Local Fractional Variational Iteration Method for Local Fractional Poisson Equations in Two Independent Variables

Abstract

The local fractional Poisson equations in two independent variables that appear in mathematical physics involving the local fractional derivatives are investigated in this paper. The approximate solutions with the nondifferentiable functions are obtained by using the local fractional variational iteration method.

1. Introduction

As it is known the Poisson equation plays an important role in mathematical physics [1, 2]; that is, it describes the electrodynamics and intersecting interface (see, e.g., [3, 4] and the cited references therein). The solution of this equation was discussed by using different methods [5–9]. We notice that recently fractional Poisson equations based on fractional derivatives were analyzed in [10] and the existence and approximations of its solutions can be found in [11]. The Legendre wavelet method was used to find the fractional Poisson equation with Dirichlet boundary conditions [12]. In [13], the Dirichlet problem for the fractional Poisson’s equation with Caputo derivatives was reported. Furthermore, the fractional Poisson equation based on the shifted Grünwald estimate was obtained in [14].

The variational iteration method structured in [15–17] was applied to deal with the following type of equations: Helmholtz [18], Burger’s and coupled Burger’s [19], Klein-Gordon [20], KdV [21], the oscillation [22], Schrodinger [23], reaction-diffusion [24], diffusion equation [25], Bernoulli equation [26], and others. The extended variational iteration method, called the fractional variational iteration method, was developed and applied to handle some fractional differential equations within the modified Riemann-Liouville derivative [27–31]. More recently, the local fractional variational iteration method, initiated in [32], was used to find the nondifferentiable solutions for the heat-conduction [32], Laplace [33], damped and dissipative wave [34], Helmholtz [35] and Fokker-Planck [36] equations, the wave equation on Cantor sets [37], and the fractal heat transfer in silk cocoon hierarchy [38] with local fractional derivative.

We mention that developing a numerical algorithm for local fractional differential equations on Cantor set is not straightforward. Thus, in this paper, we deal with the local fractional Poisson equation in two independent variables, namely, 𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑦^2𝛼,∈[]×[],=𝑓𝑥,𝑦𝑥,𝑦0,+∞0,𝑙 (1) where the nondifferentiable functions 𝑢(𝑥,𝑦) and 𝑓(𝑥,𝑦) are adopted the local fractional differential operators and 𝛼 denotes the fractal dimension, subject to the initial and boundary conditions 𝑢()𝑢()𝑢𝑦,𝜕𝑥,0=0,𝑥,𝑙=0,0,𝑦=𝜑_𝛼𝜕𝑥^𝛼𝑢0,𝑦=𝜙𝑦. (2) We recall that the local fractional Laplace equation presented in [33] is a special case of the local fractional Poisson equation with source term 𝑓(𝑥,𝑦)=0. Taking all the above thinks into account, the aim of this paper is to find the nondifferentiable solutions for (1) with different conditions by utilizing the local fractional variational iteration algorithm.

The paper has the following organization. In Section 2 the concepts of local fractional complex derivatives and integrals are briefly reviewed. In Section 3 the local fractional variational iteration method is recalled. In Section 4 the nondifferentiable solutions for local fractional Poisson equations are presented. Finally, Section 5 outlines the main conclusions.

2. A Brief Review of the Local Fractional Calculus

Definition 1 (see [32–38]). Let the function 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏), if it satisfies the condition 𝑓(𝑥)−𝑓𝑥₀<𝜀^𝛼, (3) where |𝑥−𝑥₀|<𝛿, for 𝜀>0, 0<𝛼<1, and 𝜀∈𝑅.

Definition 2 (see [32–38]). Let 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏). The local fractional derivative of 𝑓(𝑥) of order 𝛼 is defined as 𝑑^𝛼𝑓𝑥₀𝑑𝑥^𝛼=Δ^𝛼𝑓(𝑥)−𝑓𝑥₀𝑥−𝑥₀^𝛼, (4) where Δ^𝛼𝑓(𝑥)−𝑓𝑥₀≅Γ(1+𝛼)𝑓(𝑥)−𝑓𝑥₀. (5) The formulas of local fractional derivatives of special functions [37] used in the paper are as follows: 𝑑^𝛼𝑑𝑥^𝛼𝑥^𝑛𝛼Γ()=𝑥1+𝑛𝛼Γ((^{(𝑛−1)𝛼})𝛼)𝐷1+𝑛−1,𝑛∈𝑁,_𝑥(𝛼)𝑎𝑔(𝑥)=𝑎𝐷_𝑥^(𝛼)𝑔(𝑥),𝐷_𝑥^(𝛼)𝐷_𝑥^(𝛼)𝑓(𝑥)=𝐷_𝑥^(2𝛼)𝑓(𝑥), (6) where 𝑔(𝑥) is a local fractional continuous function, 𝑎 is a constant, and 𝑁 is a set of positive integers.

Definition 3 (see [32–38]). Let 𝑓(𝑥)∈𝐶_𝛼[𝑎,𝑏]. The local fractional integral of 𝑓(𝑥) of order 𝛼 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, (7) where the partitions of the interval [𝑎,𝑏] are denoted by (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,Δ𝑡_𝑗,…}.
The formulas of local fractional integrals of special functions used in the paper are presented as follows [37]: ₀𝐼_𝑥^(𝛼)𝑎𝑔(𝑥)=𝑎₀𝐼_𝑥^(𝛼)𝑔(𝑥),₀𝐼_𝑡^(𝛼)(𝑡−𝑠)^𝛼𝑡^𝑛𝛼Γ()Γ()=𝑡1+𝛼1+𝑛𝛼Γ((^(𝑛+2)𝛼)𝛼),1+𝑛+2₀𝐼_𝑥^(𝛼)𝑥^{(𝑛−1)𝛼}Γ(()𝛼)=𝑥1+𝑛−1Γ(1+^𝑛𝛼𝑛𝛼),𝑛∈𝑁, (8) where 𝑔(𝑥) is a local fractional continuous function, 𝑎 is a constant, and 𝑁 is a set of positive integers.

3. Analysis of the Method

The local fractional variational iteration method structured in [32] was applied to deal with the local fractional differential equations arising in mathematical physics (see, e.g., [33–38]). In this section, we introduce the idea of the local fractional variational iteration method.

Let us consider the local fractional operator equation in the form 𝐿_𝛼𝑢+𝑁_𝛼𝑢=𝑔(𝑡), (9) where 𝐿_𝛼 and 𝑁_𝛼 are linear and nonlinear local fractional operators, respectively, and 𝑔(𝑡) is the source term within the nondifferentiable function.

Local fractional variational iteration algorithm reads as 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+₀𝐼_𝑡^(𝛼)𝜂𝐿_𝛼𝑢_𝑛(𝑠)+𝑁_𝛼𝑢_𝑛(𝑠)−𝑔(𝑠), (10) where 𝜂 is a fractal Lagrange multiplier and 𝐿_𝛼=𝜕^𝑛𝛼/𝜕𝑢^𝑛𝛼.

Therefore, a local fractional correction functional was structured as follows: 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+₀𝐼_𝑡^(𝛼)𝜉^𝛼𝐿_𝛼𝑢_𝑛(𝑠)+𝑁_𝛼𝑢_𝑛(𝑠)−𝑔(𝑠), (11) where 𝑢_𝑛 is considered as a restricted local fractional variation and 𝜂 is a fractal Lagrange multiplier. That is, 𝛿^𝛼𝑢_𝑛=0 [27, 30].

After the fractal Lagrangian multiplier is determined, for 𝑛≥0, the successive approximations 𝑢_𝑛+1 of the solution 𝑢 can be readily given by using any selective local fractional function 𝑢₀. Consequently, we obtain the solution in the following form: 𝑢=lim_𝑛→∞𝑢_𝑛. (12)

The local fractional variational method was compared with the fractional series expansion and decomposition technologies.

If 𝐿_𝛼=𝜕^2𝛼/𝜕𝑢^2𝛼, then we have the local fractional variational iteration formula [32–34, 36, 37] as follows: 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)−₀𝐼_𝑡^(𝛼)×(𝑠−𝑡)^𝛼Γ()𝐿1+𝛼^𝛼𝑢_𝑛(𝑠)+𝑁_𝛼𝑢_𝑛(𝑠)−𝑔(𝑠). (13) The above formula plays an important role in dealing with the 2𝛼-order local fractional differential equation with either linearity or nonlinearity.

4. The Nondifferentiable Solutions for Local Fractional Poisson Equations

In this section we investigate the nondifferentiable solutions for the local fractional Poisson equations in two independent variables with different initial-boundary conditions.

Example 1. We analyze the local fractional Poisson equation in the following form: 𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑦^2𝛼=𝑦^𝛼Γ(),1+𝛼 (14) subject to the initial and boundary conditions, namely, 𝑢(𝑥,0)=0,𝑢(𝑥,𝑙)=0,𝑢=𝑦0,𝑦^3𝛼Γ(),1+3𝛼𝜕^𝛼𝜕𝑥^𝛼𝑢0,𝑦=sin_𝛼𝑦^𝛼. (15)(16)(17)(18) In view of (17) and (18), we take the initial value given by 𝑢₀=𝑦𝑥,𝑦^3𝛼Γ(1+3𝛼)+sin_𝛼𝑦^𝛼𝑥^𝛼Γ().1+𝛼 (19) From (13), the local fractional iteration procedure is given by 𝑢_𝑛+1𝑥,𝑦=𝑢_𝑛𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢_𝑛𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢_𝑛(𝑠,𝑥)𝜕𝑦^2𝛼−𝑦^𝛼Γ().1+𝛼 (20) Making use of (19) and (20), we get the first approximation as follows: 𝑢₁𝑥,𝑦=𝑢₀𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₀𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₀(𝑠,𝑥)𝜕𝑦^2𝛼−𝑦^𝛼Γ()=𝑦1+𝛼_3𝛼Γ(1+3𝛼)+sin_𝛼𝑦^𝛼1_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼).1+2𝑖+1 (21) The second approximation can be written as 𝑢₂𝑥,𝑦=𝑢₁𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₁𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₁(𝑠,𝑥)𝜕𝑦^2𝛼−𝑦^𝛼Γ()=𝑦1+𝛼_3𝛼Γ(1+3𝛼)+sin_𝛼𝑦^𝛼2_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼).1+2𝑖+1 (22) The third approximation reads as 𝑢₃𝑥,𝑦=𝑢₂𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₂𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₂(𝑠,𝑥)𝜕𝑦^2𝛼−𝑦^𝛼Γ()=𝑦1+𝛼_3𝛼Γ(1+3𝛼)+sin_𝛼𝑦^𝛼3_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼).1+2𝑖+1 (23) The fourth approximation is as follows: 𝑢₄𝑥,𝑦=𝑢₃𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₃𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₃(𝑠,𝑥)𝜕𝑦^2𝛼−𝑦^𝛼Γ()=𝑦1+𝛼_3𝛼Γ(1+3𝛼)+sin_𝛼𝑦^𝛼4_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼)⋮1+2𝑖+1 (24) and so on.
Finally, by direct calculations we obtain 𝑢_𝑛=𝑦𝑥,𝑦^3𝛼Γ(1+3𝛼)+sin_𝛼𝑦^𝛼×^𝑛_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼).1+2𝑖+1 (25) Hence, we report the nondifferentiable solution of (14) 𝑢𝑥,𝑦=lim_𝑛→∞𝑢_𝑛=𝑦𝑥,𝑦_3𝛼Γ(1+3𝛼)+sin_𝛼𝑦^𝛼sin_𝛼𝑥^𝛼 (26) and its graph is shown in Figure 1.

Figure 1 

Plot of the nondifferentiable solution of (14) with the parameter 𝛼=ln2/ln3.

Example 2. Next we discuss the local fractional Poisson equations as 𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑦^2𝛼=𝐸_𝛼𝑦^𝛼, (27) with the initial and boundary conditions given as follows: 𝑢()𝑢()𝑢𝑥,0=0,𝑥,𝑙=0,0,𝑦=𝐸_𝛼𝑦^𝛼,𝜕^𝛼𝜕𝑥^𝛼𝑢0,𝑦=cos_𝛼𝑦^𝛼. (28) In view of (13), the local fractional iteration procedure becomes 𝑢_𝑛+1𝑥,𝑦=𝑢_𝑛𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢(𝑠,𝑥)𝜕𝑦^2𝛼−𝐸_𝛼𝑦^𝛼, (29) where the initial value is given by 𝑢₀𝑥,𝑦=𝐸_𝛼𝑦^𝛼+𝑥^𝛼Γ(1+𝛼)cos_𝛼𝑦^𝛼. (30) Making use of (29) and (30), the first approximation reads as follows: 𝑢₁𝑥,𝑦=𝑢₀𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₀𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₀(𝑠,𝑥)𝜕𝑦^2𝛼−𝐸_𝛼𝑦^𝛼=𝐸_𝛼𝑦^𝛼+cos_𝛼𝑦^𝛼1_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼).1+2𝑖+1 (31) The expression of the second approximation is as follows: 𝑢₂𝑥,𝑦=𝑢₁𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₁𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₁(𝑠,𝑥)𝜕𝑦^2𝛼−𝐸_𝛼𝑦^𝛼=𝐸_𝛼𝑦^𝛼+cos_𝛼𝑦^𝛼2_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼).1+2𝑖+1 (32) The third approximation becomes 𝑢₃𝑥,𝑦=𝑢₂𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₂𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₂(𝑠,𝑥)𝜕𝑦^2𝛼−𝐸_𝛼𝑦^𝛼=𝐸_𝛼𝑦^𝛼+cos_𝛼𝑦^𝛼3_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼).1+2𝑖+1 (33) The fourth approximation is given by 𝑢₄𝑥,𝑦=𝑢₃𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₃𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₃(𝑠,𝑥)𝜕𝑦^2𝛼−𝐸_𝛼𝑦^𝛼=𝐸_𝛼𝑦^𝛼+cos_𝛼𝑦^𝛼4_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼)⋮1+2𝑖+1 (34) Therefore, we get the nondifferentiable solution of (27) 𝑢𝑥,𝑦=lim_𝑛→∞𝑢_𝑛𝑥,𝑦=𝐸_𝛼𝑦^𝛼+cos_𝛼𝑦^𝛼sin_𝛼𝑥^𝛼, (35) and the corresponding graph is depicted in Figure 2.

Figure 2 

Plot of the nondifferentiable solution of (27) with the parameter 𝛼=ln2/ln3.

Example 3. The next particular case is the local fractional Poisson equations as follows: 𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑦^2𝛼=sin_𝛼𝑥^𝛼, (36) subject to the initial and boundary conditions 𝑢()𝑢()𝑢𝑥,0=0,𝑥,𝑙=0,0,𝑦=sin_𝛼𝑦^𝛼,𝜕^𝛼𝜕𝑥^𝛼𝑢0,𝑦=𝐸_𝛼−𝑦^𝛼. (37) We start with the initial value as follows: 𝑢₀(𝑥,𝑡)=sin_𝛼𝑦^𝛼+𝑥^𝛼Γ()𝐸1+𝛼^𝛼−𝑦^𝛼. (38) The local fractional iteration procedure leads us to 𝑢_𝑛+1𝑥,𝑦=𝑢_𝑛+𝑥,𝑦₀𝐼_𝑥^(𝛼)(𝑠−𝑥)^𝛼Γ()×𝜕1+𝛼_2𝛼𝑢𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢(𝑠,𝑥)𝜕𝑦^2𝛼−sin_𝛼𝑦^𝛼. (39) In view of (38) and (39), we obtain the following successive approximations: 𝑢₁𝑥,𝑦=𝑢₀𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₀𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₀(𝑠,𝑥)𝜕𝑦^2𝛼−sin_𝛼𝑦^𝛼=sin_𝛼𝑦^𝛼+𝐸_𝛼−𝑦^𝛼1_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼),𝑢1+2𝑖+1₂𝑥,𝑦=𝑢₁𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₁𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₁(𝑠,𝑥)𝜕𝑦^2𝛼−sin_𝛼𝑦^𝛼=sin_𝛼𝑦^𝛼+𝐸_𝛼−𝑦^𝛼2_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼),𝑢1+2𝑖+1₃𝑥,𝑦=𝑢₂𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₂𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₂(𝑠,𝑥)𝜕𝑦^2𝛼−sin_𝛼𝑦^𝛼=sin_𝛼𝑦^𝛼+𝐸_𝛼−𝑦^𝛼3_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼),𝑢1+2𝑖+1₄𝑥,𝑦=𝑢₃𝑥,𝑦+₀𝐼_𝑥^(𝛼)×(𝑠−𝑥)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₃𝑠,𝑦𝜕𝑠^2𝛼+𝜕^2𝛼𝑢₃(𝑠,𝑥)𝜕𝑦^2𝛼−sin_𝛼𝑦^𝛼=sin_𝛼𝑦^𝛼+𝐸_𝛼−𝑦^𝛼4_𝑖=0(−1)^𝑖𝑥^{(2𝑖+1)𝛼}Γ(()𝛼)⋮1+2𝑖+1 (40) and so on.
Thus, the nondifferentiable solution of (36) has the form 𝑢𝑥,𝑦=lim_𝑛→∞𝑢_𝑛𝑥,𝑦=sin_𝛼𝑦^𝛼+𝐸_𝛼−𝑦^𝛼sin_𝛼𝑥^𝛼, (41) and its graph is shown in Figure 3.

Figure 3 

Plot of the nondifferentiable solution of (36) with the parameter 𝛼=ln2/ln3.

5. Conclusions

The local fractional operators started to be deeply investigated during the last few years. One of the major problems is to find new methods and techniques to solve some given important local fractional partial differential equations on Cantor set. In this line of thought we consider that three local fractional Poisson equations with differential initial and boundary values were solved by using the local fractional variational iteration method. The graphs of the nondifferentiable solutions were also obtained.

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 the Natural Science Foundation of Henan Province, China.