A General Iteration Formula of VIM for Fractional Heat- and Wave-Like Equations

Abstract

A general iteration formula of variational iteration method (VIM) for fractional heat- and wave-like equations with variable coefficients is derived. Compared with previous work, the Lagrange multiplier of the method is identified in a more accurate way by employing Laplace’s transform of fractional order. The fractional derivative is considered in Jumarie’s sense. The results are more accurate than those obtained by classical VIM and the same as ADM. It is shown that the proposed iteration formula is efficient and simple.

1. Introduction

Fractional differential equations (FDEs) have been proved to be a valuable tool in the modelling of many phenomena in the fields of applied sciences. This is because of the fact that fractional derivatives provide an excellent instrument to describe the memory and hereditary properties of various materials and processes. With the increasing applications of FDEs, considerable attentions have been paid to provide efficient methods for finding the exact and numerical solutions of FDEs. Recently, some approximate methods such as Adomian’s decomposition method [1–5], homotopy perturbation method [6–10], variational iteration method [11–27], and homotopy analysis method [28, 29] are given to find an analytical approximation to FDEs.

The variational iteration method (VIM) was first proposed by He et al. [11–15] and has been shown to be efficient for handling nonlinear problems. Thus, there has been a great deal of interest in FDEs by using the VIM [17–27]. For VIM, the key points are the construction of correct function and the identification of the Lagrange multiplier. However, the previous work either avoids the term of fractional derivative and handles them as a restricted variation or identifies the Lagrange multipliers by an approximate method, resulting in a poor convergence and inaccuracy. So, it is urgent to find a new method which can identify the Lagrange multiplier in a more accurate way.

In this paper, we will use a correct function described in [27] and give a new method to evaluate the Lagrange multipliers by employing Laplace’s transform of fractional order. Then, we develop a new framework of VIM for the three-dimensional fractional heat- and wave-like equations of the form [3]: 𝜕^𝛼𝑇𝜕𝑡^𝛼=𝑓𝑥,𝑦,𝑧𝑇_𝑥𝑥+𝑔𝑥,𝑦,𝑧𝑇_𝑦𝑦+ℎ𝑥,𝑦,𝑧𝑇_𝑧𝑧0<𝑥<𝑎,0<𝑦<𝑏,0<𝑧<𝑐,𝑛<𝛼<𝑛+1,𝑡>0, (1) subject to the boundary conditions 𝑇0,𝑦,𝑧,𝑡=𝑓₁𝑦,𝑧,𝑡,𝑇_𝑥𝑎,𝑦,𝑧,𝑡=𝑓₂,𝑇()𝑦,𝑧,𝑡𝑥,0,𝑧,𝑡=𝑔₁(𝑥,𝑧,𝑡),𝑇_𝑦(𝑥,𝑏,𝑧,𝑡)=𝑔₂(),𝑇𝑥,𝑧,𝑡𝑥,𝑦,0,𝑡=ℎ₁𝑥,𝑦,𝑡,𝑇_𝑧𝑥,𝑦,𝑐,𝑡=ℎ₂𝑥,𝑦,𝑡, (2) and the initial conditions 𝑇𝑥,𝑦,𝑧,0=𝜓𝑥,𝑦,𝑧,𝑇_𝑡𝑥,𝑦,𝑧,0=𝜂𝑥,𝑦,𝑧, (3) where 𝛼 is a parameter describing the fractional derivative. In the case of 0<𝑎≤1, (1) reduces to the fractional heat-like equation with variable coefficients and to the fractional wave-like equation that models anomalous diffusive and subdiffusive systems in the case of 1<𝑎≤2. The approximate solutions of (1) have been studied by using the ADM [3], VIM [17], FVIM [23], and modified VIM [24].

The remainder of the paper is organized as follows. In Section 2, we describe some necessary preliminaries of the fractional calculus and the Laplace transform for our subsequent development. Section 3 is devoted to the derivation of general iteration formula for the fractional heat- and wave-like equations. In Section 4, four examples are given to demonstrate our conclusions. Finally, a brief summary is presented.

2. Preliminaries

In this section, we cover some preliminaries. First, we list some basic definitions about fractional calculus. Second, the Laplace transforms of fractional integral and derivative are described. For more details, see [30–39].

2.1. Fractional Calculus

Definition 1. A real function ℎ(𝑡), 𝑡>0, is said to be in the space 𝐶_𝜇, 𝜇∈𝑅, if there exist a real number 𝑝>𝜇, such that ℎ(𝑡)=𝑡^𝑝ℎ₁(𝑡), where ℎ₁(𝑡)∈𝐶(0,∞), and it is said to be in the space 𝐶^𝑛_𝜇 if and only if ℎ^(𝑛)∈𝐶_𝜇, 𝑛∈𝑁.

Definition 2. The Riemann-Liouville fractional integral operator (𝐽^𝛼) of order 𝛼≥0, of a function 𝑓∈𝐶_𝜇, 𝜇≥−1, is defined as 𝐽^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝐽𝑑𝜏,𝑡>0,₀𝑓(𝑡)=𝑓(𝑡), (4) when 0<𝛼≤1, 𝐽^𝛼𝑓(𝑡)=(1/Γ(𝛼))∫^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏=(1/Γ(1+𝛼))∫^𝑡₀𝑓(𝜏)(𝑑𝜏)^𝛼.

If we denote the Riemann-Liouville fractional derivative by 𝐷^𝛼, then the next equation define the Riemann-Liouville fractional derivative of order 𝑚𝐷^𝛼𝑓(𝑥)=𝑑^𝑚𝑑𝑥^𝑚𝐽^𝑚−𝛼𝑓(𝑥), (5) where 𝑚−1<𝛼≤𝑚, 𝑚∈𝑁.

Jumarie (see [31–37]) proposed a simple alternative definition to the Riemann-Liouville derivative. Since his modified Riemann-Liouville derivative is defined for arbitrary continuous (nondifferentiable) functions and the fractional derivative of a constant is equal to zero, it has the advantages of both the standard Riemann-Liouville and the Caputo fractional derivatives. Now, we present some notations, definitions, and preliminary facts of the modified Riemann-Liouville derivative which will be used later in this work.

Definition 3. Jumarie’s fractional derivative is a modified Riemann-Liouville derivative defined by the expression [37, 38] 𝑓^(𝛼)(𝑥)1∶=Γ(−𝛼)^𝑥₀(𝑥−𝜉)^−𝛼−1𝑓(𝜉)𝑑𝜉,𝛼<0. (6) For positive 𝛼, one will set 𝑓^(𝛼)(𝑥)∶=𝑓^(𝛼−1)(𝑥)^′=1Γ()𝑑1−𝛼×𝑑𝑥_𝑥0(𝑥−𝜉)^−𝛼𝑓(𝜉)(0)𝑓−𝑓𝑑𝜉,0<𝛼<0,_(𝛼)(𝑥)∶=𝑓^{(𝛼−𝑛)}(𝑥)^(𝑛),𝑛≤𝛼<𝑛+1,𝑛≥1. (7)

In addition, we want to give some properties of Jumarie’s fractional derivative.

Theorem 4 (the fractional Leibniz product rule [36]). If 𝑓 and 𝑔 are two continuous functions on [0,1], then 𝑓(𝑥)𝑔(𝑥)^𝛼=𝑓(𝑥)^𝛼𝑔(𝑥)+𝑓(𝑥)𝑔(𝑥)^𝛼. (8)

Theorem 5 (the fractional Barrow’s formula [37]). For a continuous function 𝑓, one has ^𝑥₀𝑓^(𝛼)(𝑡)(𝑑𝑡)^(𝛼)=𝛼!𝑓(𝑥)−𝑓(0), (9) where 𝛼!=Γ(1+𝛼).

From Theorems 4 and 5, the formula of integration by parts is given as ¹₀𝑢^(𝛼)(𝑥)𝑣(𝑥)(𝑑𝑥)^(𝛼)=¹₀(𝑢(𝑥)𝑣(𝑥))^(𝛼)(𝑑𝑥)^(𝛼)−¹₀𝑢(𝑥)𝑣^(𝛼)(𝑥)(𝑑𝑥)^(𝛼)=𝛼![𝑢(𝑥)𝑣(𝑥)]¹₀−¹₀𝑢(𝑥)𝑣^(𝛼)(𝑥)(𝑑𝑥)^(𝛼). (10)

Definition 6. Fractional derivative of compounded functions [37, 38] is defined as 𝑑^𝛼𝑓≅Γ(1+𝛼)𝑑𝑓,0<𝛼<1. (11)

Definition 7. The integral with respect to (𝑑𝑥)^𝛼 [37, 38] is defined as the solution of the fractional differential equation 𝑑𝑦≅𝑓(𝑥)(𝑑𝑥)^𝛼,𝑥≥0,𝑦(0)=0,0<𝛼<1. (12)

Lemma 8. Let 𝑓(𝑥) denote a continuous function [37, 38], then the solution 𝑦(𝑥), 𝑦(0)=0 of (8) is defined by the equality 𝑦=^𝑥₀𝑓(𝜉)(𝑑𝜉)^𝛼=𝛼^𝑥₀(𝑥−𝜉)^𝛼𝑓(𝜉)𝑑𝜉,0<𝛼<1. (13)

Lemma 9. Let 𝑛−1<𝛼≤𝑛, 𝑛∈𝑁, 𝑡>0, ℎ∈𝐶^𝑛_𝜇, 𝜇≥−1, then 𝐽^𝛼𝐷^𝛼ℎ(𝑡)=ℎ(𝑡)−^𝑛−1_𝑘=0ℎ^(𝑘)0⁺𝑡^𝑘.𝑘! (14)

2.2. The Laplace Transform

Definition 10. The Laplace transform 𝐿{𝑓(𝑥)}∶=𝐹(𝑠), 𝑠∈𝐶, of a 𝑅→𝐶 function 𝑓(𝑥) is defined by the integral 𝐿𝑓(𝑥)∶=∶𝐹(𝑠)∶=^∞₀𝑒^𝑠𝑥𝑓(𝑥)𝑑𝑥. (15)

Lemma 11. The Laplace transform of the fractional derivative (the Riemann-Liouville derivative) is 𝐿𝐷^𝛼_𝑥𝑓(𝑥);𝑠=𝑠^𝛼𝐹(𝑠)−^𝑛_𝑘=1𝑠^𝑘−1𝐷^𝛼−𝑘𝑓(0+). (16)

Definition 12. The inverse Laplace transform is defined by the complex integral 𝑓(𝑡)=𝐿⁻¹{𝐹(𝑠=1)}2𝜋𝑖lim_𝑇→∞^{𝜌+𝑖𝑇}_{𝜌−𝑖𝑇}𝑒^𝑠𝑡𝐹(𝑠)𝑑𝑠, (17) where the integration is done along the vertically Re(𝑠)=𝜌 in the complex plane such that 𝜌 is greater than the real part of all singularities of 𝐹(𝑠). This ensures that the contour path is in the region of convergence. If all singularities are in the left half-plane, or 𝐹(𝑠) is a smooth function on −∞<Re(𝑠)<∞ (i.e., no singularities), then 𝜌 can be set to zero and the above inverse integral formula becomes identical to the inverse Fourier transform.

Now, we will introduce the definition of the fractional Laplace transform derived by Jumarie [38] for the first time, and some results of the fractional Laplace transform are also presented.

Definition 13. Let 𝑓(𝑥) denote a function which vanishes for negative values of 𝑥. Its Laplace's transform 𝐿_𝛼{𝑓(𝑥)} of order 𝛼 (or its 𝛼th fractional Laplace's transform) is defined by the following expression, when it is finite: 𝐿_𝛼𝑓(𝑥)∶=∶𝐹_𝛼(𝑠)=^∞₀𝐸_𝛼−𝑠^𝛼𝑥^𝛼𝑓(𝑥)(𝑑𝑥)^𝛼,0<𝛼≤1, (18) where 𝑠∈𝐶 and 𝐸_𝛼(𝑢) is the Mittag-Leffler function ∑^∞_𝑘=0(𝑢^𝑘/Γ(1+𝑘𝛼)).

Lemma 14. If one defines the convolution of order 𝛼 of the two functions 𝑓(𝑥) by the expression 𝑓(𝑥)∗𝑔(𝑥)_𝛼∶=^𝑥₀𝑓(𝑥−𝑢)𝑔(𝑢)(𝑑𝑢)^𝛼,0<𝛼≤1. (19) Then, one has the equality 𝐿_𝛼𝑓(𝑥)∗𝑔(𝑥)_𝛼=𝐿_𝛼𝑓(𝑥)𝐿_𝛼𝑔(𝑥). (20)

Corollary 15. Given the Laplace transform that one has the inversion formula 𝑓(𝑥)=1𝑀_𝛼^+𝑖∞_−𝑖∞𝐸_𝛼𝑠^𝛼𝑥^𝛼𝐹_𝛼(𝑠)(𝑑𝑠)^𝛼,0<𝛼≤1, (21) where 𝑀_𝛼 is the period of the complex-valued Mittag-Leffler function defined by the equality 𝐸_𝛼(𝑖(𝑀_𝛼)^𝛼)=1.

Lemma 16. The fractional Laplace transform of the fractional derivative (the modified Riemann-Liouville derivative) is 𝐿_𝛼𝑓^(𝛼)(𝑥)=𝑠^𝛼𝐿_𝛼𝑓(𝑥)−Γ(1+𝛼)𝑓(0),0<𝛼≤1. (22)
Some properties of the fractional Laplace transform are given as follows [38]: 𝐿_𝛼𝑥^𝛼𝑓(𝑥)=−𝐷^𝛼_𝑠𝐿𝑓(𝑥,𝐿_𝛼)𝑓(𝑎𝑥)_𝑠=1𝑎^𝛼𝐿_𝛼𝑓(𝑥)_𝑠/𝛼,𝐿_𝛼𝑓(𝑥−𝑏)=𝐸_𝛼−𝑠^𝛼𝑏^𝛼𝐿_𝛼𝑓(𝑥,𝐿_𝛼)𝐸_𝛼−𝑐^𝛼𝑥^𝛼𝑓(𝑥)_𝑠=𝐿_𝛼𝑓(𝑥)_𝑠+𝑐,𝐿_𝛼−𝑥^𝛼𝑓(𝑥)=𝐷^𝛼_𝑠𝐿_𝛼𝑓(𝑥,𝐿_𝛼𝑥0)𝑓(𝑢)(𝑑𝑢)^𝛼=Γ⁻¹(1+𝛼)𝑠^−𝛼𝐿_𝛼𝑓(𝑥). (23)

3. Fractional Variational Iteration Method

In order to illustrate the solution procedure of the variational iteration method, we consider the following fractional differential equation: 𝐷^𝛼_𝑡𝑢𝑥,𝑦,𝑡+𝑁𝑢𝑥,𝑦,𝑡+𝑅𝑢𝑥,𝑦,𝑡=𝑔𝑥,𝑦,𝑡,𝑡>0,𝑚−1<𝛼≤𝑚, (24) where 𝑅 is the linear operator, 𝑁 is the nonlinear operator, and 𝐷^𝛼_𝑡 is the modified Rieman-Liouville derivative of order 𝛼.

Subject to the initial condition, 𝑢^(𝑘)𝑥,𝑦,0=𝑐_𝑘𝑥,𝑦,𝑘=0,1,…,𝑚 (25) Let 𝑢_𝑘(𝑡)=𝑢_𝑘(𝑥,𝑦,𝑡), 𝑔_𝑘(𝑡)=𝑔_𝑘(𝑥,𝑦,𝑡).

Before our solution, we will describe the fractional variational iteration method described in [23, 25, 26], which construct a correct function for (24) as 𝑢_𝑘+1(𝑡)=𝑢_𝑘(𝑡)+𝐽^𝛼_𝜏𝜆(𝑡,𝜏)𝐷^𝛼_𝜏𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏)=𝑢_𝑘(𝑡)+1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝜆()×𝐷𝑡,𝜏_𝛼𝜏𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏)𝑑𝜏. (26)

Then, by using Lemma 8 proposed by Jumarie [37, 38] for the first time, one gets a correction function 𝑢_𝑘+1(𝑡)=𝑢_𝑘(𝑡)+1Γ(1+𝛼)×^𝑡₀𝜆(𝑡,𝜏)𝐷^𝛼_𝜏𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏)(𝑑𝜏)^𝛼. (27) But, unfortunately, (27) holds true only when 0<𝛼≤1. In the case of 𝑚−1<𝛼≤𝑚, 𝑚=2,3,4,…, some modifications must be made. For example, replacing the fractional order 𝛼 (𝑚−1<𝛼≤𝑚) by the order 𝑚𝛼(0<𝛼≤1), whether there is a general iteration formula for (24). Certainly, there is. In the remainder section, we will use two methods to derive a general iteration formula of VIM for (24). Unlike the previous work, which calculates the Lagrange multiplier by some approximate methods, we will use a more accurate way by employing the properties of Laplace’s transform.

3.1. The Laplace Transform Method

According to VIM [27], we can construct a correction functional as follows: 𝑢_𝑘+1(𝑡)=𝑢_𝑘(𝑡)+𝐽^𝛽_𝜏𝜆(𝑡,𝜏)𝐷^𝛼_𝜏𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏)=𝑢_𝑘(𝑡)+1Γ𝛽^𝑡₀(𝑡−𝜏)^𝛽−1𝜆()×𝐷𝑡,𝜏_𝛼𝜏𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏)𝑑𝜏, (28) where 𝐽^𝛽_𝜏 is the Riemann-Liouville fractional integral operator of order 𝛽=𝛼−floor(𝛼), that is, 𝛽=𝛼+1−𝑚, with respect to the variable 𝑡, and 𝜆 is a general Lagrange multiplier, and 𝑢_𝑘(𝜏) is a restricted variation, that is, 𝛿𝑢(𝜏)=0.

By taking the Laplace transform on the both sides of (28), we have 𝐿𝑢_𝑘+1(𝑡𝑢)=𝐿_𝑘(𝑡1)+𝐿Γ𝛽^𝑡₀(𝑡−𝜏)^𝛽−1𝜆()×𝐷𝑡,𝜏_𝛼𝜏𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏)𝑑𝜏, (29) where 𝐿 is the operator of the Laplace transform. 𝐿𝑢_𝑘+1(𝑡)=𝐿𝑢_𝑘(𝑡+1)Γ𝛽𝐿(𝑡−𝜏)^𝛽−1𝜆(𝐷𝑡,𝜏)×𝐿^𝛼_𝜏𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏). (30)

From the Lemma 11, we get 𝐿𝑢_𝑘+1(𝑡)=𝐿𝑢_𝑘(𝑡+1)Γ𝛽𝐿(𝑡−𝜏)^𝛽−1𝜆(𝑠𝑡,𝜏)×𝐿^𝛼𝐿𝑢_𝑘(𝜏)−^𝑛_𝑘=1𝑠^𝑘−1𝐷^𝛼−𝑘𝑢_𝑘(0)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏). (31)

Taking the variation derivative 𝛿 on the both sides of (31), we can derive 𝐿𝑢_𝑘+1(𝑡)=𝐿𝑢_𝑘(𝑡+1)Γ𝛽𝐿(𝑡−𝜏)^𝛽−1𝜆(𝑠𝑡,𝜏)×𝛿^𝛼𝐿𝑢_𝑘(𝜏)+𝐿𝑅𝑢_𝑘(𝜏). (32)

Without loss of generality, assuming that 𝛿𝑅[𝑢_𝑘(𝜏)]=0, then we have 𝛿𝐿𝑢_𝑘(𝑡1)∶1+Γ𝛽𝑠^𝛼𝐿(𝑡−𝜏)^𝛽−1𝜆(𝑡,𝜏)=0. (33)

Set the coefficient of 𝛿𝐿{𝑢_𝑘(𝑡)} to zeros, we obtain 𝐿(𝑡−𝜏)^𝛽−1𝜆(𝑡,𝜏)=−Γ𝛽𝑠^−𝛼. (34)

By employing the inverse Laplace transform, we have (𝑡−𝜏)^𝛽−1𝜆()()𝑡,𝜏=−𝑡−𝜏^𝛼−1Γ𝛽Γ(𝛼). (35)

Substituting (35) into (28), we get the iteration formula as follows: 𝑢_𝑘+1(𝑡)=𝑢_𝑘(𝑡)−𝐽^𝛼_𝜏𝐷^𝛼_𝑡𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏). (36)

3.2. The Fractional Laplace Transform Method

In order to illustrate our method, we replace the fractional order 𝛼(𝑚−1<𝛼≤𝑚) by the order 𝑚𝛼(0<𝛼≤1). According to VIM [27], we can construct a correction function as follows: 𝑢_𝑘+1(𝑡)=𝑢_𝑘(𝑡)+𝐽^𝛼_𝜏𝜆(𝑡,𝜏)𝐷_𝜏^𝑚𝛼𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏). (37)

Taking the fractional Laplace transform on the both sides of (37), we have 𝐿_𝛼𝑢_𝑘+1(𝑡)=𝐿_𝛼𝑢_𝑘(𝑡)+𝐿_𝛼𝐽^𝛼_𝜏𝜆(𝑡,𝜏)𝐷_𝜏^𝑚𝛼𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏), (38) where 𝐿_𝛼 is the fractional Laplace transform of order 𝛼.

By assuming that 𝜆(𝑡,𝜏) has the form as 𝜆(𝑡,𝜏)=𝜆(𝑡−𝜏) and using (10), then we get 𝐽^𝛼_𝜏𝜆𝐷_𝜏^𝑚𝛼𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)(𝜏=1−𝑔)Γ()×𝜆()∗𝐷1+𝛼𝑡,𝜏_𝜏^𝑚𝛼𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏)_𝛼. (39)

From (20), we have 𝐿_𝛼𝐽^𝛼_𝜏𝜆𝐷_𝜏^𝑚𝛼𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)(𝜏=1−𝑔)Γ()𝐿1+𝛼^𝛼{𝜆(𝑡,𝜏)}×𝐿_𝛼𝐷_𝜏^𝑚𝛼𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏). (40)

Substituting (40) into (38) and then taking the variation derivative 𝛿 on the both sides of (35), we can derive 𝐿_𝛼𝑢_𝑘+1(𝑡)=𝐿_𝛼𝑢_𝑘(𝑡+1)Γ()𝐿1+𝛼^𝛼{𝜆(𝐿𝐷𝑡−𝜏)}×𝛿_𝜏^𝑚𝛼𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏). (41)

Without loss of generality, assuming that 𝛿[𝑅𝑢_𝑘(𝜏)]=0, then we have 𝛿𝐿_𝛼𝑢_𝑘(𝑡1)∶1+Γ()𝐿1+𝛼^𝛼{𝜆(𝑡−𝜏)}𝑠^𝑚𝛼=0. (42)

Set the coefficient of 𝛿𝐿_𝛼{𝑢_𝑘(𝑡)} to zeros, we have 𝐿_𝛼{𝜆(Γ()𝑡−𝜏)}=−1+𝛼𝑠^𝑚𝛼. (43)

By employing the inverse fractional Laplace transform, we have 𝜆()Γ()𝑡,𝜏=−1+𝛼Γ(()𝛼)()1+𝑚−1𝑡−𝜏^{(𝑚−1)𝛼}. (44)

Substituting (40) into (26), we get the iteration formula as follows: 𝑢_𝑘+1(𝑡)=𝑢_𝑘(𝑡)+𝐽^𝛼_𝜏−Γ(1+𝛼)Γ(()𝛼)()1+𝑚−1𝑡−𝜏^{(𝑚−1)𝛼}×𝐷_𝜏^𝑚𝛼𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏)=𝑢_𝑘(𝑡)−𝐽_𝜏^𝑚𝛼𝐷_𝜏^𝑚𝛼𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)+𝑅𝑢_𝑘(𝜏)−𝑔(𝜏). (45)

After replacing the fractional order 𝑚𝛼(0<𝛼≤1) by the order 𝛼(𝑚−1<𝛼≤𝑚), we could find that (36) and (45) are the same. So, we could get a general iteration formula for (1) as follows: 𝑇_𝑘+1(𝑡)=𝑇_𝑘(𝑡)−𝐽^𝛼_𝜏𝐷^𝛼_𝜏𝑇_𝑘(𝜏)−𝑓𝜕𝑥,𝑦,𝑧²𝑇_𝑘𝜕𝑥²𝜕+𝑔𝑥,𝑦,𝑧²𝑇_𝑘𝜕𝑦²𝜕+ℎ𝑥,𝑦,𝑧²𝑇_𝑘𝜕𝑧². (46)

By using the Lemma 11, we finally have 𝑇_𝑘+1(𝑡)=^𝑚−1_𝑖=0𝑇_𝑘^(𝑖)0⁺𝑡^𝑖𝑖!+𝐽^𝛼_𝜏𝑓𝜕𝑥,𝑦,𝑧²𝑇_𝑘𝜕𝑥²𝜕+𝑔𝑥,𝑦,𝑧²𝑇_𝑘𝜕𝑦²𝜕+ℎ𝑥,𝑦,𝑧²𝑇_𝑘𝜕𝑧². (47)

4. Applications and Results

Example 17. Consider the following fractional model for heat conduction in polar bear hairs proposed by Qing-Li et al. [40]: 𝐷^𝛼_𝑡𝜕𝑇+𝐷²𝑇𝜕𝑥²=0,𝑥∈(0,1),𝑡>0,0<𝛼≤1, (48) where 𝐷^𝛼_𝑡𝑇 is the modified Rieman-Liouville derivative and 𝐷 is a constant, with the initial condition 𝑇(𝑥,0)=𝑎−𝑎−𝑏()+1−exp−1/𝑘𝐷𝑎−𝑏()−𝑥1−exp−1/𝑘𝐷exp,𝑘𝐷 (49) where 𝑎 is the body temperature, 𝑏 is the environment temperature, and 𝑘 is a constant. When 𝛼=1, the exact solution of (46) is 𝑇(𝑥,𝑡)=𝑎−𝑎−𝑏()+1−exp−1/𝑘𝐷𝑎−𝑏()−𝑥1−exp−1/𝑘𝐷exp−𝑡𝑘𝐷𝐷𝑘². (50)
According to the general iteration formula of VIM, one can get the iteration formulation as follows: 𝑇_𝑛+1(𝑥,𝑡)=𝑇(𝑥,0)−𝐽^𝛼𝐷𝜕²𝑇_𝑛𝜕𝑥². (51)
For the convenience, let 𝑐=(𝑎−𝑏)/(1−exp(−1/𝑘𝐷)) and 𝑑=𝑎−𝑐, then 𝑇₀−𝑥=𝑑+𝑐exp.𝑘𝐷 (52)
Starting with the initial value as shown in (52), we can derive 𝑇₁()−𝑥𝑥,𝑡=𝑑+𝑐exp1𝑘𝐷1−𝐷𝑘²Γ(1)Γ()𝑡1+𝛼^𝛼,𝑇₂()−𝑥𝑥,𝑡=𝑑+𝑐exp×1𝑘𝐷1−𝐷𝑘²Γ(1)Γ()𝑡1+𝛼^𝛼+1𝐷𝑘²²Γ(1)Γ()𝑡1+2𝛼^2𝛼, (53) and then 𝑇_𝑛()−𝑥𝑥,𝑡=𝑑+𝑐exp𝑘𝐷^𝑛_𝑚=01Γ()()1+𝑚𝛼−𝑡^𝛼𝐷𝑘^2𝑚. (54) So, the solution is 𝑇(𝑥,𝑡)=lim_𝑛→∞𝑇_𝑛()−1𝑥,𝑡=𝑑+𝑐exp𝑥𝐸𝑘𝐷^𝛼−𝑡^𝛼𝐷𝑘², (55) which is the exact solution.

Example 18. Consider the one-dimensional fractional heat-like equation: 𝜕^𝛼𝑢𝜕𝑡^𝛼=12𝑥²𝜕²𝑢𝜕𝑥²,𝑡>0,0<𝑥<1,1<𝛼≤2 (56) subject to the initial condition 𝑢()()𝑥,0=𝑥,𝜕𝑢𝑥,0𝜕𝑡=𝑥², (57) when 𝛼=2 and the exact solution is 𝑢(𝑥,𝑡)=𝑥+𝑥²sinh(𝑡).

According to the general iteration formula of VIM, one can get the following formula of iteration: 𝑢_𝑘+1(𝑥,𝑡)=𝑢_𝑘(𝑥,0)+𝜕𝑢_𝑘(𝑥,0)𝜕𝑡𝑡+𝐽^𝛼12𝑥²𝜕²𝑢_𝑘𝜕𝑥². (58)

Starting with an initial approximation 𝑢₀=𝑥+𝑥²𝑡, one can obtain 𝑢₁=𝑥+𝑥²1𝑡+Γ()𝑥2+𝛼²𝑡^1+𝛼,𝑢₂=𝑥+𝑥²𝑡𝑡+^1+𝛼Γ()+𝑡2+𝛼Γ(^1+2𝛼),𝑢2+2𝛼₃=𝑥+𝑥²𝑡𝑡+^1+𝛼Γ()+𝑡2+𝛼Γ(^1+2𝛼)+𝑡2+2𝛼Γ(^1+3𝛼).2+3𝛼 (59)

The solution in a series form is given by 𝑢(𝑥,𝑡)=lim_𝑘→∞𝑢_𝑘(𝑥,𝑡)=𝑥+𝑥²𝑡^∞_𝑘=0𝑡^𝑘𝛼Γ(𝑘𝛼+2)=𝑥+𝑥²𝑡𝐸_𝛼,2𝑡^𝛼, (60) where 𝐸_𝛼,2(𝑡^𝛼) denotes the two-parameter Mittag-Leffler function. The result obtained in (56) is exactly the same result, obtained by Momani [3] and Faraz et al. [23].

Example 19. Consider the one-dimensional fractional heat-like equation: 𝜕^2𝛼𝑢𝜕𝑡^2𝛼=1𝑥12²𝑢_𝑥𝑥+𝑦²𝑢_𝑦𝑦,0<𝑥,𝑦<1,0.5<𝛼≤1,𝑡>0, (61) subject to the boundary conditions 𝑢𝑢()()0,𝑦,𝑡=0,𝑢1,𝑦,𝑡=4cosh𝑡,𝑥,0,𝑡=0,𝑢𝑥,1,𝑡=4sinh𝑡, (62) and the initial condition 𝑢𝑥,𝑦,0=𝑥⁴,𝑢_𝑡𝑥,𝑦,0=𝑦⁴. (63) The exact solution 𝛼=1 was found to be [41] 𝑢𝑥,𝑦,𝑡=𝑥⁴cosh𝑡+𝑦⁴sinh𝑡. (64)
According to the general iteration formula of VIM, one can get the following formula of iteration: 𝑢_𝑘+1(𝑥,𝑡)=𝑢_𝑘(𝑥,𝑡)−𝐽^2𝛼𝜕^2𝛼𝑢𝜕𝑡^2𝛼−1𝑥12²𝑢_𝑥𝑥+𝑦²𝑢_𝑦𝑦. (65)
Starting with an initial approximation 𝑢₀(𝑥,𝑦,𝑡)=𝑢(𝑥,𝑦,0)=𝑥⁴+𝑦⁴𝑡, one can obtain the following successive approximate: 𝑢₁=𝑥⁴11+Γ()𝑡1+2𝛼^2𝛼+𝑦⁴𝑡11+Γ()𝑡2+2𝛼^2𝛼,𝑢₂=𝑥⁴11+Γ()𝑡1+2𝛼^2𝛼+1Γ()𝑡1+4𝛼^4𝛼+𝑦⁴𝑡11+Γ()𝑡2+2𝛼^2𝛼+1Γ()𝑡2+4𝛼^4𝛼,𝑢₃=𝑥⁴11+Γ()𝑡1+2𝛼^2𝛼+1Γ()𝑡1+4𝛼^4𝛼+1Γ()𝑡1+6𝛼^6𝛼+𝑦⁴𝑡11+Γ()𝑡2+2𝛼^2𝛼+1Γ()𝑡2+4𝛼^4𝛼+1Γ()𝑡1+6𝛼^6𝛼. (66)
The solution in a series form is given by 𝑢(𝑥,𝑡)=lim_𝑘→∞𝑢_𝑘(𝑥,𝑡)=𝑥^4∞_𝑘=0𝑡^2𝑘𝛼Γ(2𝑘𝛼+1)+𝑦⁴𝑡^∞_𝑘=0𝑡^2𝑘𝛼Γ(2𝑘𝛼+2)=𝑥⁴𝑡𝐸_2𝛼,1𝑡^2𝛼+𝑦⁴𝑡𝐸_2𝛼,2𝑡^2𝛼, (67) where 𝐸_2𝛼,2(𝑡^𝛼) denotes the two-parameter Mittag-Leffler function.

Example 20. Consider the one-dimensional fractional heat-like equation: 𝜕^2𝛼𝑢𝜕𝑡^2𝛼=𝑥²𝑦²𝑧²+12𝑥²𝑢_𝑥𝑥+𝑦²𝑢_𝑦𝑦+𝑧²𝑢_𝑧𝑧,0<𝑥,𝑦,𝑧<1,0.5<𝛼≤1,𝑡>0, (68) subject to the boundary conditions 𝑢0,𝑦,𝑧,𝑡=𝑦²𝑒^𝑡−1+𝑧²𝑒^−𝑡,𝑢=−11,𝑦,𝑧,𝑡1+𝑦²𝑒^𝑡−1+𝑧²𝑒^−𝑡,𝑢()−1𝑥,0,𝑧,𝑡=𝑥²𝑒^𝑡−1+𝑧²𝑒^−𝑡,𝑢()=−1𝑥,1,𝑧,𝑡1+𝑥²𝑒^𝑡−1+𝑧²𝑒^−𝑡,𝑢=𝑥−1𝑥,𝑦,0,𝑡²+𝑦²𝑒^𝑡,𝑢=𝑥−1𝑥,𝑦,1,𝑡²+𝑦²𝑒^𝑡−1+𝑒^−𝑡−1, (69) and the initial condition 𝑢𝑥,𝑦,𝑧,0=0,𝑢_𝑡𝑥,𝑦,𝑧,0=𝑥²+𝑦²−𝑧². (70) The exact solution 𝛼=1 was found to be [41] 𝑢𝑥,𝑦,𝑡=𝑥²+𝑦²𝑒^𝑡+𝑧²𝑒^−𝑡−𝑥²+𝑦²+𝑧². (71)
According to the general iteration formula of VIM, one can get the following formula of iteration: 𝑢_𝑘+1(𝑥,𝑡)=𝑢_𝑘(𝑥,𝑡)−𝐽^2𝛼𝜕^2𝛼𝑢𝜕𝑡^2𝛼−𝑥²𝑦²𝑧²−12𝑥²𝑢_𝑥𝑥+𝑦²𝑢_𝑦𝑦+𝑧²𝑢_𝑧𝑧. (72)
Starting with an initial approximation, 𝑢₀𝑥,𝑦,𝑧,𝑡=𝑥²+𝑦²𝑡𝑡+^2𝛼Γ(2𝛼+1)+𝑧²𝑡−𝑡+^2𝛼Γ(),2𝛼+1 (73) which was given by [41]. One can obtain the following successive approximate: 𝑢₁=𝑥²+𝑦²×𝑡1+𝑡+^2𝛼Γ()+𝑡1+2𝛼Γ(^1+2𝛼)+𝑡2+2𝛼Γ(1+^4𝛼4𝛼)+𝑧²𝑡1−𝑡+^2𝛼Γ()−𝑡1+2𝛼Γ(^1+2𝛼)+𝑡2+2𝛼Γ(1+^4𝛼4𝛼)−𝑥²𝑦²𝑧²,𝑢₂=𝑥²+𝑦²𝑡1+𝑡+^2𝛼Γ()+𝑡1+2𝛼Γ(^1+2𝛼)+𝑡2+2𝛼_4𝛼Γ()+𝑡1+4𝛼Γ(^1+4𝛼)+𝑡2+4𝛼Γ(1+^6𝛼6𝛼)+𝑧²𝑡1−𝑡+^2𝛼Γ()−𝑡1+2𝛼Γ(^1+2𝛼)+𝑡2+2𝛼Γ(^4𝛼)−𝑡1+4𝛼_1+4𝛼Γ()+𝑡2+4𝛼Γ(1+^6𝛼6𝛼)−𝑥²𝑦²𝑧², (74) when 𝛼=1 and the exact solution for (68) is 𝑢𝑥,𝑦,𝑡=𝑥²+𝑦²𝑒^𝑡+𝑧²𝑒^−𝑡−𝑥²+𝑦²+𝑧². (75) From above procedure of solution, one can conclude that the result obtained by the fractional variational iteration method (FVIM) is the same as the decomposition method [3].

5. Conclusion

VIM has been known as a powerful tool for solving many fractional differential equations. In this paper, we derive a general iteration formula of VIM for fractional heat- and wave-like equations with variable coefficients. There are two points to make here. First, the Lagrange multiplier of the method is identified in a more accurate way by employing the Laplace transform. Second, our iteration formula still holds true in the case of 𝛼>1. All the examples show that the results of the proposed method are more accurate than those obtained by the classical VIM and the same as the ADM but not require the calculation of Adomian’s polynomials.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 41105063). The authors are very grateful to reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper.