Abstract and Applied Analysis

Volume 2014 (2014), Article ID 872318, 5 pages

http://dx.doi.org/10.1155/2014/872318

Research Article

Local Fractional Derivative Boundary Value Problems for Tricomi Equation Arising in Fractal Transonic Flow

Xiao-Feng Niu,¹ Cai-Li Zhang,¹ Zheng-Biao Li,² and Yang Zhao³

¹College of Materials Science and Engineering, Taiyuan University of Technology, Taiyuan 030024, China

²College of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan 655011, China

³Department of Electronic and Information Technology, Jiangmen Polytechnic, Jiangmen 529090, China

Received 19 June 2014; Accepted 26 June 2014; Published 13 July 2014

Academic Editor: Xiao-Jun Yang

Copyright © 2014 Xiao-Feng Niu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The local fractional decomposition method is applied to obtain the nondifferentiable numerical solutions for the local fractional Tricomi equation arising in fractal transonic flow with the local fractional derivative boundary value conditions.

1. Introduction

The Tricomi equation [1] is the second-order linear partial differential equations of mixed type, which had been applied to describe the theory of plane transonic flow [2–7]. The Tricomi equation was used to describe the differentiable problems for the theory of plane transonic flow. However, for the fractal theory of plane transonic flow with nondifferentiable terms, the Tricomi equation is not applied to describe them. Recently, the local fractional calculus [8] was applied to describe the nondifferentiable problems, such as the fractal heat conduction [8, 9], the damped and dissipative wave equations in fractal strings [10], the local fractional Schrödinger equation [11], the wave equation on Cantor sets [12], the Navier-Stokes equations on Cantor sets [13], and others [14–19]. Recently, the local fractional Tricomi equation arising in fractal transonic flow was suggested in the form [19]

where the quantity is the nondifferentiable function, and the local fractional operator denotes [8]

where

The local fractional decomposition method [12] was used to solve the diffusion equation on Cantor time-space. The aim of this paper is to use the local fractional decomposition method to solve the local fractional Tricomi equation arising in fractal transonic flow with the local fractional derivative boundary value conditions. The structure of this paper is as follows. In Section 2, the local fractional integrals and derivatives are introduced. In Section 3, the local fractional decomposition method is suggested. In Section 4, the nondifferentiable numerical solutions for local fractional Tricomi equation with the local fractional derivative boundary value conditions are given. Finally, the conclusions are shown in Section 5.

2. Local Fractional Integrals and Derivatives

In this section, we introduce the basic theory of the local fractional integrals and derivatives [8–19], which are applied in the paper.

Definition 1 (see [8–19]). For , for and , we give the function , when

is valid.

Definition 2 (see [8–19]). Let , , , and with and , be a partition of the interval . The local fractional integral of in the interval is defined as

As the inverse operator of (6), local fractional derivative of of the order in the interval is presented as [8–19]

where

The formulas of local fractional derivative and integral, which appear in the paper, are valid [8]:

3. Analysis of the Method

In this section, we give the local fractional decomposition method [12]. We consider the following local fractional operator equation in the form

where is linear local fractional operators of the order with respect to and is the linear local fractional operators of order less than . We write (9) as

where the -th local fractional differential operator denotes

and the linear local fractional operators of order less than denote

Define the -fold local fractional integral operator

so that we obtain the local fractional iterative formula as follows:

which leads to

Therefore, for , we obtain the recurrence formula in the form

Finally, the solution of (9) reads

4. The Nondifferentiable Numerical Solutions

In this section, we discuss the nondifferentiable numerical solutions for the local fractional Tricomi equation arising in fractal transonic flow with the local fractional derivative boundary value conditions.

Example 1. We consider the initial-boundary value conditions for the local fractional Tricomi equation in the form [19]

Using (20)-(21), we structure the recurrence formula in the form

Hence, for , the first term of (22) reads

For the second term of (22) is given as

Hence, we obtain

Finally, the solution of (9) with the local fractional derivative boundary value conditions (19)–(21) can be written as

which is in accordance with the result from the local fractional variational iteration method [19].

Example 2. Let us consider the initial-boundary value conditions for the local fractional Tricomi equation in the form

In view of (27), we set up the recurrence formula in the form

Hence, from (28) we get the following equations:

Finally, we obtain the solution of (9) with the local fractional derivative boundary value conditions (27), namely,

whose graph is shown in Figure 1.

Figure 1: The plot of the solution of (9) with the local fractional derivative boundary value conditions (27) when .

Example 3. Let us consider the initial-boundary value conditions for the local fractional Tricomi equation in the form

Making use of (31), the recurrence formula can be written as

Appling (32) gives the following equations:

Finally, the solution of (9) with the local fractional derivative boundary value conditions (31) reads

and its graph is shown in Figure 2.

Figure 2: The plot of the solution of (9) with the local fractional derivative boundary value conditions (31) when .

5. Conclusions

In this work we discussed the nondifferentiable numerical solutions for the local fractional Tricomi equation arising in fractal transonic flow with the local fractional derivative boundary value conditions by using the local fractional decomposition method and their plots were also shown in the MatLab software.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by NSF110361048 of China and Yunnan province NSF Grant no. 2011FB090.

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