Local Fractional Laplace Variational Iteration Method for Solving Linear Partial Differential Equations with Local Fractional Derivative

Abstract

The local fractional Laplace variational iteration method was applied to solve the linear local fractional partial differential equations. The local fractional Laplace variational iteration method is coupled by the local fractional variational iteration method and Laplace transform. The nondifferentiable approximate solutions are obtained and their graphs are also shown.

1. Introduction

Fractional calculus [1] has successfully been used to study the mathematical and physical problems arising in science and engineering. Fractional differential equations are applied to describe the dynamical systems in physics and engineering (see [2, 3]). It is one of the hot topics for finding the solutions for the fractional differential equations for scientists and engineers. There are many analytical and numerical methods for solving them, such as the spectral Legendre-Gauss-Lobatto collocation method [4], the shifted Jacobi-Gauss-Lobatto collocation method [5], the variation iteration method [6], the heat-balance integral method [7], the Adomian decomposition method [8], the finite element method [9], and the finite difference method [10].

The above methods did not deal with some nondifferentiable problems arising in mathematics and physics (see [11–13]). Local fractional calculus (see [12–14] and the cited references) is the best choice to deal with them. Some methods for solving the local fractional differential equations were suggested, such as the Cantor-type cylindrical-coordinate method [15], the local fractional variational iteration method [16, 17], the local fractional decomposition method [18], the local fractional series expansion method [19], the local fractional Laplace transform method [20], and local fractional function decomposition method [21, 22]. More recently, the coupling schemes for local fractional variational iteration method and Laplace transform were suggested in [23]. However, the results are very little. In this paper, our aim is to use the local fractional Laplace variational iteration method to solve the linear local fractional partial differential equations. The structure of the paper is suggested as follows. In Section 2 the basic theory of local fractional calculus and local fractional Laplace transform are introduced. Section 3 is devoted to the local fractional Laplace variational iteration method. In Section 4, the four examples for the local fractional partial differential equations are given. Finally, the conclusions are considered in Section 5.

2. Local Fractional Calculus and Local Fractional Laplace Transform

In this section, we present the basic theory of local fractional calculus and concepts of local fractional Laplace transform.

The local fractional derivative of 𝑓(𝑥) of order 𝛼 is defined as [12–15] 𝑑^𝛼𝑓𝑥₀𝑑𝑥^𝛼=Δ^𝛼𝑓(𝑥)−𝑓𝑥₀𝑥−𝑥₀^𝛼, (1) where Δ^𝛼𝑓(𝑥)−𝑓𝑥₀≅Γ(1+𝛼)𝑓(𝑥)−𝑓𝑥₀. (2) The local fractional integral of 𝑓(𝑥) of order 𝛼 in the interval [𝑎,𝑏] is defined as [12–14, 16–23] _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, (3) where the partitions of the interval [𝑎,𝑏] are (𝑡_𝑗,𝑡_𝑗+1), with Δ𝑡_𝑗=𝑡_𝑗+1−𝑡_𝑗, 𝑡₀=𝑎, 𝑡_𝑁=𝑏, and Δ𝑡=max{Δ𝑡₀,Δ𝑡₁,Δ𝑡_𝑗,…}, 𝑗=0,…,𝑁−1.

The local fractional series of nondifferentiable function used in this paper are presented as follows [12–14]: 𝐸_𝛼𝑥^𝛼=^∞_𝑘=0𝑥^𝛼𝑘Γ(),1+𝑘𝛼sin_𝛼𝑥^𝛼=^∞_𝑘=0(−1)^𝑘𝑥^{(2𝑘+1)𝛼}Γ[()𝛼],1+2𝑘+1cos_𝛼𝑥^𝛼=^∞_𝑘=0(−1)^𝑘𝑥^2𝛼𝑘Γ(),1+2𝛼𝑘sinh_𝛼𝑥^𝛼=^∞_𝑘=0𝑥^{(2𝑘+1)𝛼}Γ[()𝛼],1+2𝑘+1cosh_𝛼𝑥^𝛼=^∞_𝑘=0𝑥^2𝛼𝑘Γ().1+2𝛼𝑘 (4) The properties of local fractional derivatives and integral of nondifferentiable functions are given by [12, 13] 𝑑^𝛼𝑑𝑥^𝛼𝑥^𝑛𝛼Γ()=𝑥1+𝑛𝛼Γ((^{(𝑛−1)𝛼})𝛼),𝑑1+𝑛−1_𝛼𝑑𝑥^𝛼𝐸_𝛼𝑥^𝛼=𝐸_𝛼𝑥^𝛼,𝑑^𝛼𝑑𝑥^𝛼sin_𝛼𝑥^𝛼=cos_𝛼𝑥^𝛼,𝑑^𝛼𝑑𝑥^𝛼cos_𝛼𝑥^𝛼=−sin_𝛼𝑥^𝛼,𝑑^𝛼𝑑𝑥^𝛼sinh_𝛼𝑥^𝛼=cosh_𝛼𝑥^𝛼,𝑑^𝛼𝑑𝑥^𝛼cosh_𝛼𝑥^𝛼=−sinh_𝛼𝑥^𝛼,₀𝐼_𝑥^(𝛼)𝑥^𝑛𝛼Γ()=𝑥1+𝑛𝛼Γ((^(𝑛+1)𝛼)𝛼).1+𝑛+1 (5) The local fractional Laplace transform is defined as [12, 20–22] 𝐿_𝛼𝑓(𝑥)=𝑓_𝑠^𝐿,𝛼(𝑠)=1Γ()×1+𝛼_∞0𝐸_𝛼−𝑠^𝛼𝑥^𝛼𝑓(𝑥)(𝑑𝑥)^𝛼,0<𝛼≤1, (6) where 𝑓(𝑥) is local fractional continuous and 𝑠^𝛼=𝛽^𝛼+𝑖^𝛼∞^𝛼.

The inverse formula of local fractional Laplace transform is defined as [12, 20–22] 𝑓(𝑥)=𝐿_𝛼⁻¹𝑓_𝑠^𝐿,𝛼(𝑠=1)(2𝜋)^{𝛼𝛽+𝑖∞}_{𝛽−𝑖∞}𝐸_𝛼𝑠^𝛼𝑥^𝛼𝑓_𝑠^𝐿,𝛼(𝑠)(𝑑𝑠)^𝛼, (7) where 𝑠^𝛼=𝛽^𝛼+𝑖^𝛼∞^𝛼 and Re(𝑠^𝛼)=𝛽^𝛼.

The local fractional convolution of two functions is defined as [12, 20–22] 𝑓₁(𝑥)∗𝑓₂(𝑥)=1Γ(1+𝛼)^∞_−∞𝑓₁(𝑡)𝑓₂(𝑥−𝑡)(𝑑𝑡)^𝛼. (8) The properties for local fractional Laplace transform used in the paper are given as [12] 𝐿_𝛼𝑎𝑓(𝑥)+𝑏𝑔(𝑥)=𝑎𝐿_𝛼𝑓(𝑥)+𝑏𝐿_𝛼𝑔(𝑥,𝐿_𝛼)𝑓^(𝑛𝛼)(𝑥)=𝑠^𝑛𝛼𝐿_𝛼𝑓(𝑥−_𝑛𝑘=1)𝑠^{(𝑘−1)𝛼}𝑓^{(𝑛−𝑘)𝛼}(0),𝐹_𝛼𝑓₁(𝑥)∗𝑓₂(𝑥)=𝑓^𝐹,𝛼_𝜔,1(𝜔)𝑓^𝐹,𝛼_𝜔,2(𝜔),𝐿_𝛼sin_𝛼𝑐^𝛼𝑥^𝛼=𝑐^𝛼𝑠^2𝛼+𝑐^2𝛼,𝐿_𝛼cos_𝛼𝑐^𝛼𝑥^𝛼=𝑠^𝛼𝑠^2𝛼+𝑐^2𝛼,𝐿_𝛼𝑥^𝑘𝛼=Γ(1+𝑘𝛼)𝑠^(𝑘+1)𝛼. (9)

3. Analysis of the Method

In this section, we introduce the idea of local fractional variational iteration method [16, 17], which is coupled by the local fractional variational iteration method and Laplace transform.

Let us consider the following nonlinear operator with local fractional derivative: 𝐿_𝛼𝑢−𝑁_𝛼𝑢=0, (10) where the linear local fractional differential operator denotes 𝐿_𝛼=(𝑑^𝑘𝛼/𝑑𝑠^𝑘𝛼) and 𝑢(𝑥) is a source term of the nondifferential function.

Following the local fractional Laplace variational iteration method [23], we have the local fractional functional formula as follows: 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+₀𝐼_𝑥^(𝛼)𝜆(𝑡)^𝛼Γ()𝐿1+𝛼^𝛼𝑢_𝑛(𝑡)−𝑁_𝛼𝑢_𝑛, (11) which leads to 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+₀𝐼_𝑥^(𝛼)𝜆(𝑥−𝑡)^𝛼Γ()𝐿1+𝛼^𝛼𝑢_𝑛(𝑡)−𝑁_𝛼𝑢_𝑛. (12) Using the local fractional Laplace transform, from (12), we get 𝐿_𝛼𝑢_𝑛+1(𝑥)=𝐿_𝛼𝑢_𝑛(𝑥+𝐿_𝛼)𝜆(𝑥)^𝛼Γ()𝐿1+𝛼^𝛼𝐿_𝛼𝑢_𝑛(𝑥)−𝑁_𝛼𝑢_𝑛(𝑥). (13) Taking the local fractional variation [21], we obtain 𝛿^𝛼𝐿_𝛼𝑢_𝑛+1(𝑥)=𝛿^𝛼𝐿_𝛼𝑢_𝑛(𝑥)+𝛿^𝛼𝐿_𝛼𝜆(𝑥)^𝛼Γ()𝐿1+𝛼^𝛼𝐿_𝛼𝑢_𝑛(𝑥)−𝑁_𝛼𝑢_𝑛(𝑥) (14) which leads to 𝛿^𝛼𝐿_𝛼𝑢_𝑛+1(𝑥)=𝛿^𝛼𝐿_𝛼𝑢_𝑛(𝑥+𝐿_𝛼)𝜆(𝑥)^𝛼Γ()𝛿1+𝛼^𝛼𝐿_𝛼𝐿_𝛼𝑢_𝑛(𝑥)=0, (15) where 𝐿_𝛼𝐿_𝛼𝑢_𝑛(𝑥)=𝑠^𝑘𝛼𝐿_𝛼𝑢_𝑛(𝑥)−𝑠^{(𝑘−1)𝛼}𝑢_𝑛(0)−𝑠^{(𝑘−2)𝛼}𝑢_𝑛^(𝛼)(0)−⋯−𝑢_𝑛^{((𝑘−1)𝛼)}(0)=𝑠^𝑘𝛼𝐿_𝛼𝑢_𝑛(𝑥). (16) Hence, from (15) and (16) we get 1+𝐿_𝛼𝜆(𝑥)^𝛼Γ()𝑠1+𝛼^𝑘𝛼=0, (17) which yields 𝐿_𝛼𝜆(𝑥)^𝛼Γ()11+𝛼=−𝑠^𝑘𝛼. (18) Hence, we get the new iteration algorithm as follows: 𝐿_𝛼𝑢_𝑛+1(𝑥)=𝐿_𝛼𝑢_𝑛(𝑥−1)𝑠^𝑘𝛼𝐿_𝛼𝐿_𝛼𝑢_𝑛(𝑥)−𝑁_𝛼𝑢_𝑛(𝑥)=𝐿_𝛼𝑢_𝑛(𝑥−1)𝑠^𝑘𝛼𝐿_𝛼𝐿_𝛼𝑢_𝑛(𝑥+1)𝑠^𝑘𝛼𝐿_𝛼𝑁_𝛼𝑢_𝑛(𝑥), (19) where the initial value is suggested as 𝑠^{(𝑘−1)𝛼}𝑢(0)+𝑠^{(𝑘−2)𝛼}𝑢^(𝛼)(0)+⋯+𝑢^{((𝑘−1)𝛼)}(0)𝑠^𝑘𝛼=0. (20) Therefore, we have the local fractional series solution of (10) 𝐿_𝛼{𝑢}=lim_𝑛→∞𝐿_𝛼𝑢_𝑛 (21) so that 𝑢=lim_𝑛→∞𝐿_𝛼⁻¹𝐿_𝛼𝑢_𝑛. (22) The above process is called the local fractional variational iteration method.

4. The Nondifferentiable Solutions for Linear Local Fractional Differential Equations

In this section, we present the examples for linear local fractional differential equations of high order.

Example 1. The local fractional differential equation is presented as 𝜕^3𝛼𝑢(𝑥,𝑡)𝜕𝑡^3𝛼=𝜕^2𝛼𝑢(𝑥,𝑡)𝜕𝑥^2𝛼, (23) subject to the initial value 𝜕^𝛼𝑢(0,𝑡)𝜕𝑥^𝛼=0,𝑢(0,𝑡)=𝐸_𝛼−𝑡^𝛼. (24) From (19) and (23) we obtain 𝐿_𝛼𝑢_𝑛+1(=𝐿𝑥,𝑡)_𝛼𝑢_𝑛(−1𝑥,𝑡)𝑠^2𝛼𝐿_𝛼𝜕^2𝛼𝑢_𝑛(𝑥,𝑡)𝜕𝑥^2𝛼−𝜕^3𝛼𝑢_𝑛(𝑥,𝑡)𝜕𝑡^3𝛼=𝐿_𝛼𝑢_𝑛(−1𝑥,𝑡)𝑠^2𝛼𝑠^2𝛼𝑢_𝑛(𝑠,𝑡)−𝑠^𝛼𝑢_𝑛(0,𝑡)−𝑢_𝑛^(𝛼)()−𝜕0,𝑡_3𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑡^3𝛼=𝐿_𝛼𝑢_𝑛(−1𝑥,𝑡)𝑠^2𝛼𝑠^2𝛼𝑢_𝑛(𝑠,𝑡)−𝑠^𝛼𝑢_𝑛()−𝜕0,𝑡^3𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑡^3𝛼, (25) where the initial value is given by 𝐿_𝛼𝑢₀(𝑥,𝑡)=𝑢₀(𝑠,𝑡)=𝐿_𝛼𝐸_𝛼−𝑡^𝛼=𝐸_𝛼−𝑡^𝛼𝑠^𝛼. (26) Therefore, the successive approximations are 𝐿_𝛼𝑢₁(=𝐿𝑥,𝑡)_𝛼𝑢₀(−1𝑥,𝑡)𝑠^2𝛼𝑠^2𝛼𝑢₀(𝑠,𝑡)−𝑠^𝛼𝑢₀()−𝜕0,𝑡^3𝛼𝑢₀(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼−𝑡^𝛼𝑠^𝛼−𝐸_𝛼−𝑡^𝛼𝑠^3𝛼=𝐸_𝛼−𝑡^𝛼1_𝑘=0(−1)^𝑘1𝑠^{(2𝑘+1)𝛼},𝐿_𝛼𝑢₂(=𝐿𝑥,𝑡)_𝛼𝑢₁(−1𝑥,𝑡)𝑠^2𝛼𝑠^2𝛼𝑢₁(𝑠,𝑡)−𝑠^𝛼𝑢₁()−𝜕0,𝑡^3𝛼𝑢₁(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼−𝑡^𝛼𝑠^𝛼−𝐸_𝛼−𝑡^𝛼𝑠^3𝛼+𝐸_𝛼−𝑡^𝛼𝑠^5𝛼=𝐸_𝛼−𝑡^𝛼2_𝑘=0(−1)^𝑘1𝑠^{(2𝑘+1)𝛼},𝐿_𝛼𝑢₃(=𝐿𝑥,𝑡)_𝛼𝑢₂(−1𝑥,𝑡)𝑠^2𝛼𝑠^2𝛼𝑢₂(𝑠,𝑡)−𝑠^𝛼𝑢₂()−𝜕0,𝑡^3𝛼𝑢₂(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼−𝑡^𝛼𝑠^𝛼−𝐸_𝛼−𝑡^𝛼𝑠^3𝛼+𝐸_𝛼−𝑡^𝛼𝑠^5𝛼−𝐸_𝛼−𝑡^𝛼𝑠^7𝛼=𝐸_𝛼−𝑡^𝛼3_𝑘=0(−1)^𝑘1𝑠^{(2𝑘+1)𝛼},𝐿_𝛼𝑢₄(=𝐿𝑥,𝑡)_𝛼𝑢₃(−1𝑥,𝑡)𝑠^2𝛼𝑠^2𝛼𝑢₃(𝑠,𝑡)−𝑠^𝛼𝑢₃()−𝜕0,𝑡^3𝛼𝑢₃(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼−𝑡^𝛼𝑠^𝛼−𝐸_𝛼−𝑡^𝛼𝑠^3𝛼+𝐸_𝛼−𝑡^𝛼𝑠^5𝛼−𝐸_𝛼−𝑡^𝛼𝑠^7𝛼+𝐸_𝛼−𝑡^𝛼𝑠^9𝛼=𝐸_𝛼−𝑡^𝛼4_𝑘=0(−1)^𝑘1𝑠^{(2𝑘+1)𝛼},⋮𝐿_𝛼𝑢_𝑛(=𝐿𝑥,𝑡)_𝛼𝑢_𝑛−1(−1𝑥,𝑡)𝑠^2𝛼𝑠^2𝛼𝑢_𝑛−1(𝑠,𝑡)−𝑠^𝛼𝑢_𝑛−1()−𝜕0,𝑡_3𝛼𝑢_𝑛−1(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼−𝑡^𝛼𝑛_𝑘=0(−1)^𝑘1𝑠^{(2𝑘+1)𝛼}. (27) Hence, we get 𝑢(𝑥,𝑡)=lim_𝑛→∞𝐿_𝛼⁻¹𝐿_𝛼𝑢_𝑛=lim_𝑛→∞𝐿_𝛼⁻¹𝐸_𝛼−𝑡^𝛼𝑛_𝑘=0(−1)^𝑖1𝑠^{(2𝑘+1)𝛼}=𝐸_𝛼−𝑡^𝛼𝑛_𝑘=0(−1)^𝑘𝑥^2𝑘𝛼Γ(1+2𝑘𝛼)=𝐸_𝛼−𝑡^𝛼cos_𝛼𝑥^𝛼 (28) and its graph is shown in Figure 1.

Figure 1 

Plot of 𝑢(𝑥,𝑡) with the fractal dimension 𝛼=ln2/ln3.

Example 2. We report the following local fractional partial differential equation: 𝜕^2𝛼𝑢(𝑥,𝑡)𝜕𝑡^2𝛼=𝜕^3𝛼𝑢(𝑥,𝑡)𝜕𝑥^3𝛼. (29) The initial value is given by 𝜕^2𝛼𝑢(0,𝑡)𝜕𝑥^2𝛼𝜕=0,^𝛼𝑢(0,𝑡)𝜕𝑥^𝛼=𝐸_𝛼𝑡^𝛼,𝑢(0,𝑡)=0. (30) In view of (19) and (29) the local fractional iteration algorithm can be written as follows: 𝐿_𝛼𝑢_𝑛+1(=𝐿𝑥,𝑡)_𝛼𝑢_𝑛(−1𝑥,𝑡)𝑠^3𝛼𝐿_𝛼𝜕^3𝛼𝑢_𝑛(𝑥,𝑡)𝜕𝑥^3𝛼−𝜕^2𝛼𝑢_𝑛(𝑥,𝑡)𝜕𝑡^2𝛼=𝐿_𝛼𝑢_𝑛(−1𝑥,𝑡)𝑠^3𝛼𝑠^3𝛼𝑢_𝑛(𝑠,𝑡)−𝑠^2𝛼𝑢_𝑛(0,𝑡)−𝑠^𝛼𝑢_𝑛^(𝛼)(0,𝑡)−𝑢_𝑛^(2𝛼)()−𝜕0,𝑡^3𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑡^3𝛼=𝐿_𝛼𝑢_𝑛(−1𝑥,𝑡)𝑠^3𝛼𝑠^3𝛼𝑢_𝑛(𝑠,𝑡)−𝑠^𝛼𝑢_𝑛^(𝛼)()−𝜕0,𝑡^3𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑡^3𝛼, (31) where the initial value is 𝐿_𝛼𝑢₀(𝑥,𝑡)=𝑢₀()=𝐿𝑠,𝑡_𝛼𝑥^𝛼Γ()𝐸1+𝛼^𝛼𝑡^𝛼=𝐸_𝛼𝑡^𝛼𝑠^2𝛼. (32) Making use of (31) and (32), the successive approximate solutions are shown as follows: 𝐿_𝛼𝑢₁(=𝐿𝑥,𝑡)_𝛼𝑢₀(−1𝑥,𝑡)𝑠^3𝛼𝑠^3𝛼𝑢₀(𝑠,𝑡)−𝑠^𝛼𝑢₀^(𝛼)()−𝜕0,𝑡^3𝛼𝑢₀(𝑠,𝑡)𝜕𝑡^3𝛼=𝐿_𝛼𝑢₀(−1𝑥,𝑡)𝑠^3𝛼𝑠^3𝛼𝑢₀(𝑠,𝑡)−𝑠^𝛼𝐸_𝛼𝑡^𝛼−𝜕^3𝛼𝑢₀(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼𝑡^𝛼𝑠^2𝛼+𝐸_𝛼𝑡^𝛼𝑠^5𝛼=𝐸_𝛼𝑡^𝛼1_𝑘=01𝑠^{(3𝑘+2)𝛼},𝐿_𝛼𝑢₂(=𝐿𝑥,𝑡)_𝛼𝑢₁(−1𝑥,𝑡)𝑠^3𝛼𝑠^3𝛼𝑢₁(𝑠,𝑡)−𝑠^𝛼𝑢₁^(𝛼)()−𝜕0,𝑡^3𝛼𝑢₁(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼𝑡^𝛼𝑠^2𝛼+𝐸_𝛼𝑡^𝛼𝑠^5𝛼+𝐸_𝛼𝑡^𝛼𝑠^8𝛼=𝐸_𝛼𝑡^𝛼2_𝑘=01𝑠^{(3𝑘+2)𝛼},𝐿_𝛼𝑢₃(=𝐿𝑥,𝑡)_𝛼𝑢₂(−1𝑥,𝑡)𝑠^3𝛼𝑠^3𝛼𝑢₂(𝑠,𝑡)−𝑠^𝛼𝑢₂^(𝛼)()−𝜕0,𝑡^3𝛼𝑢₂(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼𝑡^𝛼𝑠^2𝛼+𝐸_𝛼𝑡^𝛼𝑠^5𝛼+𝐸_𝛼𝑡^𝛼𝑠^8𝛼+𝐸_𝛼𝑡^𝛼𝑠^11𝛼=𝐸_𝛼𝑡^𝛼3_𝑘=01𝑠^{(3𝑘+2)𝛼},𝐿_𝛼𝑢₄(=𝐿𝑥,𝑡)_𝛼𝑢₃(−1𝑥,𝑡)𝑠^3𝛼𝑠^3𝛼𝑢₃(𝑠,𝑡)−𝑠^𝛼𝑢₃^(𝛼)()−𝜕0,𝑡^3𝛼𝑢₃(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼𝑡^𝛼𝑠^2𝛼+𝐸_𝛼𝑡^𝛼𝑠^5𝛼+𝐸_𝛼𝑡^𝛼𝑠^8𝛼+𝐸_𝛼𝑡^𝛼𝑠^11𝛼+𝐸_𝛼𝑡^𝛼𝑠^14𝛼=𝐸_𝛼𝑡^𝛼4_𝑘=01𝑠^{(3𝑘+2)𝛼},⋮𝐿_𝛼𝑢_𝑛(=𝐿𝑥,𝑡)_𝛼𝑢_𝑛−1(−1𝑥,𝑡)𝑠^3𝛼𝑠^3𝛼𝑢_𝑛−1()−𝜕𝑠,𝑡^3𝛼𝑢_𝑛−1(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼𝑡^𝛼𝑛_𝑘=01𝑠^{(2𝑘+1)𝛼}. (33) Therefore, the nondifferentiable solution of (29) reads 𝑢(𝑥,𝑡)=lim_𝑛→∞𝐿_𝛼⁻¹𝐿_𝛼𝑢_𝑛=lim_𝑛→∞𝐿_𝛼⁻¹𝐸_𝛼𝑡^𝛼𝑛_𝑘=01𝑠^{(3𝑘+2)𝛼}=𝐸_𝛼𝑡^𝛼𝑛_𝑘=0𝑥^{(3𝑘+2)𝛼}Γ(1+(3𝑘+2)𝛼) (34) and its plot is presented in Figure 2.

Figure 2 

Plot of 𝑢(𝑥,𝑡) with the parameters 𝛼=ln2/ln3 and 𝑘=50.

Example 3. The following local fractional partial differential equation 𝜕^2𝛼𝑢(𝑥,𝑡)𝜕𝑡^2𝛼−𝜕^4𝛼𝑢(𝑥,𝑡)𝜕𝑥^4𝛼=0 (35) is considered and its initial value is 𝜕^3𝛼𝑢(0,𝑡)𝜕𝑥^3𝛼=𝐸_𝛼𝑡^𝛼,𝜕^2𝛼𝑢(0,𝑡)𝜕𝑥^2𝛼𝜕=0,^𝛼𝑢(0,𝑡)𝜕𝑥^𝛼𝑢()=0,0,𝑡=0. (36) Making use of (19) and (35) the local fractional iteration algorithm reads 𝐿_𝛼𝑢_𝑛+1(=𝐿𝑥,𝑡)_𝛼𝑢_𝑛(−1𝑥,𝑡)𝑠^4𝛼𝐿_𝛼𝜕^4𝛼𝑢_𝑛(𝑥,𝑡)𝜕𝑥^4𝛼−𝜕^2𝛼𝑢_𝑛(𝑥,𝑡)𝜕𝑡^2𝛼=𝐿_𝛼𝑢_𝑛(−1𝑥,𝑡)𝑠^4𝛼𝑠^4𝛼𝑢_𝑛(𝑠,𝑡)−𝑠^3𝛼𝑢_𝑛(0,𝑡)−𝑠^2𝛼𝑢_𝑛^(𝛼)(−10,𝑡)𝑠^4𝛼−𝑠^𝛼𝑢_𝑛^(2𝛼)(0,𝑡)−𝑢_𝑛^(3𝛼)()−𝜕0,𝑡^3𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑡^3𝛼=𝐿_𝛼𝑢_𝑛(−1𝑥,𝑡)𝑠^4𝛼𝑠^4𝛼𝑢_𝑛(𝑠,𝑡)−𝑢_𝑛^(3𝛼)()−𝜕0,𝑡^3𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑡^3𝛼, (37) where the initial value is suggested as 𝐿_𝛼𝑢₀(𝑥,𝑡)=𝑢₀()=𝐿𝑠,𝑡_𝛼𝑥^3𝛼Γ()𝐸1+3𝛼^𝛼𝑡^𝛼=𝐸_𝛼𝑡^𝛼𝑠^4𝛼. (38) From (38) we have the successive approximations as follows: 𝐿_𝛼𝑢₁(=𝐿𝑥,𝑡)_𝛼𝑢₀(−1𝑥,𝑡)𝑠^4𝛼𝑠^4𝛼𝑢₀(𝑠,𝑡)−𝑢₀^(3𝛼)()−𝜕0,𝑡^3𝛼𝑢₀(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼𝑡^𝛼𝑠^4𝛼+𝐸_𝛼𝑡^𝛼𝑠^8𝛼=𝐸_𝛼𝑡^𝛼1_𝑘=01𝑠^{4(𝑘+1)𝛼},𝐿_𝛼𝑢₂(=𝐿𝑥,𝑡)_𝛼𝑢₁(−1𝑥,𝑡)𝑠^4𝛼𝑠^4𝛼𝑢₁(𝑠,𝑡)−𝑢₁^(3𝛼)()−𝜕0,𝑡^3𝛼𝑢₁(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼𝑡^𝛼𝑠^4𝛼+𝐸_𝛼𝑡^𝛼𝑠^8𝛼+𝐸_𝛼𝑡^𝛼𝑠^12𝛼=𝐸_𝛼𝑡^𝛼2_𝑘=01𝑠^{4(𝑘+1)𝛼},𝐿_𝛼𝑢₃(=𝐿𝑥,𝑡)_𝛼𝑢₂(−1𝑥,𝑡)𝑠^4𝛼𝑠^4𝛼𝑢₂(𝑠,𝑡)−𝑢₂^(3𝛼)()−𝜕0,𝑡^3𝛼𝑢₂(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼𝑡^𝛼𝑠^4𝛼+𝐸_𝛼𝑡^𝛼𝑠^8𝛼+𝐸_𝛼𝑡^𝛼𝑠^12𝛼+𝐸_𝛼𝑡^𝛼𝑠^16𝛼=𝐸_𝛼𝑡^𝛼3_𝑘=01𝑠^{4(𝑘+1)𝛼},𝐿_𝛼𝑢₄(=𝐿𝑥,𝑡)_𝛼𝑢₃(−1𝑥,𝑡)𝑠^4𝛼𝑠^4𝛼𝑢₃(𝑠,𝑡)−𝑢₃^(3𝛼)()−𝜕0,𝑡^3𝛼𝑢₃(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼𝑡^𝛼𝑠^4𝛼+𝐸_𝛼𝑡^𝛼𝑠^8𝛼+𝐸_𝛼𝑡^𝛼𝑠^12𝛼+𝐸_𝛼𝑡^𝛼𝑠^16𝛼+𝐸_𝛼𝑡^𝛼𝑠^20𝛼=𝐸_𝛼𝑡^𝛼4_𝑘=01𝑠^{4(𝑘+1)𝛼},⋮𝐿_𝛼𝑢_𝑛(=𝐿𝑥,𝑡)_𝛼𝑢_𝑛−1(−1𝑥,𝑡)𝑠^4𝛼𝑠^4𝛼𝑢_𝑛−1(𝑠,𝑡)−𝑢^(3𝛼)_𝑛−1()−𝜕0,𝑡^3𝛼𝑢_𝑛−1(𝑠,𝑡)𝜕𝑡^3𝛼=𝐸_𝛼𝑡^𝛼𝑠^4𝛼+𝐸_𝛼𝑡^𝛼𝑠^8𝛼+𝐸_𝛼𝑡^𝛼𝑠^12𝛼+𝐸_𝛼𝑡^𝛼𝑠^16𝛼+𝐸_𝛼𝑡^𝛼𝑠^20𝛼=𝐸_𝛼𝑡^𝛼𝑛_𝑘=01𝑠^{4(𝑘+1)𝛼}. (39) Hence, the approximate solution of (35) is given by 𝑢(𝑥,𝑡)=lim_𝑛→∞𝐿_𝛼⁻¹𝐿_𝛼𝑢_𝑛=lim_𝑛→∞𝐿_𝛼⁻¹𝐸_𝛼𝑡^𝛼𝑛_𝑘=01𝑠^{4(𝑘+1)𝛼}=𝐸_𝛼𝑡^𝛼4_𝑘=0𝑥^{4(𝑘+1)𝛼}Γ(1+4(𝑘+1)𝛼) (40) and its graph is presented in Figure 3.

Figure 3 

Plot of 𝑢(𝑥,𝑡) with the parameters 𝛼=ln2/ln3 and 𝑘=50.

5. Conclusions

Local fractional calculus was successfully applied to deal with the nondifferentiable problems arising in mathematical physics. In this work we considered the coupling method of the local fractional variational iteration method and Laplace transform to solve the linear local fractional partial differential equations and their nondifferentiable solutions were obtained. The results are efficient implement of the local fractional Laplace variational iteration method to solve the partial differential equations with local fractional derivative.

Conflict of Interests

The authors declare that they have no competing interests in this paper.

Acknowledgments

This work was supported by National Scientific and Technological Support Projects (no. 2012BAE09B00), the National Natural Science Foundation of China (no. 51274270), and the National Natural Science Foundation of Hebei Province (no. E2013209215).