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

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] 𝑦^𝛼Γ()𝜕1+𝛼^2𝛼𝑢𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑦^2𝛼=0, (1) where the quantity 𝑢(𝑥,𝑦) is the nondifferentiable function, and the local fractional operator denotes [8] 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=Δ^𝛼𝑢(𝑥,𝑡)−𝑢𝑥,𝑡₀𝑡−𝑡₀^𝛼, (2) where Δ^𝛼𝑢(𝑥,𝑡)−𝑢𝑥,𝑡₀≅Γ(1+𝛼)𝑢(𝑥,𝑡)−𝑢𝑥,𝑡₀. (3) 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 |𝑥−𝑥₀|<𝛿, for 𝜀,𝛿>0 and 𝜀∈𝑅, we give the function 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏), when 𝑓(𝑥)−𝑓𝑥₀<𝜀^𝛼,0<𝛼≤1, (4) is valid.

Definition 2 (see [8–19]). Let (𝑡_𝑗,𝑡_𝑗+1), 𝑗=0,…,𝑁−1, 𝑡₀=𝑎, and 𝑡_𝑁=𝑏 with Δ𝑡_𝑗=𝑡_𝑗+1−𝑡_𝑗 and Δ𝑡=max{Δ𝑡₀,Δ𝑡₁,…}, be a partition of the interval [𝑎,𝑏]. The local fractional integral of 𝑓(𝑥) in the interval [𝑎,𝑏] is defined as _𝑎𝐼_𝑏^(𝛼)𝑓(𝑥)=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑡)(𝑑𝑡)^𝛼=1Γ(1+𝛼)lim_Δ𝑡→0^{𝑗=𝑁−1}_𝑗=0𝑓𝑡_𝑗Δ𝑡_𝑗^𝛼. (5) As the inverse operator of (6), local fractional derivative of 𝑓(𝑥) of the order 𝛼 in the interval (𝑎,𝑏) is presented as [8–19] 𝑑^𝛼𝑓𝑥₀𝑑𝑥^𝛼=𝐷_𝑥^(𝛼)𝑓𝑥₀=Δ^𝛼𝑓(𝑥)−𝑓𝑥₀𝑥−𝑥₀^𝛼, (6) where Δ^𝛼𝑓(𝑥)−𝑓𝑥₀≅Γ(1+𝛼)𝑓(𝑥)−𝑓𝑥₀. (7) The formulas of local fractional derivative and integral, which appear in the paper, are valid [8]: 𝑑^𝛼𝑑𝑥^𝛼𝑥^𝑛𝛼Γ()=𝑥1+𝑛𝛼Γ((^{(𝑛−1)𝛼})𝛼)𝑑1+𝑛−1,𝑛∈𝑁,_𝛼𝑑𝑥^𝛼𝐸_𝛼𝑥^𝛼=𝐸_𝛼𝑥^𝛼,𝑑^𝛼𝑑𝑥^𝛼sin_𝛼𝑥^𝛼=cos_𝛼𝑥^𝛼,𝑑^𝛼𝑑𝑥^𝛼cos_𝛼𝑥^𝛼=−sin_𝛼𝑥^𝛼,₀𝐼_𝑥^(𝛼)𝑥^{(𝑛−1)𝛼}Γ(()𝛼)=𝑥1+𝑛−1Γ(1+^𝑛𝛼𝑛𝛼),𝑛∈𝑁,₀𝐼_𝑥^(𝛼)cos_𝛼𝑥^𝛼=sin_𝛼𝑥^𝛼. (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 𝐿_𝛼⁽²⁾𝑢+𝑅_𝛼𝑢=0, (9) where 𝐿_𝛼⁽²⁾ is linear local fractional operators of the order 2𝛼 with respect to 𝑥 and 𝑅_𝛼 is the linear local fractional operators of order less than 2𝛼. We write (9) as 𝐿^(2𝛼)_𝑥𝑥𝑢+𝑅_𝛼𝑢=0, (10) where the 2𝛼-th local fractional differential operator denotes 𝐿_𝛼^(𝑛)=𝐿^(2𝛼)_𝑥𝑥=𝜕^2𝛼𝜕𝑥^2𝛼, (11) and the linear local fractional operators of order less than 2𝛼 denote 𝑅_𝛼=Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑦^2𝛼. (12) Define the 2𝛼-fold local fractional integral operator 𝐿_𝛼^(−2𝛼)𝑚(𝑠)=₀𝐼_𝑥0^(𝛼)𝐼_𝑥^(𝛼)𝑚(𝑠) (13) so that we obtain the local fractional iterative formula as follows: 𝐿_𝛼^(−2𝛼)𝐿^(2𝛼)_𝑥𝑥𝑢+𝐿_𝛼^(−2𝛼)𝐿_𝛼^(−2𝛼)𝑅_𝛼𝑢=0, (14) which leads to 𝑢(𝑥)=𝑢₀(𝑥)+𝐿_𝛼^(−2𝛼)𝐿_𝛼^(−2𝛼)𝑅_𝛼𝑢. (15) Therefore, for 𝑛≥0, we obtain the recurrence formula in the form 𝑢_𝑛+1(𝑥)=𝐿_𝛼⁽⁻²⁾𝑅_𝛼𝑢_𝑛(𝑥),𝑢₀(𝑥)=𝑟(𝑥). (16) Finally, the solution of (9) reads 𝑢(𝑥)=lim_𝑛→∞𝜙_𝑛(𝑥)=lim^∞_{𝑛→∞𝑛=0}𝑢_𝑛(𝑥). (17)

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] 𝑢0,𝑦=0,𝑢𝑙,𝑦=0,𝑢()=𝑥𝑥,0^𝛼Γ(),1+𝛼𝜕^𝛼𝑢(𝑥,0)𝜕𝑥^𝛼=𝑥^𝛼Γ().1+𝛼 (18)(19)(20)(21) Using (20)-(21), we structure the recurrence formula in the form 𝑢_𝑛+1𝑥,𝑦=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢_𝑛𝑥,𝑦𝜕𝑦^2𝛼,𝑢₀=𝑥𝑥,𝑦^𝛼Γ()+𝑥1+𝛼Γ(^𝛼)𝑦1+𝛼Γ(^𝛼).1+𝛼 (22) Hence, for 𝑛=0, the first term of (22) reads 𝑢₁𝑥,𝑦=0. (23) For 𝑛=1 the second term of (22) is given as 𝑢₂𝑥,𝑦=0. (24) Hence, we obtain 𝑢₀𝑥,𝑦=𝑢₁𝑥,𝑦=⋯=𝑢_𝑛𝑥,𝑦=0. (25) Finally, the solution of (9) with the local fractional derivative boundary value conditions (19)–(21) can be written as 𝑢𝑥,𝑦=lim_𝑛→∞𝜙_𝑛𝑥,𝑦=lim^∞_{𝑛→∞𝑛=0}𝑢_𝑛=𝑥𝑥,𝑦_𝛼Γ()+𝑥1+𝛼Γ(^𝛼)𝑦1+𝛼Γ(1+^𝛼𝛼) (26) 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 𝑢𝑢𝑢()𝜕0,𝑦=0,𝑙,𝑦=0,𝑥,0=0,_𝛼𝑢(𝑥,0)𝜕𝑥^𝛼=cos_𝛼𝑥^𝛼. (27) In view of (27), we set up the recurrence formula in the form 𝑢_𝑛+1𝑥,𝑦=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢_𝑛𝑥,𝑦𝜕𝑦^2𝛼,𝑢₀𝑥,𝑦=cos_𝛼𝑥^𝛼𝑦^𝛼Γ().1+𝛼 (28) Hence, from (28) we get the following equations: 𝑢₁𝑥,𝑦=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢₀𝑥,𝑦𝜕𝑦^2𝛼=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝜕𝑦^2𝛼cos_𝛼𝑥^𝛼𝑦^𝛼Γ()𝑢1+𝛼=0,₂𝑥,𝑦=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢₁𝑥,𝑦𝜕𝑦^2𝛼=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝜕𝑦^2𝛼(0𝑢)=0,₃𝑥,𝑦=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢₂𝑥,𝑦𝜕𝑦^2𝛼=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝜕𝑦^2𝛼(0𝑢)=0,₄𝑥,𝑦=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢₃𝑥,𝑦𝜕𝑦^2𝛼=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝜕𝑦^2𝛼(0⋮𝑢)=0,_𝑛𝑥,𝑦=0. (29) Finally, we obtain the solution of (9) with the local fractional derivative boundary value conditions (27), namely, 𝑢𝑥,𝑦=lim_𝑛→∞𝜙_𝑛𝑥,𝑦=lim^∞_{𝑛→∞𝑛=0}𝑢_𝑛𝑥,𝑦=cos_𝛼𝑥^𝛼𝑦^𝛼Γ(),1+𝛼 (30) 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 𝛼=ln2/ln3.

Example 3. Let us consider the initial-boundary value conditions for the local fractional Tricomi equation in the form 𝑢𝑢𝑢()=𝑥0,𝑦=0,𝑙,𝑦=0,𝑥,0^𝛼Γ(),𝜕1+𝛼_𝛼𝑢(𝑥,0)𝜕𝑥^𝛼=sin_𝛼𝑥^𝛼. (31) Making use of (31), the recurrence formula can be written as 𝑢_𝑛+1𝑥,𝑦=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢_𝑛𝑥,𝑦𝜕𝑦^2𝛼,𝑢₀=𝑥𝑥,𝑦^𝛼Γ(1+𝛼)+sin_𝛼𝑥^𝛼𝑦^𝛼Γ().1+𝛼 (32) Appling (32) gives the following equations: 𝑢₁𝑥,𝑦=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢₀𝑥,𝑦𝜕𝑦^2𝛼=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝜕𝑦^2𝛼×𝑥^𝛼Γ(1+𝛼)+sin_𝛼𝑥^𝛼𝑦^𝛼Γ()𝑢1+𝛼=0,₂𝑥,𝑦=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢₁𝑥,𝑦𝜕𝑦^2𝛼=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝜕𝑦^2𝛼(0𝑢)=0,₃𝑥,𝑦=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢₂𝑥,𝑦𝜕𝑦^2𝛼=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝜕𝑦^2𝛼(0𝑢)=0,₄𝑥,𝑦=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝑢₃𝑥,𝑦𝜕𝑦^2𝛼=𝐿_𝛼⁽⁻²⁾Γ(1+𝛼)𝑦^𝛼𝜕^2𝛼𝜕𝑦^2𝛼(0⋮𝑢)=0,_𝑛𝑥,𝑦=0. (33) Finally, the solution of (9) with the local fractional derivative boundary value conditions (31) reads 𝑢𝑥,𝑦=lim_𝑛→∞𝜙_𝑛𝑥,𝑦=lim^∞_{𝑛→∞𝑛=0}𝑢_𝑛=𝑥𝑥,𝑦_𝛼Γ(1+𝛼)+sin_𝛼𝑥^𝛼𝑦^𝛼Γ(),1+𝛼 (34) 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 𝛼=ln2/ln3.

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.