Solving Initial-Boundary Value Problems for Local Fractional Differential Equation by Local Fractional Fourier Series Method

and Applied Analysis 3 Definition 5. Local fractional trigonometric Fourier series of f(t) is given by [6, 12–16]


Introduction
In various fields of physics, mathematics, and engineering, because of the different operators, there are classical differential equations [1], fractional differential equation [2][3][4], and local fractional differential equations [5,6].There are more techniques to achieve analytical approximations to the solutions to differential equations in mathematical physics, such as the decomposition method [7], the variational iteration method [8], the homotopy perturbation method [9], the heatbalance integral method [10], the Fourier transform [11], the Laplace transform [11], and the references therein.
Recently, a new Fourier series (local fractional Fourier series) via local fractional operator was proposed [6] and had various applications in the applied fields such as fractal wave problems in fractal string [12,13] and the heat-conduction problems arising in fractal heat transfer [14,15].For a detailed description of the local fractional Fourier series method, we refer the readers to the recent works [14][15][16].This is the main advantage of local fractional differential equations in comparison with classical integer-order and fractional-order models.
In the present paper we consider the local fractional differential equation: subject to the initial-boundary value conditions: where the operators are described by the local fractional differential operators [5,6,[12][13][14][15].The paper is organized as follows.In Section 2, the basic theory of the local fractional calculus and local fractional Fourier series is presented.In Section 3, we discuss the initial-boundary problems for local fractional differential equation.Finally, Section 4 is devoted to the conclusions.

Analysis of the Method
In this section, we present the basic theory of the local fractional calculus and analyze the local fractional Fourier series method.
Definition 1.Let  be a subset of the real line and be a fractal.The mass function   [, , ] can be written as [5]   [, , ] = lim The following properties are present as follows [5].If  : (, ) → (Ω  ,   ) is a bi-Lipschitz mapping, then we have [5,12] such that In view of ( 5), we have where  is the fractal dimension of .This result is directly deduced from fractal geometry and relates to the fractal coarse-grained mass function   [, , ], which reads [5,13] with where   is  dimensional Hausdorff measure.
The local fractional Fourier coefficients of ( 19) can be computed by If  0 = 1, then we get where the local fractional Fourier coefficients can be computed by

The Initial-Boundary Problems for the Local Fractional Differential Equation
In this section, we consider (1) with the various initialboundary conditions.
Example 6.The initial-boundary condition (2) becomes Let  =  in (1).Separation of the variables yields Setting we obtain Hence, we have their solutions, which read Therefore, a solution is written in the form where  = ,  = .
For the given condition there is  = 0, so that For the given condition we obtain sin  (    ) = 0, (32) Thus, from (33) we deduce that To satisfy the condition (23), (34) is written in the form Then, we derive ) . (40)

Conclusions
In this work, the initial-boundary value problems for the local fractional differential equation are discussed by using the local fractional Fourier series method.Analytical solutions for the local fractional differential equation with the nondifferentiable conditions are obtained.