The Nondifferentiable Solution for Local Fractional Tricomi Equation Arising in Fractal Transonic Flow by Local Fractional Variational Iteration Method

which is structured by Tricomi [10]. The Tricomi equation was used to describe the transonic flow [10–22]. Local fractional variational iteration method first structured in [4] was an efficient tool to solve the local fractional differential equations, such as the fractal heat equation [4], the damped and dissipative wave equation in fractal strings [5], the wave equation on Cantor sets [6], the local fractional Poisson equation [7], the local fractional Laplace equation [8], and the local fractional Helmholtz equation [9]. The aim of this paper is to use the local fractional variational iteration method to deal with the local fractional Tricomi equation which arises in fractal transonic flow. The paper is organized as follows. In Section 2, the local fractional calculus theory is introduced. In Section 3, the local fractional variational iteration method is presented. In Section 4, the local fractional Tricomi equation is discussed. Finally, the conclusions are presented in Section 5.

Local fractional variational iteration method first structured in [4] was an efficient tool to solve the local fractional differential equations, such as the fractal heat equation [4], the damped and dissipative wave equation in fractal strings [5], the wave equation on Cantor sets [6], the local fractional Poisson equation [7], the local fractional Laplace equation [8], and the local fractional Helmholtz equation [9].The aim of this paper is to use the local fractional variational iteration method to deal with the local fractional Tricomi equation which arises in fractal transonic flow.The paper is organized as follows.In Section 2, the local fractional calculus theory is introduced.In Section 3, the local fractional variational iteration method is presented.In Section 4, the local fractional Tricomi equation is discussed.Finally, the conclusions are presented in Section 5.

Local Fractional Calculus Theory
In this section, we present the local fractional calculus theory, which is used in the present paper.

Local Fractional Variational Iteration Method
In this section, we introduce the local fractional variational iteration method.In order to show it, we consider the following local fractional operator equation: where  (2)   denotes the linear local fractional differential operator and   denotes the linear local fractional differential operators of order less than  (2)   .

The Initial-Boundary Value Problems for Local Fractional Tricomi Equation
In this section, we discuss the initial-boundary value problems for local fractional Tricomi equation.
(, ) = 0, From ( 18), (20), and ( 21), we have where the initial value is given by From ( 22), we present the first approximate formula as follows: and its graph is shown in Figure 1.The second approximate term is and its graph is given in Figure 2. The third approximation is presented as follows: and its graph is illustrated in Figure 3.
The fourth approximation reads as follows:

= 𝑥
and its graph is presented in Figure 4.The fifth approximation is as follows: and its graph is shown in Figure 5.
After successive iterative processes, we obtain the nondifferentiable series solution as follows: which is the local fractional divergent series.Therefore, we can obtain the approximate solution.
Example 2. The initial-boundary value conditions for the local fractional Tricomi equation are presented as follows: (, ) = 0, (31) In view of ( 16), (32), and (33), we obtain the local fractional iterative formula as follows: with the initial value suggested as follows:  and its graph is shown in Figure 6.

Conclusions
The initial-boundary value problems for local fractional Tricomi equation arising in fractal transonic flow based upon the local fractional derivatives are discussed.The solutions with nondifferentiable terms are obtained by using the local fractional variational iteration method and their graphs are also given to show the implement of the present method.

Example 1 .
Let us consider the initial-boundary value conditions for the local fractional Tricomi equation as follows: