Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

Abstract

We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

1. Introduction

Fractional calculus theory [1–3] has been applied to a wide class of complex problems encompassing physics, biology, mechanics, and interdisciplinary areas [4–9]. Various methods, for example, the Adomian decomposition method [10], the Rach-Adomian-Meyers modified decomposition method [11], the variational iteration method [12, 13], the homotopy perturbation method [13, 14], the fractal Laplace and Fourier transforms [15], the homotopy analysis method [16], the heat-balance integral method [17–19], the fractional variational iteration method [20–22], the fractional subequation method [23, 24], and the generalized Exp-function method [25], have been utilized to solve fractional differential equations [3, 15].

The characteristics of fractal materials have local and fractal behaviors well described by nondifferential functions. However, the classic fractional calculus is not valid for differential equation on Cantor sets due to its no-local nature. In contrast, the local fractional calculus is one of the best candidates for dealing with such problems [26–44]. The local fractional calculus theory has played crucial applications in several fields, such as theoretical physics, transport problems in fractal media described by nondifferential functions. There are some versions of the local fractional calculus where different approaches in definition of the local fractional derivative exist, among them the local fractional derivative of Kolwankar et al. [32–38], the fractal derivative of Chen et al. [39, 40], the fractal derivative of Parvate et al. [41, 42], the modified Riemann-Liouville of Jumarie [43, 44], and versions described in [45–52].

In order to deal with local fractional ordinary and partial differential equations, there are some developed technologies, for example, the local fractional variational iteration method [45, 46], the local fractional Fourier series method [47, 48], the Cantor-type cylindrical-coordinate method [49], the Yang-Fourier transform [50, 51], and the Yang-Laplace transform [52].

The local fractional derivative is defined as follows [26–31, 45–52]: 𝑓^(𝛼)𝑥₀=𝑑^𝛼𝑓(𝑥)𝑑𝑥^𝛼_0𝑥=𝑥=lim_𝑥→𝑥0Δ^𝛼𝑓(𝑥)−𝑓𝑥₀𝑥−𝑥₀^𝛼, (1) where Δ^𝛼(𝑓(𝑥)−𝑓(𝑥₀))≅Γ(1+𝛼)Δ(𝑓(𝑥)−𝑓(𝑥₀)), and 𝑓(𝑥) is satisfied with the condition [26, 47] 𝑓(𝑥)−𝑓𝑥₀≤𝜏^𝛼𝑥−𝑥₀^𝛼 (2) so that [26–31] 𝑓(𝑥)−𝑓𝑥₀<𝜀^𝛼 (3) with 𝑈∶|𝑥−𝑥₀|<𝛿, for 𝜀,𝛿>0 and 𝜀,𝛿∈𝑅.

The main idea of this paper is to present the local fractional series expansion method for effective solutions of wave and diffusion equations on Cantor sets involving local fractional derivatives. The paper has been organized as follows. Section 2 gives a local fractional series expansion method. Some illustrative examples are shown in Section 3. The conclusions are presented in Section 4.

2. Analysis of the Method

Let us consider the local fractional differential equation 𝑢_𝑡^𝑛𝛼=𝐿_𝛼𝑢, (4) where 𝐿 is a linear local operator with respect to 𝑥, 𝑛∈{1,2}.

In accordance with the results in [28, 47], there are multiterm separated functions of independent variables 𝑡 and 𝑥, namely, 𝑢(𝑥,𝑡)=^∞_𝑖=0𝑇_𝑖(𝑡)𝑋_𝑖(𝑥), (5) where 𝑇_𝑖(𝑡) and 𝑋_𝑖(𝑥) are local fractional continuous functions.

Moreover, there is a nondifferential series term 𝑇_𝑖(𝑡)=𝑝_𝑖𝑡^𝑖𝛼Γ(),1+𝑖𝛼 (6) where 𝑝_𝑖 is a coefficient.

In view of (6), we may present the solution in the form 𝑢(𝑥,𝑡)=^∞_𝑖=0𝑝_𝑖𝑡^𝑖𝛼Γ()𝑋1+𝑖𝛼^𝑖(𝑥). (7) Then, following (7), we have 𝑢(𝑥,𝑡)=^∞_𝑖=0𝑡^𝑖𝛼Γ()𝑋1+𝑖𝛼^𝑖(𝑥). (8) Hence, 𝑢_𝑡^𝑛𝛼=^∞_𝑖=01Γ()𝑡1+𝑖𝛼^𝑖𝛼𝑋_𝑖+1(𝑥)=^∞_𝑖=01Γ()𝑡1+𝑖𝛼^𝑖𝛼𝑋_𝑖+𝑛(𝑥),𝐿_𝛼𝑢=𝐿_𝛼^∞_𝑖=0𝑡^𝑖𝛼Γ()𝑋1+𝑖𝛼^𝑖(𝑥)=^∞_𝑖=0𝑡^𝑖𝛼Γ()𝐿1+𝑖𝛼^𝛼𝑋_𝑖(𝑥). (9) In view of (9), we have ^∞_𝑖=01Γ()𝑡1+𝑖𝛼^𝑖𝛼𝑋_𝑖+𝑛(𝑥)=^∞_𝑖=0𝑡^𝑖𝛼Γ()𝐿1+𝑖𝛼^𝛼𝑋_𝑖(𝑥). (10) Hence, from (10) we can obtain a recursion; namely, 𝑋_𝑖+𝑛(𝑥)=𝐿_𝛼𝑋_𝑖(𝑥), (11) with 𝑛=1; we arrive at the following relation: 𝑋_𝑖+1(𝑥)=𝐿_𝛼𝑋_𝑖(𝑥), (12) with 𝑛=2; we may rewrite (11) as 𝑋_𝑖+2(𝑥)=𝐿_𝛼𝑋_𝑖(𝑥). (13) By the recursion formulas, we can obtain the solution of (4) as 𝑢(𝑥,𝑡)=^∞_𝑖=0𝑡^𝑖𝛼Γ()𝑋1+𝑖𝛼^𝑖(𝑥). (14) The convergent condition is lim_𝑛→∞𝑡^𝑖𝛼Γ()𝑋1+𝑖𝛼^𝑖(𝑥)=0. (15) This approach is termed the local fractional series expansion method (LFSEM)

3. Applications to Wave and Diffusion Equations on Cantor Sets

In this section, four examples for wave and diffusion equations on Cantor sets will demonstrate the efficiency of LFSEM.

Example 1. Let us consider the diffusion equation on Cantor set 𝑢^𝛼_𝑡(𝑥,𝑡)−𝑢_𝑥^2𝛼(𝑥,𝑡)=0,0<𝛼≤1 (16) with the initial condition 𝑢()=𝑥𝑥,0^𝛼Γ().1+𝛼 (17) Following (12), we have recursive formula 𝑋_𝑖+1(x)=𝜕^2𝛼𝑋_𝑖(𝑥)𝜕𝑥^2𝛼,𝑋₀(𝑥)=𝑥^𝛼Γ().1+𝛼 (18) Hence, we get 𝑋₀(𝑥)=𝑥^𝛼Γ(),𝑋1+𝛼₁(𝑥)𝑋=0,₂(𝑥)⋮=0, (19) and so on.
Therefore, through (19) we get the solution 𝑢()=𝑥𝑥,𝑡^𝛼Γ().1+𝛼 (20)

Example 2. Let us consider the diffusion equation on Cantor set 𝑢^𝛼_𝑡()−𝑥𝑥,𝑡^2𝛼Γ(1+2𝛼)⋅𝑢_𝑥^2𝛼(𝑥,𝑡)=0,0<𝛼≤1 (21) with the initial condition 𝑢()=𝑥𝑥,0^2𝛼Γ().1+2𝛼 (22) Following (12), we get 𝑋_𝑖+1(𝑥)=𝑥^2𝛼Γ()𝜕1+2𝛼^2𝛼𝑋_i(𝑥)𝜕𝑥^2𝛼,𝑋₀(𝑥)=𝑥^2𝛼Γ().1+2𝛼 (23) By using the recursive formula (23), we get consequently 𝑋₀(𝑥)=𝑥^2𝛼Γ(),𝑋1+2𝛼₁(𝑥)=𝑥^2𝛼Γ(),𝑋1+2𝛼₂(𝑥)=𝑥^2𝛼Γ(),⋮1+2𝛼 (24) As a direct result of these recursive calculations, we arrive at 𝑢()=𝑥𝑥,𝑡^2𝛼Γ(1+2𝛼)^∞_𝑖=0𝑡^𝑖𝛼Γ()=𝑥1+𝑖𝛼Γ(^2𝛼)𝐸1+2𝛼^𝛼𝑡^𝛼. (25)

Example 3. Let us consider the following wave equation on Cantor sets: 𝑢_𝑡^2𝛼()−𝑥𝑥,𝑡^2𝛼Γ(1+2𝛼)⋅𝑢_𝑥^2𝛼(𝑥,𝑡)=0,0<𝛼≤1 (26) with the initial condition 𝑢()=𝑥𝑥,0^𝛼Γ().1+𝛼 (27) In view of (14), we obtain 𝑋_i+2(𝑥)=𝑥^2𝛼Γ()𝜕1+2𝛼^2𝛼𝑋_𝑖(𝑥)𝜕𝑥^2𝛼,𝑋₀(𝑥)()=𝑥=𝑢𝑥,0^𝛼Γ(),𝑋1+𝛼_𝑖+2(𝑥)=𝑥^2𝛼Γ()𝜕1+2𝛼^2𝛼𝑋_𝑖(𝑥)𝜕x^2𝛼,𝑋₁(𝑥)=𝑢_𝑥^(𝛼)(𝑥,0)=1. (28) Hence, using the relations (29), the recursive calculations yield 𝑋₀(𝑥)=𝑥^𝛼Γ(),𝑋1+𝛼₁(𝑥)𝑋=1,₂(𝑥)𝑋=0,₃(𝑥)𝑋=0,₄(𝑥)⋮=0, (29)(30) and so on.
Finally, we obtain 𝑢()=𝑥𝑥,𝑡^𝛼Γ()+𝑡1+𝛼Γ(^2𝛼).1+2𝛼 (31)

Example 4. Let us consider the wave equation on Cantor sets [26, 30] 𝑢_𝑡^2𝛼(𝑥,𝑡)−cu_𝑥^2𝛼(𝑥,𝑡)=0,0<𝛼≤1, (32) where c is a constant.
The initial condition is 𝑢(𝑥,0)=𝐸_𝛼𝑥^𝛼. (33) By using (14) we have 𝑋_𝑖+2(𝑥)𝜕=𝑐^2𝛼𝑋_𝑖(𝑥)𝜕𝑥^2𝛼,𝑋₀(𝑥)=𝑢(𝑥,0)=𝐸_𝛼𝑥^𝛼,𝑋_𝑖+2(𝑥)𝜕=𝑐^2𝛼𝑋_𝑖(𝑥)𝜕𝑥^2𝛼,𝑋₀(𝑥)=𝑢_𝑥^(𝛼)(𝑥,0)=𝐸_𝛼𝑥^𝛼. (34) Then, through the iterative relations (35), we have 𝑋₀(𝑥)=𝐸_𝛼𝑥^𝛼,𝑋₁(𝑥)=𝐸_𝛼𝑥^𝛼,𝑋₂(𝑥)=𝑐𝐸_𝛼𝑥^𝛼,𝑋₃(𝑥)=𝑐𝐸_𝛼𝑥^𝛼,𝑋₄(𝑥)=𝑐²𝐸_𝛼𝑥^𝛼,⋮ (35)(36) Therefore, we obtain 𝑢(𝑥,𝑡)=𝐸_𝛼𝑥^𝛼∞_𝑖=0𝑐^𝑖𝑡^2𝑖𝛼Γ(1+2𝑖𝛼)+𝐸_𝛼𝑥^𝛼∞_𝑖=0𝑐^𝑖𝑡^{(2𝑖+1)𝛼}Γ(1+(2𝑖+1)𝛼)=𝐸_𝛼𝑥^𝛼cosh_𝛼𝑐𝑡^𝛼+sin_𝛼𝑐𝑡^𝛼, (37) where cosh_𝛼𝑡^𝛼=^∞_𝑖=0𝑡^2𝑖𝛼Γ(),1+2𝑖𝛼sinh_𝛼𝑡^𝛼=^∞_𝑖=0𝑡^{(2𝑖+1)𝛼}Γ(()𝛼).1+2𝑖+1 (38) For more details concerning (38), we refer to [26–28].

4. Conclusions

In this work, the local fractional series expansion method is demonstrated as an effective method for solutions of a wide class of problems. Analytical solutions of the wave and diffusion equations on Cantor sets involving local fractional derivatives are successfully developed by recurrence relations resulting in convergent series solutions. In this context, the suggested method is a potential tool for development of approximate solutions of local fractional differential equations with fractal initial value conditions, which, of course, draws new problems beyond the scope of the present work.

Acknowledgments

The first author was supported by the National Scientific and Technological Support Projects (no. 2012BAE09B00), the National Natural Science Foundation of China (no. 11126213 and no. 61170317), and the National Natural Science Foundation of Hebei Province (no. E2013209123). The third author is supported in part by NSF11061028 of China and Yunnan Province NSF Grant no. 2011FB090.