝐿,𝛼(𝑠)=1Γ(1+𝛼)^∞₀𝐸_𝛼−𝑠^𝛼𝑥^𝛼𝑓(𝑥)(𝑑𝑥)^𝛼,0<𝛼≤1. (11) The inverse formula local fractional Laplace transform of 𝑓(𝑥) is given as [14, 16, 18, 22] 𝑓(𝑥)=𝐿_𝛼⁻¹𝑓_𝑠^𝐿,𝛼(𝑠=1)(2𝜋)^{𝛼𝛽+𝑖∞}_{𝛽−𝑖∞}𝐸_𝛼𝑠^𝛼𝑥^𝛼𝑓_𝑠^𝐿,𝛼(𝑠)(𝑑𝑠)^𝛼, (12) where 𝑓(𝑥) is local fractional continuous, 𝑠^𝛼=𝛽^𝛼+𝑖^𝛼∞^𝛼 and Re(𝑠)=𝛽>0.

There is the following formula [14, 16, 18, 22]: 𝐿_𝛼𝑦^(2𝛼)(𝑥)=𝑠^2𝛼𝐿_𝛼𝑦(𝑥)−𝑠^𝛼𝑦(0)−𝑓^(𝛼)(0). (13) The basic properties of the local fractional calculus and the local fractional Laplace transform were listed in [1, 14, 16, 18, 22].

3. Analysis of the Local Fractional Functional Method

In this section, we introduce the local fractional functional method for the local fractional diffusion equations defined on Cantor sets [11, 22].

Let us consider the nondifferentiable decomposition of the function with the nondifferentiable systems {sin_𝛼𝑛^𝛼(𝜋𝑡/𝑙)^𝛼}. There are the following functional coefficients of (1) and (2), which are given as follows: 𝑢(𝑥,𝑡)=^∞_𝑛=1𝐴_𝑛(𝑥)sin_𝛼𝑛^𝛼𝜋𝑡𝑙^𝛼,𝑔(𝑡)=^∞_𝑛=1𝐶_𝑛sin_𝛼𝑛^𝛼𝜋𝑡𝑙^𝛼,𝑓(𝑡)=^∞_𝑛=1𝐷_𝑛sin_𝛼𝑛^𝛼𝜋𝑡𝑙^𝛼, (14) where 𝐴_𝑛(𝑥)=2𝑙^𝛼1₀𝑢(𝑥,𝑡)sin_𝛼𝑛^𝛼𝜋𝑡𝑙^𝛼(𝑑𝑡)^𝛼,𝐶_𝑛=2𝑙^𝛼1₀𝑔(𝑡)sin_𝛼𝑛^𝛼𝜋𝑡𝑙^𝛼(𝑑𝑡)^𝛼,𝐷_𝑛=2𝑙^𝛼1₀𝑓(𝑡)sin_𝛼𝑛^𝛼𝜋𝑡𝑙^𝛼(𝑑𝑡)^𝛼. (15) If we submit (14) into (1) and (2), then we have 𝑑^2𝛼𝑑𝑥^2𝛼𝐴_𝑛(𝑥)=1𝑎^2𝛼𝐴_𝑛(𝑥)𝑛𝜋𝑙^𝛼,𝐴_𝑛^(𝛼)(0)=𝐶_𝑛,𝐴_𝑛(0)=𝐷_𝑛. (16) Taking the local fractional Laplace transform of (16) gives 𝐿_𝛼𝑑^2𝛼𝑑𝑥^2𝛼𝐴_𝑛(𝑥)=𝑠^2𝛼𝐿_𝛼𝐴_𝑛(𝑥)−𝑠^𝛼𝐴_𝑛(0)−𝐴_𝑛^(𝛼)(0), (17) which leads to 𝑠^2𝛼𝐿_𝛼𝐴_𝑛(𝑥)−𝑠^𝛼𝐷_𝑛−𝐶_𝑛=1𝑎^2𝛼𝑛𝜋𝑙^𝛼𝐿_𝛼𝐴_𝑛(𝑥). (18) We can rewrite (18) as 𝐴_𝑛(𝑠)=𝐿_𝛼𝐴_𝑛(𝑥=𝑠_𝛼)𝐷_𝑛+𝐶_𝑛𝑠^2𝛼−1/𝑎^2𝛼(𝑛𝜋/𝑙)^𝛼=𝐷_𝑛𝑠^𝛼−(1/𝑎^𝛼)(𝑛𝜋/𝑙)^𝛼/2+𝐶_𝑛−1/𝑎^𝛼(𝑛𝜋/𝑙)^𝛼/2𝐷_𝑛𝑠^2𝛼−1/𝑎^2𝛼(𝑛𝜋/𝑙)^𝛼=𝐷_𝑛𝑠^𝛼−(1/𝑎^𝛼)(𝑛𝜋/𝑙)^𝛼/21+𝑃𝑠^𝛼−(1/𝑎^𝛼)(𝑛𝜋/𝑙)^𝛼/2−1𝑠^𝛼+(1/𝑎^𝛼)(𝑛𝜋/𝑙)^𝛼/2, (19) where 𝐶𝑃=^𝑛−1/𝑎^𝛼(𝑛𝜋/𝑙)^𝛼/2𝐷_𝑛(1/𝑎^𝛼)(𝑛𝜋/𝑙)^𝛼/2. (20) The inverse formula local fractional Laplace transform of (19) gives 𝐴_𝑛(𝑥)=𝐿_𝛼⁻¹𝐴_𝑛(𝑠=1)(2𝜋)^{𝛼𝛽+𝑖∞}_{𝛽−𝑖∞}𝐸_𝛼𝑠^𝛼𝑥^𝛼𝐴_𝑛(𝑠)(𝑑𝑠)^𝛼=1(2𝜋)^𝛼×^𝛽+𝑖∞_{𝛽−𝑖∞}𝐸_𝛼𝑠^𝛼𝑥^𝛼𝐷_𝑛𝑠^𝛼−(1/𝑎^𝛼)(𝑛𝜋/𝑙)^𝛼/21+𝑃𝑠^𝛼−(1/𝑎^𝛼)(𝑛𝜋/𝑙)^𝛼/2−1𝑠^𝛼+(1/𝑎^𝛼)(𝑛𝜋/𝑙)^𝛼/2(𝑑𝑠)^𝛼=𝐷_𝑛𝐸_𝛼1𝑎^𝛼𝑛𝜋𝑙^𝛼/2𝑥^𝛼+𝑃𝐸_𝛼1𝑎^𝛼𝑛𝜋𝑙^𝛼/2𝑥^𝛼−𝑃𝐸_𝛼−1𝑎^𝛼𝑛𝜋𝑙^𝛼/2𝑥^𝛼, (21) where 𝐶𝑃=^𝑛−1/𝑎^𝛼(𝑛𝜋/𝑙)^𝛼/2𝐷_𝑛(1/𝑎^𝛼)(𝑛𝜋/𝑙)^𝛼/2. (22) Hence, the solution of (1) reads as follows: 𝑢()=𝑥,𝑡_∞𝑛=1𝐴_𝑛(𝑥)sin_𝛼𝑛^𝛼𝜋𝑡𝑙^𝛼=^∞_𝑛=1𝐷_𝑛𝐸_𝛼1𝑎^𝛼𝑛𝜋𝑙^𝛼/2𝑥^𝛼sin_𝛼𝑛^𝛼𝜋𝑡𝑙^𝛼+^∞_𝑛=1𝑃𝐸_𝛼1𝑎^𝛼𝑛𝜋𝑙^𝛼/2𝑥^𝛼sin_𝛼𝑛^𝛼𝜋𝑡𝑙^𝛼−^∞_𝑛=1𝑃𝐸_𝛼−1𝑎^𝛼𝑛𝜋𝑙^𝛼/2𝑥^𝛼sin_𝛼𝑛^𝛼𝜋𝑡𝑙^𝛼, (23) where 𝐶𝑃=^𝑛−1/𝑎^𝛼(𝑛𝜋/𝑙)^𝛼/2𝐷_𝑛(1/𝑎^𝛼)(𝑛𝜋/𝑙)^𝛼/2. (24)

4. The Exact Solutions for Local Fractional Diffusion Equations Defined on Cantor Sets

In this section we give two examples for initial boundary problems for local fractional diffusion equations defined on Cantor sets.

Example 6. The initial-boundary values of (1) read as follows: 𝜕^𝛼𝜕𝑥^𝛼𝑢(0,𝑡)=sin𝑡^𝛼()𝑢()(),𝑢0,𝑡=0,𝑥,0=𝑢𝑥,𝜋=0. (25) Making use of (14), we obtain the following formulas: sin𝑡^𝛼=^∞_𝑛=1𝐶_𝑛sin_𝛼𝑛^𝛼𝑡^𝛼,0=^∞_𝑛=1𝐷_𝑛sin_𝛼𝑛^𝛼𝑡^𝛼, (26) which lead to the following parameters: 𝐶_𝑛𝐶=1,𝑛=1,_𝑛𝐷=0,𝑛>1,_𝑛=0,𝑛≥1. (27) Therefore, (23) gives the nondifferentiable solution of (1) with initial-boundary values (25) 𝑢()=𝑥,𝑡_∞𝑛=1𝐴_𝑛(𝑥)sin_𝛼𝑛^𝛼𝑡^𝛼=^∞_𝑛=1𝐷_𝑛𝐸_𝛼1𝑎^𝛼𝑥^𝛼sin_𝛼𝑛^𝛼𝑡^𝛼+^∞_𝑛=1𝑃𝐸_𝛼1𝑎^𝛼𝑥^𝛼sin_𝛼𝑛^𝛼𝑡^𝛼−^∞_𝑛=1𝑃𝐸_𝛼−1𝑎^𝛼𝑥^𝛼sin_𝛼𝑛^𝛼𝑡^𝛼=𝑎^𝛼𝐸_𝛼1𝑎^𝛼𝑥^𝛼sin_𝛼𝑡^𝛼−𝑎^𝛼𝐸_𝛼−1𝑎^𝛼𝑥^𝛼sin_𝛼𝑡^𝛼. (28) When 𝑎=1, we get the nondifferentiable solution 𝑢(𝑥,𝑡)=𝐸_𝛼𝑥^𝛼sin_𝛼𝑡^𝛼−𝐸_𝛼−𝑥^𝛼sin_𝛼𝑡^𝛼, (29) and its graph is shown in Figure 1.

Figure 1 

The solution of (1) with initial-boundary value (25) when 𝑎=1 and 𝛼=ln2/ln3.

Example 7. We present the initial-boundary values of (1) as 𝜕^𝛼𝜕𝑥^𝛼𝑢(0,𝑡)=sin𝑡^𝛼,𝑢(0,𝑡)=sin𝑡^𝛼,𝑢(𝑥,0)=𝑢(𝑥,𝜋)=0. (30) Using the relation (14), we get0=^∞_𝑛=1𝐶_𝑛sin_𝛼𝑛^𝛼𝑡^𝛼,sin𝑡^𝛼=^∞_𝑛=1𝐷_𝑛sin_𝛼𝑛^𝛼𝑡^𝛼, (31) which reduce to 𝐶_𝑛𝐷=0,𝑛≥1,_𝑛𝐷=1,𝑛=1,_𝑛=0,𝑛>1. (32) Using (23), we hence have the nondifferentiable solution of (1) with initial-boundary values (30), which is given as 𝑢()=𝑥,𝑡_∞𝑛=1𝐴_𝑛(𝑥)sin_𝛼𝑛^𝛼𝑡^𝛼=^∞_𝑛=1𝐷_𝑛𝐸_𝛼1𝑎^𝛼𝑥^𝛼sin_𝛼𝑛^𝛼𝑡^𝛼+^∞_𝑛=1𝑃𝐸_𝛼1𝑎^𝛼𝑥^𝛼sin_𝛼𝑛^𝛼𝑡^𝛼−^∞_𝑛=1𝑃𝐸_𝛼−1𝑎^𝛼𝑥^𝛼sin_𝛼𝑛^𝛼𝑡^𝛼=𝐸_𝛼1𝑎^𝛼𝑥^𝛼sin_𝛼𝑡^𝛼−𝑎^𝛼𝐸_𝛼1𝑎^𝛼𝑥^𝛼sin_𝛼𝑡^𝛼+𝑎^𝛼𝐸_𝛼−1𝑎^𝛼𝑥^𝛼sin_𝛼𝑡^𝛼. (33) For 𝑎=1, the nondifferentiable solution rewrites as follows: 𝑢(𝑥,𝑡)=𝐸_𝛼−𝑥^𝛼sin_𝛼𝑡^𝛼, (34) and its graph is illustrated in Figure 2.

Figure 2 

The solution of (1) with initial-boundary value (30) when 𝑎=1 and 𝛼=ln2/ln3.

5. Conclusions

Local fractional calculus was applied to describe the physical problems because of nondifferentiable characteristics. In this work, the initial-boundary value problems for the diffusion equation on Cantor sets within the local fractional derivatives were investigated by using the local fractional functional method, which is a coupling method for local fractional Fourier series and Laplace transform based upon the nondifferentiable decomposition of the function with the nondifferentiable systems. The two examples are given to express the efficiency of the presented method and their graphs are also obtained. The results of this paper could provide the theory support to the problems diffusion equations with the nondifferentiable terms in health monitor of high-speed trains and their control systems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.