Variational Iteration Transform Method for Fractional Differential Equations with Local Fractional Derivative

Abstract

We propose the variational iteration transform method in the sense of local fractional derivative, which is derived from the coupling method of local fractional variational iteration method and differential transform method. The method reduces the integral calculation of the usual variational iteration computations to more easily handled differential operation. And the technique is more orderly and easier to analyze computing result as compared with the local fractional variational iteration method. Some examples are illustrated to show the feature of the presented technique.

1. Introduction

Fractional differential equation has been considered with great importance due to its demonstrated applications in various areas such as electrical networks, fluid flow, biology, and dynamical systems [1–8]. And many classical differential equations possess a fractional analogy. As a result, a substantial number of methods for solving the fractional differential equations are developed to solve fractional differential equations [9–25].

Recently, local fractional derivative and calculus theory has been introduced by the researcher in [18, 19], which is set up on fractal geometry and which is the best method for describing the nondifferential function defined on Cantor sets. The physical explanation of the local fractional derivative can be seen in [26, 27]. A great deal of research work has been directed for the nondifferentiable phenomena in fractal domain concerning local fractional derivative, for example, [11, 12, 15, 18–25].

Motivated by the ongoing research method of local fractional differential equation, we present variational iteration transform method, which is the coupling method of local fractional variational iteration method and differential transform method. It is worth mentioning that since the method makes the complex iteration calculation more orderly, it may be considered as an efficient modification of variational iteration method. The rest of the paper is organized as follows. In Section 2, the basic mathematical fundamentals are presented briefly. In Section 3, the local fractional function differential transform method for solving the differential equations with local fractional derivative is investigated. In Section 4, several examples are illustrated. Finally, in Section 5, the conclusion is given.

2. Mathematical Fundamentals

In this section, we introduce the basic notions of local fractional derivative, local fractional integral, and local fractional differential transform method.

2.1. Local Fractional Derivatives and Integrals

The local fractional derivative of 𝑢(𝑥) of order 𝛼(0<𝛼≤1) at 𝑥₀ is defined by [18, 19] 𝑢^(𝛼)𝑥₀=lim_𝑥→𝑥0Δ^𝛼𝑢(𝑥)−𝑢𝑥₀𝑥−𝑥₀^𝛼=𝐷_𝑥^(𝛼)𝑢𝑥₀=𝑑^𝛼𝑢(𝑥)𝑑𝑥^𝛼_𝑥=𝑥0, (1) where Δ^𝛼(𝑢(𝑥)−𝑢(𝑥₀))≅Γ(1+𝛼)(𝑢(𝑥)−𝑢(𝑥₀)).

In the interval [𝑎,𝑏], local fractional integral of 𝑢(𝑥) of order 𝛼 is given by [18, 19] _𝑎𝐼_𝑏^(𝛼)𝑢(𝑥)=1Γ(1+𝛼)lim_Δ𝑡→0^{𝑗=𝑁−1}_𝑗=0𝑢𝑡_𝑗Δ𝑡_𝑗^𝛼=1Γ(1+𝛼)^𝑏_𝑎𝑢(𝑡)(𝑑𝑡)^𝛼, (2) where  Δ𝑡_𝑗=𝑡_𝑗+1−𝑡_𝑗, Δ𝑡=max{Δ𝑡₁,Δ𝑡₂,Δ𝑡_𝑗,…} and [𝑡_𝑗,𝑡_𝑗+1], 𝑗=0,…,𝑁−1, and 𝑡₀=𝑎,𝑡_𝑁=𝑏 is a partition of the interval [𝑎,𝑏].

2.2. Local Fractional Differential Transform Method

Similarly in [28–34], the local fractional different transform 𝑈(𝑥,𝑡) (or 𝑢(𝑥), resp.) of the function 𝑢(𝑥,𝑡) (or 𝑢(𝑥), resp.) is defined by the following formula: 𝑈()=1𝑥,𝑘Γ()𝜕1+𝑘𝛼𝜕𝑡^{𝑘𝛼𝑘𝛼}𝑢(𝑥,𝑡)_𝑡=0(𝑘)=1or𝑈Γ()𝑑1+𝑘𝛼^𝑘𝛼𝑢(𝑡)𝑑𝑡^𝑘𝛼_𝑡=0,resp.. (3) Obviously 𝑢(𝑥,𝑡)=^∞_𝑘=0𝑈(𝑥,𝑘)𝑡^𝑘𝛼or𝑢(𝑡)=^∞_𝑘=0𝑈(𝑘)𝑡^𝑘𝛼,resp.. (4) If we let 𝑈(𝑥,𝑘)=𝐷𝑇_𝑘{𝑢((𝑘)𝑥,𝑡)}or𝑈=𝐷𝑇_𝑘{𝑢(𝑡)},resp. (5) and let 𝐷𝑇_𝑘⁻¹{𝑈(=𝑢𝑥,𝑘)}^(𝑘𝛼)(𝑥,0)Γ()𝑡1+𝑘𝛼Γ(1+^𝑘𝛼𝑘𝛼)or𝐷𝑇_𝑘⁻¹{𝑈(𝑘=𝑢)}^(𝑘𝛼)(0)Γ()𝑡1+𝑘𝛼Γ(1+^𝑘𝛼𝑘𝛼),resp. (6) then we have 𝑢(𝑥,𝑡)=^∞_𝑘=0𝐷𝑇_𝑘⁻¹{𝑈((𝑡)=𝑥,𝑘)}or𝑢^∞_𝑘=0𝐷𝑇_𝑘⁻¹{𝑈(𝑘)},resp.. (7)

In Table 1, we list a few operations concerning local fractional differential transform, where 𝑚 is a nonnegative integer.

Table 1 

3. Local Fractional Variational Iteration Transform Method

To introduce the basic ideas of the variational iteration transform method, we consider the following nonlinear fractional differential equation in the sense of local fractional derivative: 𝜕^𝑘𝑜𝛼𝜕𝑡^𝑘0𝑇(𝑥,𝑡)+𝐿_𝛼𝑇(𝑥,𝑡)+𝑅_𝛼𝑇(𝑥,𝑡)=𝑔(𝑥,𝑡), (8) where 𝐿_𝛼 is the linear local fractional operator of order less than 𝑘₀(𝑘₀≥2), 𝑅_𝛼 is the nonlinear local fractional operator of order less than 𝑘₀, and 𝑔(𝑥,𝑡) is a source term of the nondifferential function.

According to the local fractional variational iteration method [20, 35], we can construct local fractional variational iteration algorithm: 𝑇_𝑛+1(𝑥,𝑡)=𝑇_𝑛(𝑥,𝑡)+₀𝐼_𝑡^{((𝑘𝑜−1)𝛼)}×𝜆^𝛼(𝜏)Γ()𝜕1+𝛼𝜕𝑡^{𝑘𝑜𝛼𝑘0}𝑇_𝑛(𝑥,𝜏)+𝐿_𝛼𝑇_𝑛(𝑥,𝜏)+𝑅_𝛼𝑇_𝑛(𝑥,𝜏)−𝑔(𝑥,𝜏), (9) where 𝜆^𝛼(𝜏)/Γ(1+𝛼) is a fractal Lagrange multiplier.

Suppose 𝛿^𝛼𝑇_𝑛(𝑥,𝜏) is a restricted local fractional variation; that is, 𝛿^𝛼𝑇_𝑛(𝑥,𝜏)=0 [20, 35]; we get [21] 𝜆^𝛼(𝜏)Γ()=()1+𝛼𝜏−𝑡^{(𝑘0−1)𝛼}Γ1+𝑘₀𝛼.−1 (10) Substituting (10) into (9), a fractional iteration procedure is constructed as follows: 𝑇_𝑛+1(𝑥,𝑡)=𝑇_𝑛(𝑥,𝑡)+₀𝐼_𝑡^{((𝑘𝑜−1)𝛼)}×(𝑥−𝑡)^{(𝑘0−1)𝛼}Γ1+𝑘₀𝛼×𝜕−1_𝑘𝑜𝛼𝜕𝑡^𝑘0𝑇_𝑛(𝑥,𝜏)+𝐿_𝛼𝑇_𝑛(𝑥,𝜏)+𝑅_𝛼𝑇_𝑛(𝑥,𝜏)−𝑔(𝑥,𝜏). (11)

Applying local fractional differential transform with respect to 𝑡 on both sides of (11) and using local fractional derivative algorithm [18], we obtain 𝐷𝑇_𝑘𝑇_𝑛+1(𝑥,𝑡)=𝐷𝑇_𝑘𝑇₀(𝑥,𝑡),0≤𝑘≤𝑘₀−1 (12) and also 𝐷𝑇_𝑘𝑇_𝑛+1(𝑥,𝑡)=−𝐷𝑇_𝑘0𝐼_𝑡^(𝑘𝛼)𝐿_𝛼𝑇_𝑛(𝑥,𝜏)+𝑅_𝛼𝑇_𝑛(𝑥,𝜏)−𝑔(𝑥,𝜏),𝑘≥𝑘₀, (13) where 𝑇₀(𝑥,𝑡) is an initial value.

By virtue of (5), (12), (13), and the operation displayed in Table 1, we can yield 𝑇_𝑛+1(𝑥,𝑡)=−^∞_𝑘=𝑘0𝐷𝑇_𝑘⁻¹Γ1+𝑘−𝑘₀𝛼Γ()𝑊1+𝑘𝛼^𝑛𝑥,𝑘−𝑘₀+^𝑘0−1_𝑘=0𝐷𝑇_𝑘⁻¹𝐷𝑇_𝑘𝑇₀(𝑥,𝑘)=−^∞_𝑘=𝑘0𝐷𝑇_𝑘⁻¹Γ1+𝑘−𝑘₀𝛼Γ()𝑊1+𝑘𝛼^𝑛𝑥,𝑘−𝑘₀+^𝑘0−1_𝑘=0𝑇₀^(𝑘𝛼)(𝑥,0)Γ()𝑡1+𝑘𝛼Γ(^𝑘𝛼),1+𝑘𝛼 (14) where 𝑊_𝑛(𝑥,𝑘)=𝐷𝑇_𝑘⁻¹𝑤_𝑛(𝑥,𝑡) (15) and where 𝑤_𝑛(𝑥,𝑡)=𝐿_𝛼𝑇_𝑛(𝑥,𝑡)+𝑅_𝛼𝑇_𝑛(𝑥,𝑡)−𝑔(𝑥,𝑡). (16)

So, we have the exact solution of (8) in the form: 𝑇(𝑥,𝑡)=lim_𝑛→∞𝑇_𝑛+1(𝑥,𝑡)=lim_𝑛→∞−^∞_𝑘=𝑘0𝐷𝑇_𝑘⁻¹×Γ1+𝑘−𝑘₀𝛼Γ()𝑊1+𝑘𝛼^𝑛𝑥,𝑘−𝑘₀+^𝑘0−1_𝑘=0𝑇₀^(𝑘𝛼)(𝑥,0)Γ()𝑡1+𝑘𝛼Γ(^𝑘𝛼).1+𝑘𝛼 (17)

Now we discuss the solution of the following quadratic linear fractional differential equations which is the simple example of (8): 𝜕^2𝛼𝑇(𝑥,𝑡)𝜕𝑡^2𝛼+𝑝₁𝜕^2𝛼𝑇(𝑥,𝑡)𝜕𝑥^2𝛼+𝑝₂𝜕^𝛼𝑇(𝑥,𝑡)𝜕𝑡^𝛼+𝑝₃𝜕^𝛼𝑇(𝑥,𝑡)𝜕𝑥^𝛼+𝑝₄𝑇(𝑥,𝑡)=0, (18) where 𝑝₁, 𝑝₂, 𝑝₃, and 𝑝₄ are all constant numbers.

According to (9), the correction iteration algorithm is given as follows: 𝑇_𝑛+1(𝑥,𝑡)=𝑇_𝑛(𝑥,𝑡)+₀𝐼^𝛼_𝑡×(𝜏−𝑡)^(𝛼)Γ()𝐿1+𝛼^{(2𝛼)𝑡𝑡}𝑇_𝑛+𝑝₁𝐿^(2𝛼)_𝑥𝑥𝑇_𝑛+𝑝₂𝐿_𝑡^(𝛼)𝑇_𝑛+𝑝₃𝐿_𝑥^(𝛼)𝑇_𝑛+𝑝₄𝑇_𝑛. (19) Taking local fractional differential transform with respect to 𝑡 on (19), we deduce 𝐷𝑇_𝑘𝑇_𝑛+1(𝑥)=−𝐷𝑇_𝑘−2𝑝₁𝐿^(2𝛼)_𝑥𝑥𝑇_𝑛+𝑝₂𝐿_𝑡^(𝛼)𝑇_𝑛+𝑝₃𝐿_𝑥^(𝛼)𝑇_𝑛+𝑝₄𝑇_𝑛,(𝑘≥2)𝐷𝑇_𝑘𝑇_𝑛+1(𝑥)=𝐷𝑇_𝑘𝑇₀(𝑥),(𝑘=0,1), (20) where 𝑇_𝑛(𝑥,𝑡)=𝑢_𝑛(𝑡)𝑓_𝑛(𝑥) (21) and where 𝑈_𝑛(𝑘)=𝐷𝑇_𝑘𝑢_𝑛(𝑡). (22) By using (20), (21), (22), and the results included in Table 1, we obtain 𝑈_𝑛+1(𝑘)𝑓_𝑛+1(𝑥)=−𝑝₁𝑓_𝑛^(2𝛼)(𝑥)+𝑝₃𝑓_𝑛^(𝛼)(𝑥)+𝑝₄𝑓_𝑛(𝑥)×𝐷𝑇_𝑘0𝐼_𝑡^(2𝛼)𝑢_𝑛−𝑝₂𝑓_𝑛(𝑥)𝐷𝑇_𝑘0𝐼_𝑡^(𝛼)𝑢_𝑛=−𝑝₁𝑓_𝑛^(2𝛼)(𝑥)+𝑝₃𝑓_𝑛^(𝛼)(𝑥)+𝑝₄𝑓_𝑛(𝑥)𝑈(𝑘−2)𝑘𝛼(𝑘−1)𝛼−𝑝₂𝑓_𝑛(𝑥)𝑈(𝑘−1),()𝑘𝛼𝑘≥2 (23) and also 𝑈_𝑛+1(𝑘)𝑓_𝑛+1(𝑥)=𝑈₀(𝑘)𝑓_𝑛+1(𝑥),(0≤𝑘≤1). (24) If we let 𝑝₁𝑓₀^(2𝛼)(𝑥)+𝑝₃𝑓₀^(𝛼)(𝑥)+𝑝₄𝑓₀(𝑥)=𝑝₅𝑓₀(𝑥), (25) then 𝑓_𝑛(𝑥)=𝑓₀(𝑥). (26) By (23), (24), and (26), we have 𝑈_𝑛+1(𝑘)=−𝑝₅𝑈_𝑛(𝑘−2)𝑘𝛼(𝑘−1)𝛼−𝑝₂𝑈_𝑛(𝑘−1),()𝑈𝑘𝛼𝑘≥2_𝑛+1(𝑘)=𝑈₀(𝑘),(𝑘=0,1). (27) According to (14) and (27), we get 𝑢_𝑛+1(𝑡)=^∞_𝑘=2𝐷𝑇_𝑘⁻¹−𝑝₅𝑈_𝑛(𝑘−2)𝑘𝛼(𝑘−1)𝛼−𝑝₂𝑈_𝑛(𝑘−1)+𝑘𝛼_1𝑘=0𝑢₀^(𝑘𝛼)(0)Γ()𝑡1+𝑘𝛼Γ(^𝑘𝛼).1+𝑘𝛼 (28) Taking the limit, we arrive at 𝑢(𝑡)=lim_𝑛→∞𝑢_𝑛+1(𝑡)=lim_𝑛→∞^∞_𝑘=2𝐷𝑇_𝑘⁻¹−𝑝₅𝑈_𝑛(𝑘−2)𝑘𝛼(𝑘−1)𝛼−𝑝₂𝑈_𝑛(𝑘−1)+𝑘𝛼_1𝑘=0𝑢₀^(𝑘𝛼)(0)Γ()𝑡1+𝑘𝛼Γ(^𝑘𝛼).1+𝑘𝛼 (29)

Hence, under the condition equation (25), the exact solution of (18) reads as follows: 𝑇(𝑥,𝑡)=𝑢(𝑡)𝑓₀(𝑥). (30)

4. Illustrative Examples

In porous media, there are fractals [35]. In fractal media, there are lots of mathematical physical models by the local differential fractional equation, for example, [36, 37]. In this section, we will present the solution of several equations or system of equations to assess the validity of variational iteration transform method.

Case 1. Consider the following nonlinear local fractional wave-like differential equation with variable coefficients: 𝜕^2𝛼𝑇𝜕𝑡^2𝛼−𝑥^2𝛼Γ()𝜕1+2𝛼^𝛼𝐷_𝑥^(𝛼)(𝑇)𝐷_𝑥^(2𝛼)(𝑇)𝜕𝑥^𝛼+𝑥^2𝛼Γ()𝜕1+2𝛼^2𝛼𝑇𝜕𝑥^2𝛼2(),+𝑇=0,𝑡>0,0<𝑥<1 (31) where 𝑇=𝑇(𝑥,𝑡), (32) subject to the initial conditions 𝑇()𝜕𝑥,0=0,^𝛼𝜕𝑡^𝛼𝑇()=𝑥𝑥,0^2𝛼Γ().1+2𝛼 (33)

From (33) we take the initial value, which reads 𝑇₀()=𝑥𝑥,𝑡^2𝛼Γ()𝑡1+2𝛼Γ(^𝛼).1+𝛼 (34) By using (31), we structure a local fractional iteration procedure as follows: 𝑇_𝑛+1=𝑇_𝑛+₀𝐼_𝑡^(𝛼)×(𝜏−𝑡)^𝛼Γ()𝜕1+𝛼^2𝛼𝑇_𝑛𝜕𝑡^2𝛼−𝑥^2𝛼Γ()×𝜕1+2𝛼_𝛼𝐷_𝑥^(𝛼)𝑇_𝑛𝐷_𝑥^(2𝛼)𝑇_𝑛𝜕𝑥^𝛼+𝑥^𝛼Γ()𝜕1+𝛼^2𝛼𝑇_𝑛𝜕𝑥^2𝛼2+𝑇_𝑛, (35) where 𝑇_𝑛=𝑇_𝑛(𝑥,𝑡). (36)

Let 𝑤_𝑛()𝑥𝑥,𝜏=−^2𝛼Γ()𝜕1+2𝛼^𝛼𝐷_𝑥^(𝛼)𝑇_𝑛𝐷_𝑥^(2𝛼)𝑇_𝑛𝜕𝑥^𝛼+𝑥^𝛼Γ()𝜕1+𝛼^2𝛼𝑇_𝑛𝜕𝑥^2𝛼2+𝑇_𝑛. (37)

According to (14), we can derive 𝑇₁(𝑥,𝑡)=¹_𝑘=0𝑇₀^(𝑘𝛼)(𝑥,0)Γ()𝑡1+𝑘𝛼Γ(^𝑘𝛼)−1+𝑘𝛼_∞𝑘=2𝐷𝑇_𝑘⁻¹1()𝛼𝑊𝑘𝛼⋅𝑘−1⁰(𝑥,𝑘−2)=𝑇₀(𝑥,𝑡)−^∞_𝑘=2𝐷𝑇_𝑘⁻¹1()𝛼𝑊𝑘𝛼⋅𝑘−1⁰(𝑥,𝑘−2)=𝑇₀(𝑥,𝑡)−𝐷𝑇₂⁻¹𝑊₀(𝑥,1)Γ()=𝑥1+3𝛼_2𝛼Γ()𝑡1+2𝛼Γ(^𝛼)−𝑡1+𝛼Γ(^3𝛼);𝑇1+3𝛼₂(𝑥,𝑡)=¹_𝑘=0𝑇₀^(𝑘𝛼)(𝑥,0)Γ()𝑡1+𝑘𝛼Γ(^𝑘𝛼)−1+𝑘𝛼_∞𝑘=2𝐷𝑇_𝑘⁻¹1()𝛼𝑊𝑘𝛼⋅𝑘−1¹(𝑥,𝑘−2)=𝑇₁(𝑥,𝑡)−𝐷𝑇₄⁻¹1𝑊5𝛼⋅4𝛼¹(=𝑥𝑥,3)_2𝛼Γ()𝑡1+2𝛼Γ(^𝛼)−𝑡1+𝛼Γ(^3𝛼)+𝑡1+3𝛼Γ(^5𝛼);⋮1+5𝛼 (38) Proceeding in this manner, we get 𝑇_𝑛()=𝑥𝑥,𝑡^2𝛼Γ()𝑡1+2𝛼Γ(^𝛼)−𝑡1+𝛼Γ(^3𝛼)+𝑡1+3𝛼Γ(^5𝛼)()1+5𝛼−⋯+−1^𝑛+1𝑡^{(2𝑛−1)𝛼}Γ(()𝛼).1+2𝑛−1 (39)

Thus, the exact solution of (31) is given in the form: 𝑇(𝑥,𝑡)=lim_𝑛→∞𝑇_𝑛()=𝑥𝑥,𝑡^2𝛼Γ()×𝑡1+2𝛼_𝛼Γ()−𝑡1+𝛼Γ(^3𝛼)+𝑡1+3𝛼Γ(^5𝛼)()1+5𝛼−⋯+−1^𝑛+1𝑡^{(2𝑛−1)𝛼}Γ(()𝛼).=𝑥1+2𝑛−1+⋯_2𝛼Γ(1+2𝛼)sin_𝛼𝑡^𝛼. (40)

Case 2. Consider the following local fractional equation: 𝜕^2𝛼𝑇(𝑥,𝑡)𝜕𝑡^2𝛼+2^𝛼𝜕^2𝛼𝑇(𝑥,𝑡)𝜕𝑥^2𝛼+9^𝛼𝑇(𝑥,𝑡)=0 (41) subject to the initial conditions 𝑇(𝑥,0)=sin_𝛼2𝑥^𝛼,𝜕^𝛼𝜕𝑡^𝛼𝑇(𝑥,0)=0. (42)

According to (41), we structure the following iteration formula: 𝑇_𝑛+1(𝑥,𝑡)=𝑇_𝑛(𝑥,𝑡)+₀𝐼_𝑡^(𝛼)×(𝜏−𝑡)^𝛼Γ()𝜕1+𝛼^2𝛼𝑇(𝑥,𝜏)𝜕𝑡^2𝛼+2^𝛼𝜕^2𝛼𝑇(𝑥,𝜏)𝜕𝑥^2𝛼+9^𝛼𝑇(𝑥,𝜏). (43) By using (25) and (42), we take an initial value as 𝑇₀(𝑥,𝑡)=sin_𝛼2𝑥^𝛼. (44) According to (23) and (43), we can get 𝑈_𝑛+1(𝑘)𝑈=−^𝑛(𝑘−2)()𝛼,()𝑘𝛼⋅𝑘−1𝑘≥2 (45) and moreover 𝑇_𝑛+1(𝑥,𝑡)=sin_𝛼2𝑥^𝛼1_𝑘=0𝑢₀^(𝑘𝛼)(𝑥,0)Γ()𝑡1+𝑘𝛼Γ(1+^𝑘𝛼𝑘𝛼)+sin_𝛼2𝑥^𝛼∞_𝑘=2𝐷𝑇_𝑘⁻¹−𝑈_𝑛(𝑘−2)(𝑘−1)𝛼⋅𝑘𝛼=𝑇₀(𝑥,𝑡)+sin_𝛼2𝑥^𝛼∞_𝑘=2𝐷𝑇_𝑘⁻¹−𝑈_𝑛(𝑘−2)().𝑘−1𝛼⋅𝑘𝛼 (46) Thus we have 𝑇₁(𝑥,𝑡)=𝑇₀(𝑥,𝑡)+sin_𝛼2𝑥^𝛼∞_𝑘=2𝐷𝑇_𝑘⁻¹−𝑈₀(𝑘−2)(𝑘−1)𝛼⋅𝑘𝛼=sin_𝛼2𝑥^𝛼−sin_𝛼2𝑥^𝛼𝐷𝑇₂⁻¹𝑈₀(0)𝛼⋅2𝛼=sin_𝛼2𝑥^𝛼1^𝛼−𝑡^2𝛼Γ();𝑇1+2𝛼₂(𝑥,𝑡)=𝑇₀(𝑥,𝑡)+sin_𝛼2𝑥^𝛼∞_𝑘=2𝐷𝑇_𝑘⁻¹−𝑈₁(𝑘−2)(𝑘−1)𝛼⋅𝑘𝛼=sin_𝛼2𝑥^𝛼1^𝛼−𝑡^2𝛼Γ(1+2𝛼)−𝐷𝑇₄⁻¹𝐸_𝛼𝑥^𝛼𝑈₁(2)4𝛼⋅3𝛼=sin_𝛼2𝑥^𝛼1^𝛼−𝑡^2𝛼Γ()+𝑡1+2𝛼Γ(^4𝛼);𝑇1+4𝛼₃(𝑥,𝑡)=sin_𝛼2𝑥^𝛼×1^𝛼−𝑡^2𝛼Γ()+𝑡1+2𝛼Γ(^4𝛼)−𝑡1+4𝛼Γ(^6𝛼);⋮1+6𝛼 (47) Continuing in this manner, we obtain 𝑇_𝑛(𝑥,𝑡)=sin_𝛼2𝑥^𝛼∞_𝑘=0(−1)^𝑘𝑡^2𝑘𝛼Γ().1+2𝑘𝛼 (48) Thus, the expression of the exact solution of (41) is given by 𝑇(𝑥,𝑡)=lim_𝑛→∞𝑇_𝑛(𝑥,𝑡)=sin_𝛼2𝑥^𝛼∞_𝑘=0(−1)^𝑘𝑡^2𝑘𝛼Γ(1+2𝑘𝛼)=sin_𝛼2𝑥^𝛼cos_𝛼−𝑡^𝛼. (49)

According to the ideas of local fractional variational iteration transform method, we can also deal with fractional differential equation system with a similar method.

Case 3. Consider the system of local fractional coupled Helmholtz equations 𝜕^2𝛼𝑢₁(𝑥,𝑡)𝜕𝑡^2𝛼+𝜕^2𝛼𝑢₂(𝑥,𝑡)𝜕𝑥^2𝛼−𝑢₁()𝜕𝑥,𝑡=0,_2𝛼𝑢₂(𝑥,𝑡)𝜕𝑡^2𝛼+𝜕^2𝛼𝑢₁(𝑥,𝑡)𝜕𝑥^2𝛼−𝑢₂(𝑥,𝑡)=0 (50)
with the initial conditions 𝜕^𝛼𝜕𝑡^𝛼𝑢₁(𝑥,0)=𝐸_𝛼𝑥^𝛼,𝑢₁(𝑥,0)=0. (51)

Now we can structure the similar local fractional iteration procedure.

Using (51), we take an initial value as 𝑢_1,0()=𝑡𝑥,𝑡^𝛼𝐸_𝛼𝑥^𝛼Γ().1+𝛼 (52)

According to variational iteration method [38], we can similarly construct local fractional variational iteration formula 𝑢_1,𝑛+1(𝑥,𝑡)=𝑢_1,𝑛()+𝑥,𝑡₀𝐼_𝑡^(𝛼)(𝜏−𝑡)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢_1,𝑛(𝑥,𝜏)𝜕𝜏^2𝛼+𝜕^2𝛼𝑢_2,𝑛(𝑥,𝜏)𝜕𝑥^2𝛼−𝑢_1,𝑛,𝑢_2,𝑛+1(𝑥,𝑡)=𝑢_2,𝑛()+𝑥,𝑡₀𝐼_𝑡^(𝛼)(𝜏−𝑡)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢_2,𝑛(𝑥,𝜏)𝜕𝜏^2𝛼+𝜕^2𝛼𝑢_1,𝑛(𝑥,𝜏)𝜕𝑥^2𝛼−𝑢_2,𝑛. (53)

Applying local fractional differential transform and integral transformation on (53), we can easily get 𝐷𝑇_𝑘𝑢_1,𝑛+1(𝑥,𝑡)=−𝐷𝑇_𝑘0𝐼_𝑡0^(𝛼)𝐼_𝑡^(𝛼)𝜕^2𝛼𝑢_2,𝑛(𝑥,𝜏)𝜕𝑥^2𝛼−𝑢_1,𝑛(𝑥,𝜏),𝐷𝑇_𝑘𝑢_2,𝑛+1(𝑥,𝑡)=−𝐷𝑇_𝑘0𝐼_𝑡0^(𝛼)𝐼_𝑡^(𝛼)𝜕^2𝛼𝑢_1,𝑛(𝑥,𝜏)𝜕𝑥^2𝛼−𝑢_2,𝑛(,().𝑥,𝜏)𝑘≥2 (54) Analyzing (52) and (54), we can deduce 𝐷𝑇_𝑘𝑢_1,𝑛+1(𝑥,𝑡)=−𝐷𝑇_𝑘𝑢_2,𝑛+1(𝑥,𝑡),𝐷𝑇_𝑘𝑢_2,𝑛+1(𝑥,𝑡)=2𝐷𝑇_𝑘0𝐼_𝑡0^(𝛼)𝐼_𝑡^(𝛼)𝑢_2,𝑛(𝑥,𝑡). (55) According to (54) and (55), we have 𝑈_𝑛+1()𝑈𝑥,𝑘=2^𝑛(𝑥,𝑘−2)()𝛼,().𝑘𝛼⋅𝑘−1𝑘≥2 (56) From (52), we can deduce 𝑈_𝑛+1()𝑈𝑥,𝑘=2^𝑛(𝑥,𝑘−2)𝑘𝛼⋅(𝑘−1)𝛼𝑘≥2,𝑘=2𝑘^′−1,𝑘^′,𝑈∈𝑁_𝑛+1(𝑥,𝑘)=0,𝑘≥2,𝑘=2𝑘^′,𝑘^′∈𝑁. (57) Then we get 𝑢_1,𝑛+1(𝑥,𝑡)=^∞_𝑘=2𝐷𝑇_𝑘⁻¹2𝑈_1,𝑛(𝑥,𝑘−2)()𝛼+𝑘𝛼𝑘−1_1𝑘=0𝑢^(𝑘𝛼)_1,0(𝑥,𝑘)Γ()𝑡1+𝑘𝛼Γ(^𝑘𝛼).1+𝑘𝛼 (58) By virtue of (58), we have 𝑢_1,1()=𝑥,𝑡_1𝑘=0𝑢^(𝑘𝛼)_1,0(𝑥,𝑘)Γ()𝑡1+𝑘𝛼Γ(^𝑘𝛼)+1+𝑘𝛼^∞_𝑘=2𝐷𝑇_𝑘⁻¹2𝑈_1,0(𝑥,𝑘−2)()𝛼=𝑡𝑘𝛼⋅𝑘−1_𝛼𝐸_𝛼𝑥^𝛼Γ(1+𝛼)+𝐷𝑇₃⁻¹2𝑈_1,0(𝑥,1)3𝛼⋅2𝛼=𝐸_𝛼𝑥^𝛼𝑡^𝛼Γ()+1+𝛼2𝑡^3𝛼Γ();𝑢1+3𝛼_1,2()=𝑥,𝑡_1𝑘=0𝑢^(𝑘𝛼)_1,0(𝑥,𝑘)Γ()𝑡1+𝑘𝛼Γ(^𝑘𝛼)+1+𝑘𝛼^∞_𝑘=2𝐷𝑇_𝑘⁻¹2𝑈_1,1(𝑥,𝑘−2)𝑘𝛼(𝑘−1)𝛼=𝐸_𝛼𝑥^𝛼𝑡^𝛼Γ()+1+𝛼2𝑡^3𝛼Γ(1+3𝛼)+𝐷𝑇₅⁻¹2𝑈_1,1(𝑥,3)4𝛼⋅5𝛼=𝐸_𝛼𝑥^𝛼𝑡^𝛼Γ()+1+𝛼2𝑡^3𝛼Γ()+1+3𝛼4𝑡^5𝛼Γ();𝑢1+5𝛼_1,3(𝑥,𝑡)=𝐸_𝛼𝑥^𝛼𝑡^𝛼Γ()+1+𝛼2𝑡^3𝛼Γ()+1+3𝛼4𝑡^5𝛼Γ()+1+5𝛼8𝑡^7𝛼Γ();⋮1+7𝛼 (59) Then we derive the following result: 𝑢_1,𝑛(𝑥,𝑡)=𝐸_𝛼𝑥^𝛼𝑛_𝑘=02^𝑘𝑡^{(2𝑘+1)𝛼}Γ(()𝛼).1+2𝑘+1 (60) Obviously 𝑢_2,𝑛(𝑥,𝑡)=−𝑢_1,𝑛(𝑥,𝑡)=−𝐸_𝛼𝑥^𝛼𝑛_𝑘=02^𝑘𝑡^{(2𝑘+1)𝛼}Γ(()𝛼).1+2𝑘+1 (61) Thus, the expression of the final solution of (50) reads as follows: 𝑢₁(𝑥,𝑡)=lim_𝑛→∞𝑢_1,𝑛(𝑥,𝑡)=𝐸_𝛼𝑥^𝛼sinh_𝛼√2𝑡^𝛼√2,𝑢₂(𝑥,𝑡)=lim_𝑛→∞𝑢_2,𝑛(𝑥,𝑡)=−𝐸_𝛼𝑥^𝛼sinh_𝛼√2𝑡^𝛼√2. (62)

Case 4. Consider the system of local fractional coupled Burger’s equations 𝜕^𝛼𝑢₁(𝑥,𝑡)𝜕𝑡^𝛼+𝜕^2𝛼𝑢₁(𝑥,𝑡)𝜕𝑥^2𝛼𝜕−2^𝛼𝑢₁(𝑥,𝑡)𝜕𝑥^𝛼𝑢₁()+𝜕𝑥,𝑡_𝛼𝑢₁(𝑥,𝑡)𝑢₂(𝑥,𝑡)𝜕𝑥^𝛼𝜕=0,_𝛼𝑢₂(𝑥,𝑡)𝜕𝑡^𝛼+𝜕^2𝛼𝑢₂(𝑥,𝑡)𝜕𝑥^2𝛼𝜕−2^𝛼𝑢₂(𝑥,𝑡)𝜕𝑥^𝛼𝑢₂()+𝜕𝑥,𝑡_𝛼𝑢₁(𝑥,𝑡)𝑢₂(𝑥,𝑡)𝜕𝑥^𝛼=0 (63) with the initial conditions 𝑢₁(𝑥,0)=𝑢₂(𝑥,0)=𝐸_𝛼𝑥^𝛼,𝑢₁(𝑥,0)=𝑢₁(𝑥,0)=𝐸_𝛼𝑥^𝛼. (64)

Now we can structure the similar local fractional iteration procedure.

Using (64), we take an initial value as 𝑢_1,0(𝑥,𝑡)=𝑢_2,0(𝑥,𝑡)=𝐸_𝛼𝑡^𝛼. (65)

According to variational iteration method [38], we can similarly construct local fractional variational iteration formula 𝑢_1,𝑛+1(𝑥,𝑡)=𝑢_1,𝑛()+𝑥,𝑡₀𝐼_𝑥^(𝛼)(𝜏−𝑥)^𝛼Γ()𝜕1+𝛼^𝛼𝑢_1,𝑛𝜕𝑡^𝛼+𝜕^2𝛼𝑢_1,𝑛𝜕𝑥^2𝛼𝜕−2^𝛼𝑢_1,𝑛𝜕𝑥^𝛼𝑢_1,𝑛+𝜕^𝛼𝑢_1,𝑛𝑢_2,𝑛𝜕𝑥^𝛼,𝑢_2,𝑛+1(𝑥,𝑡)=𝑢_2,𝑛()+𝑥,𝑡₀𝐼_𝑥^(𝛼)(𝜏−𝑥)^𝛼Γ()𝜕1+𝛼^𝛼𝑢_2,𝑛𝜕𝑡^𝛼+𝜕^2𝛼𝑢_2,𝑛𝜕𝑥^2𝛼𝜕−2^𝛼𝑢_2,𝑛𝜕𝑥^𝛼𝑢_2,𝑛+𝜕^𝛼𝑢_1,𝑛𝑢_2,𝑛𝜕𝑥^𝛼. (66)

Applying local fractional differential transform and integral transformation on (66), we can easily get 𝐷𝑇_𝑘𝑢_1,𝑛+1(𝑥,𝑡)=−𝐷𝑇_𝑘0𝐼_𝑥0^(𝛼)𝐼_𝑥^(𝛼)×𝜕^𝛼𝑢_1,𝑛𝜕𝑡^𝛼𝜕−2^𝛼𝑢_1,𝑛𝜕𝑥^𝛼𝑢_1,𝑛+𝜕^𝛼𝑢_1,𝑛𝑢_2,𝑛𝜕𝑥^𝛼,𝐷𝑇_𝑘𝑢_2,𝑛+1(𝑥,𝑡)=−𝐷𝑇_𝑘0𝐼_𝑥0^(𝛼)𝐼_𝑥^(𝛼)×𝜕^𝛼𝑢_2,𝑛𝜕𝑡^𝛼𝜕−2^𝛼𝑢_2,𝑛𝜕𝑥^𝛼𝑢_2,𝑛+𝜕^𝛼𝑢_1,𝑛𝑢_2,𝑛𝜕𝑥^𝛼,(𝑘≥2). (67) Analyzing (64) and (67), we can deduce 𝐷𝑇_𝑘𝑢_1,𝑛+1(𝑥,𝑡)=𝐷𝑇_𝑘𝑢_2,𝑛+1(𝑥,𝑡),𝐷𝑇_𝑘𝑢_2,𝑛+1(𝑥,𝑡)=−𝐷𝑇_𝑘0𝐼_𝑥0^(𝛼)𝐼_𝑥^(𝛼)𝑢_2,𝑛(𝑥,𝑡). (68) According to (67) and (68), we have 𝑈_1,𝑛+1()𝑈𝑘,𝑡=−^1,𝑛(𝑘−2,𝑡)()𝛼,().𝑘𝛼⋅𝑘−1𝑘≥2 (69) From (64), we can deduce 𝑈_1,𝑛+1()𝑈𝑘,𝑡=−^1,𝑛(𝑘−2,𝑡)𝑘𝛼⋅(𝑘−1)𝛼𝑘≥2,𝑘=2𝑘^′,𝑘^′,𝑈∈𝑁_𝑛+1(𝑘,𝑡)=0,𝑘≥2,𝑘=2𝑘^′−1,𝑘^′∈𝑁. (70) Then we get 𝑢_1,𝑛+1(𝑥,𝑡)=^∞_𝑘=2𝐷𝑇_𝑘⁻¹−𝑈_1,𝑛(𝑘−2,𝑡)()𝛼+𝑘𝛼𝑘−1_1𝑘=0𝑢^(𝑘𝛼)_1,0(𝑘,𝑡)Γ()𝑥1+𝑘𝛼Γ(^𝑘𝛼).1+𝑘𝛼 (71) By virtue of (71), we have 𝑢_1,1()=𝑥,𝑡_1𝑘=0𝑢^(𝑘𝛼)_1,0(𝑘,𝑡)Γ()𝑥1+𝑘𝛼Γ(^𝑘𝛼)+1+𝑘𝛼_∞𝑘=2𝐷𝑇_𝑘⁻¹−𝑈_1,0(𝑘−2,𝑡)𝑘𝛼⋅(𝑘−1)𝛼=𝐸_𝛼𝑡^𝛼+𝐷𝑇₂⁻¹−𝑈_1,0(0,𝑡)2𝛼⋅𝛼=𝐸_𝛼𝑡^𝛼1^𝛼−𝑥^2𝛼Γ();𝑢1+2𝛼_1,2()=𝑥,𝑡_1𝑘=0𝑢^(𝑘𝛼)_1,0(𝑘,𝑡)Γ()𝑥1+𝑘𝛼Γ(^𝑘𝛼)+1+𝑘𝛼_∞𝑘=2𝐷𝑇_𝑘⁻¹−𝑈_1,1(𝑘−2,𝑡)𝑘𝛼(𝑘−1)𝛼=𝐸_𝛼𝑡^𝛼1^𝛼−𝑥^2𝛼Γ(1+2𝛼)+𝐷𝑇₄⁻¹−𝑈_1,1(2,𝑡)4𝛼⋅3𝛼=𝐸_𝛼𝑡^𝛼1^𝛼−𝑥^2𝛼Γ()+𝑥1+2𝛼Γ(^4𝛼);𝑢1+4𝛼_1,3(𝑥,𝑡)=𝐸_𝛼𝑡^𝛼1^𝛼−𝑥^2𝛼Γ()+𝑥1+2𝛼Γ(^4𝛼)−𝑥1+4𝛼Γ(^6𝛼);⋮1+6𝛼 (72) Then we derive the following result: 𝑢_1,𝑛(𝑥,𝑡)=𝐸_𝛼𝑡^𝛼𝑛_𝑘=0(−1)^𝑘𝑥^2𝑘𝛼Γ().1+2𝑘𝛼 (73) Obviously 𝑢_2,𝑛(𝑥,𝑡)=𝑢_1,𝑛(𝑥,𝑡)=𝐸_𝛼𝑡^𝛼𝑛_𝑘=0(−1)^𝑘𝑥^2𝑘𝛼Γ().1+2𝑘𝛼 (74) Thus, the expression of the final solution of (63) reads as follows: 𝑢₁(𝑥,𝑡)=lim_𝑛→∞𝑢_1,𝑛(𝑥,𝑡)=𝐸_𝛼𝑡^𝛼cos_𝛼𝑥^𝛼,𝑢₂(𝑥,𝑡)=lim_𝑛→∞𝑢_2,𝑛(𝑥,𝑡)=𝐸_𝛼𝑡^𝛼cos_𝛼𝑥^𝛼. (75)

5. Conclusions

In this work, we proposed the local fractional iteration transform method. The applications of the methods for solving fractional differential equations with local fractional derivative are discussed in several cases. The method provides the variational iteration formula in the form of polynomial or Maclaurin’s series, which is more easily to deal with convergence and approximate problem. Of course, the new problems are beyond the scope of the present work.

Conflict of Interests

The authors declare that they have no conflict of interests regarding this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 609040410), the Foundation and Advanced Technology Research Program of Henan province (112300410300), and the Nanyang Normal University (NYNU2005k37). The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the paper.