Abstract
The initial-boundary value problems for the local fractional differential equation are investigated in this paper. The local fractional Fourier series solutions with the nondifferential terms are obtained. Two illustrative examples are given to show efficiency and accuracy of the presented method to process the local fractional differential equations.
1. Introduction
In various fields of physics, mathematics, and engineering, because of the different operators, there are classical differential equations [1], fractional differential equation [2–4], 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 heat-balance 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 [14–16]. 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, 12–15]. 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.
2. 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] The following properties are present as follows [5].(i)If , then .(ii)If and , then .If is a bi-Lipschitz mapping, then we have [5, 12] such that In view of ((5)), we have such that 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.
Definition 2. If there is [5, 6, 12–15]
with , for and , then is called local fractional continuous at .
If is local fractional continuous on the interval , then we can write it in the form [5, 6, 12]
Definition 3. Local fractional derivative of of order at is defined as follows [5, 6, 12–15]:
where .
From ((12)) we can rewrite the local fractional derivative as
where
Definition 4. The partition of the interval is , , and with and . Local fractional integral of of order in the interval is given by [5, 6, 12–15]
Following ((14)), we have
where
If are Cantor sets, we can get the derivative and integral on Cantor sets.
Some properties of local fractional integrals are listed as follows:
Definition 5. Local fractional trigonometric Fourier series of is given by [6, 12–16]
for and .
The local fractional Fourier coefficients of ((19)) can be computed by
If , then we get
where the local fractional Fourier coefficients can be computed by
3. The Initial-Boundary Problems for the Local Fractional Differential Equation
In this section, we consider ((1)) with the various initial-boundary 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 , so that
For the given condition
we obtain
Thus, from ((33)) we deduce that
To satisfy the condition ((23)), ((34)) is written in the form
Then, we derive
Example 7. Let us consider ((1)) with the initial-boundary value condition, which becomes Following ((35)), we have where Hence, we get
4. 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.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.