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Ai-Min Yang, Xiao-Jun Yang, Zheng-Biao Li, "Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets", Abstract and Applied Analysis, vol. 2013, Article ID 351057, 5 pages, 2013. https://doi.org/10.1155/2013/351057
Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets
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.
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 , the Rach-Adomian-Meyers modified decomposition method , the variational iteration method [12, 13], the homotopy perturbation method [13, 14], the fractal Laplace and Fourier transforms , the homotopy analysis method , 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 , 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 , the Yang-Fourier transform [50, 51], and the Yang-Laplace transform .
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 where is a linear local operator with respect to , .
Moreover, there is a nondifferential series term where is a coefficient.
In view of (6), we may present the solution in the form Then, following (7), we have Hence, In view of (9), we have Hence, from (10) we can obtain a recursion; namely, with ; we arrive at the following relation: with ; we may rewrite (11) as By the recursion formulas, we can obtain the solution of (4) as The convergent condition is 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
with the initial condition
Following (12), we have recursive formula
Hence, we get
and so on.
Therefore, through (19) we get the solution
Example 2. Let us consider the diffusion equation on Cantor set with the initial condition Following (12), we get By using the recursive formula (23), we get consequently As a direct result of these recursive calculations, we arrive at
Example 3. Let us consider the following wave equation on Cantor sets:
with the initial condition
In view of (14), we obtain
Hence, using the relations (29), the recursive calculations yield
and so on.
Finally, we obtain
Example 4. Let us consider the wave equation on Cantor sets [26, 30]
where is a constant.
The initial condition is By using (14) we have Then, through the iterative relations (35), we have Therefore, we obtain where For more details concerning (38), we refer to [26–28].
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.
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.
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