Journal of Applied Mathematics

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]: subject to the boundary conditions and the initial conditions where is a parameter describing the fractional derivative. In the case of , (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 . 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 , , is said to be in the space , , if there exist a real number , such that , where , and it is said to be in the space if and only if , .

Definition 2. The Riemann-Liouville fractional integral operator () of order , of a function , , is defined as when , .

If we denote the Riemann-Liouville fractional derivative by , then the next equation define the Riemann-Liouville fractional derivative of order where , .

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] For positive , one will set

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

Theorem 4 (the fractional Leibniz product rule [36]). If and are two continuous functions on , then

Theorem 5 (the fractional Barrow’s formula [37]). For a continuous function , one has where .

From Theorems 4 and 5, the formula of integration by parts is given as

Definition 6. Fractional derivative of compounded functions [37, 38] is defined as

Definition 7. The integral with respect to [37, 38] is defined as the solution of the fractional differential equation

Lemma 8. Let denote a continuous function [37, 38], then the solution , of (8) is defined by the equality

Lemma 9. Let , , , , , then

2.2. The Laplace Transform

Definition 10. The Laplace transform , , of a function is defined by the integral

Lemma 11. The Laplace transform of the fractional derivative (the Riemann-Liouville derivative) is

Definition 12. The inverse Laplace transform is defined by the complex integral where the integration is done along the vertically in the complex plane such that is greater than the real part of all singularities of . This ensures that the contour path is in the region of convergence. If all singularities are in the left half-plane, or is a smooth function on (i.e., no singularities), then 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 denote a function which vanishes for negative values of . Its Laplace's transform of order (or its th fractional Laplace's transform) is defined by the following expression, when it is finite: where and is the Mittag-Leffler function .

Lemma 14. If one defines the convolution of order of the two functions by the expression Then, one has the equality

Corollary 15. Given the Laplace transform that one has the inversion formula where is the period of the complex-valued Mittag-Leffler function defined by the equality .

Lemma 16. The fractional Laplace transform of the fractional derivative (the modified Riemann-Liouville derivative) is
Some properties of the fractional Laplace transform are given as follows [38]:

3. Fractional Variational Iteration Method

In order to illustrate the solution procedure of the variational iteration method, we consider the following fractional differential equation: where is the linear operator, is the nonlinear operator, and is the modified Rieman-Liouville derivative of order .

Subject to the initial condition, Let , .

Before our solution, we will describe the fractional variational iteration method described in [23, 25, 26], which construct a correct function for (24) as

Then, by using Lemma 8 proposed by Jumarie [37, 38] for the first time, one gets a correction function But, unfortunately, (27) holds true only when . In the case of , , some modifications must be made. For example, replacing the fractional order () by the order ), 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: where is the Riemann-Liouville fractional integral operator of order , that is, , with respect to the variable , and is a general Lagrange multiplier, and is a restricted variation, that is, .

By taking the Laplace transform on the both sides of (28), we have where is the operator of the Laplace transform.

From the Lemma 11, we get

Taking the variation derivative on the both sides of (31), we can derive

Without loss of generality, assuming that , then we have

Set the coefficient of to zeros, we obtain

By employing the inverse Laplace transform, we have

Substituting (35) into (28), we get the iteration formula as follows:

3.2. The Fractional Laplace Transform Method

In order to illustrate our method, we replace the fractional order by the order . According to VIM [27], we can construct a correction function as follows:

Taking the fractional Laplace transform on the both sides of (37), we have where is the fractional Laplace transform of order .

By assuming that has the form as and using (10), then we get

From (20), we have

Substituting (40) into (38) and then taking the variation derivative on the both sides of (35), we can derive

Without loss of generality, assuming that , then we have

Set the coefficient of to zeros, we have

By employing the inverse fractional Laplace transform, we have

Substituting (40) into (26), we get the iteration formula as follows:

After replacing the fractional order by the order , we could find that (36) and (45) are the same. So, we could get a general iteration formula for (1) as follows:

By using the Lemma 11, we finally have

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]: where is the modified Rieman-Liouville derivative and is a constant, with the initial condition where is the body temperature, is the environment temperature, and is a constant. When , the exact solution of (46) is
According to the general iteration formula of VIM, one can get the iteration formulation as follows:
For the convenience, let and , then
Starting with the initial value as shown in (52), we can derive and then So, the solution is which is the exact solution.