Local Fractional Functional Method for Solving Diffusion Equations on Cantor Sets

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In this paper, we consider the local fractional diffusion equations defined on Cantor sets [5] given by subject to the initial-boundary conditions      (0, ) =  () , (0, ) =  () , where the local fractional partial derivatives denote and (, ), (), and () 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.

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    (/)  }.There are the following functional coefficients of (1) and ( 2), which are given as follows: where If we submit ( 14) into ( 1) and ( 2), then we have Taking the local fractional Laplace transform of (16) gives which leads to We can rewrite (18) as where The inverse formula local fractional Laplace transform of (19) gives +  ( 1 where Hence, the solution of (1) reads as follows: where

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: Making use of ( 14), we obtain the following formulas: which lead to the following parameters: Therefore, (23) gives the nondifferentiable solution of (1) with initial-boundary values ( 25) When  = 1, we get the nondifferentiable solution and its graph is shown in Figure 1.
Example 7. We present the initial-boundary values of (1) as Using the relation ( 14), we get (32) Using (23), we hence have the nondifferentiable solution of (1) with initial-boundary values (30), which is given as For  = 1, the nondifferentiable solution rewrites as follows: and its graph is illustrated in Figure 2.

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 highspeed trains and their control systems.