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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 [

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 [

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 [

The local fractional derivative is defined as follows [

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

Let us consider the local fractional differential equation

In accordance with the results in [

Moreover, there is a nondifferential series term

In view of (

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

Let us consider the diffusion equation on Cantor set

Therefore, through (

Let us consider the diffusion equation on Cantor set

Let us consider the following wave equation on Cantor sets:

Finally, we obtain

Let us consider the wave equation on Cantor sets [

The initial condition is

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.