Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative

Abstract

We propose the local fractional function decomposition method, which is derived from the coupling method of local fractional Fourier series and Yang-Laplace transform. The forms of solutions for local fractional differential equations are established. Some examples for inhomogeneous wave equations are given to show the accuracy and efficiency of the presented technique.

1. Introduction

Fractional differential equations with arbitrary orders [1] have attracted more and more attention to their extensive applications in various areas, such as physics, applied mathematics, and biology [2–8]. As a result, great deal of methods for solving the fractional differential equations are developed [9–21], such as the heat balance integral method [9, 10], the homotopy analysis method [11], the variational iteration method [12], the homotopy decomposition method [13, 14], and the Adomian decomposition method [15, 16].

The fractional differential equations were considered in sense of the Caputo derivative, the Riemann-Liouville derivative, and the Grünwald-Letnikov derivative [17]. However, they do not deal with the nondifferentiable functions defined on Cantor sets. Local fractional derivative [18, 19] is the best method for describing the nondifferential problems defined on Cantor sets. For example, the heat equations arising in fractal transient conduction were investigated in [19–22]. The Helmholtz and diffusion equations on the Cantor sets within local fractional derivative were discussed [23]. The Navier-Stokes equations on Cantor sets were suggested in [24]. There are some methods for solving the local fractional differential equations, such as the local fractional variational iteration method [20], the Yang-Fourier transform [21], the Yang-Laplace transform [22], the local fractional Fourier series method [25], and the local fractional Adomian decomposition method [26].

In this paper, our aims are to present the coupling method of local fractional series method and Yang-Laplace transform, which is called as the local fractional function decomposition method, and to use it to solve the differential equations with local fractional derivative. The organization of the manuscript is as follows. In Section 2, the basic mathematical tools are introduced. In Section 3, the local fractional function decomposition method for solving the differential equations with local fractional derivative is investigated. In Section 4, several examples are considered. Finally, in Section 5 the conclusions are given.

2. Mathematical Fundamentals

In this section, we introduce the basic notions of local fractional continuity, local fractional derivative, local fractional Fourier series, and special function in fractal space [18, 19], which are used in the paper.

Definition 1. Suppose that there is [19] with , for and ; then is called local fractional continuous at and it is denoted by .

Definition 2. Suppose that the function satisfies condition (1), for ; it is so-called local fractional continuous on the interval , denoted by

Definition 3. In fractal space, let , local fractional derivative of of order at is given by [19] where .

Local fractional derivative of high order and local fractional partial derivative are defined, respectively, in the following forms [18, 19]:

Definition 4. In fractal space, the Mittage Leffler function, sine function, cosine function, hyperbolic sine function, and hyperbolic cosine function are, respectively, defined by [18, 19]

Definition 5. Let be -periodic. For and , the local fraction Fourier series of is defined as (see [18, 25]) where are the local fraction Fourier coefficients.

Definition 6. Let . The Yang-Laplace transforms of is given by [18, 22] where the latter integral converges and .

Definition 7. The inverse formula of the Yang-Laplace transforms of is given by [18, 22]: where ; fractal imaginary unit and .

3. Local Fractional Function Decomposition Method

In this section we will present the local fractional function decomposition method.

At first, we present the local fractional differential equation with constants , , , and with boundary and initial conditions Now we discuss the solution of (10).

According to the decomposition of the local fractional function, with respect to the system , the following functions coefficients can be given by where Substituting (12) into (10) implies that Suppose that the Yang-Laplace transforms of functions and are and , respectively. Then we obtain That is Hence, we have Let Hence, we get Then, making use of (8) and (9) and rearranging integration sequence, we have the following several formulas about and .

(I) Suppose that where

Then, we get When we get

(II) If then we have When we arrive at

(III) Let where .

Then, we have When we obtain The above results are the desired solutions.

4. Illustrative Examples

In order to illustrate the above results in Section 3, we give the following several examples.

Example 1. The local fractional Laplace differential equation is written in the following form [18, 19]: subjected to the boundary and initial conditions described by From (33), the final solution can be easily deduced as follows:

Example 2. We consider the following inhomogeneous wave equation with local fractional derivative: subjected to the boundary and initial conditions In order to find its solution, we suppose that which leads to Contrasting (37) with (35), we directly get According to (30) and (32), we can derive Conclusively, we get Thus, we obtain

Example 3. The inhomogeneous wave equation with local fractional differential operator is written in the following form: The boundary and initial conditions are described by In order to find the solution of (46), we set Hence, we get Let Making use of (21) and (23),we can write When we obtain Conclusively, we arrive at Hence, we obtain the solution of (46) in the following form:

Example 4. The inhomogeneous wave equation with local fractional differential operator is written in the form The boundary and initial conditions are presented as follows: Let We can write Obviously, we have From (25) and (27) we obtain Hence, the nondifferentiable solution of (56) reads as

5. Conclusions

In this work we proposed the local fractional function decomposition method. The applications of the methods for solving the inhomogeneous wave equations with local fractional derivative are discussed in detail. The new technique is an efficient mathematical tool for the scientists to deal with local fractional differential equations.

Conflict of Interests

The authors declare that have no conflict of interests regarding this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 609040410), the foundation and advanced technology research program of Henan province (112300410300), and the Nanyang Normal University (NYNU200749).