Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 914725, 7 pages

http://dx.doi.org/10.1155/2014/914725

Research Article

Local Fractional Laplace Variational Iteration Method for Nonhomogeneous Heat Equations Arising in Fractal Heat Flow

Shu Xu,1,2 Xiang Ling,¹ Carlo Cattani,³ Gong-Nan Xie,⁴ Xiao-Jun Yang,⁵ and Yang Zhao⁶

¹School of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing 210009, China

²School of Mechanical Engineering, Huaihai Institute of Technology, Lianyungang 222005, China

³Department of Mathematics, University of Salerno, Via Giovanni Paolo II, Fisciano, 84084 Salerno, Italy

⁴School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

⁵Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

⁶Electronic and Information Technology Department, Jiangmen Polytechnic, Jiangmen 529090, China

Received 29 May 2014; Accepted 1 July 2014; Published 7 August 2014

Academic Editor: Jun Liu

Copyright © 2014 Shu Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The local fractional Laplace variational iteration method is used for solving the nonhomogeneous heat equations arising in the fractal heat flow. The approximate solutions are nondifferentiable functions and their plots are also given to show the accuracy and efficiency to implement the previous method.

1. Introduction

Fractional calculus [1–4] was used to deal with the heat conduction equation in fractal media. Fractional heat conduction equation was studied by many researchers [5–17]. For example, Povstenko considered the thermoelasticity based on the fractional heat conduction equation [7]. Youssef suggested the generalized theory of fractional-order thermoelasticity [8]. Ezzat and El-Karamany presented the fractional-order conduction in thermoelastic medium [9]. Ezzat proposed the fractional-order heat transfer in thermoelectric fluid [10]. Sherief et al. reported the fractional-order generalized thermoelasticity with one relaxation time [11]. Vázquez et al. used the second law of thermodynamics to fractional heat conduction equation [12]. Hristov considered the inverse Stefan problem and nonlinear heat conduction with Jeffrey’s fading memory by using the heat balance integral method [13, 14]. Davey and Prosser gave the solutions of the heat transfer on fractal and prefractal domains [15]. Ostoja-Starzewski investigated thermoelasticity of fractal media [16]. Qi and Jiang discussed space-time fractional Cattaneo diffusion equation [17]. Bhrawy and Alghamdi applied the Legendre tau-spectral method to find time fractional heat equation with nonlocal conditions [18]. Atangana and Kılıçman suggested the Sumudu transform solving certain nonlinear fractional heat-like equations [19].

Recently, the local fractional calculus [20–22] was used to deal with the discontinuous problem for heat transfer in fractal media [23–25]. The nonhomogeneous heat equations arising in fractal heat flow were considered by using the local fractional Fourier series method [26]. The local fractional heat conduction equation was investigated by the local fractional variation iteration method [27]. The nondifferentiable solution of one-dimensional heat equations arising in fractal transient conduction was found by the local fractional Adomian decomposition method [28]. Local fractional Laplace variational iteration method [29, 30] was considered to deal with linear partial differential equations. In this paper, our aim is to investigate the nonhomogeneous heat equations arising in heat flow with local fractional derivative. The paper is organized as follows. Section 2 introduces the nonhomogeneous heat equations arising in heat flow with local fractional derivative. In Section 3, local fractional Laplace variational iteration method is presented. In Section 4, the nondifferentiable solutions for nonhomogeneous heat equations arising in heat flow with local fractional derivative are investigated. Finally, conclusions are shown in Section 5.

2. The Nonhomogeneous Heat Equations Arising in Heat Flow with Local Fractional Derivatives

In this section we present the one-dimensional nonhomogeneous heat equations arising in heat flow with local fractional derivatives.

Let the local fractional volume integral of the function be defined as [19]

where the elements of the volume as and the fractal dimension of the volume . The equality is the temperature at the point , time , and the total amount of heat is described as

where is the special heat of the fractal material and is the density of the fractal material.

The local fractional surface integral is defined as [19, 22]

where are elements of area with a unit normal local fractional vector , as for .

From the local fractional Fourier law of the material in fractal media [19, 23] was suggested as follows:

where is the fractal surface measure over and is the thermal conductivity of the fractal material.

In view of , the change in heat reads as follows [19, 23]:

where is the boundary of .

From we suggest the following source term [23]:

Making use of , , and , we have

such that

which leads to the nonhomogeneous local fractional heat equations [23]:

From we obtain the nonhomogeneous heat equations in the dimensionless case:

The two-dimensional case is [23]

and the one-dimensional case is [26]

3. Local Fractional Laplace Variational Iteration Method

In this section, we give the idea of local fractional Laplace variational method [29, 30] in order to investigate the one-dimensional nonhomogeneous heat equations arising in fractal heat flow.

We present the following local fractional differential operator as follows:

where the linear local fractional differential operator denotes and is a nondifferential function.

We can write the local fractional functional formula as

The local fractional Laplace transform is given as [29–32]

and the inverse formula of local fractional Laplace transform is suggested as [29–32]

where is a local fractional continuous function, , , and the local fractional integral of of order in the interval is given as [23]

with the partitions of the interval which is , for , , , and , .

From the local fractional convolution of two functions is defined as [29–32]

and we have

From we obtain

By the local fractional variation [23, 27, 29, 30], we obtain

which leads to

From we have

such that

From we get

such that local fractional iteration algorithm reads as

where the initial value is presented as follows:

Therefore, the local fractional series solution is given as

From we arrive at

4. The Nondifferentiable Solutions

In this section, we discuss the one-dimensional nonhomogeneous heat equations arising in fractal heat flow.

Example 1. The nonhomogeneous local fractional heat equation with the nondifferentiable sink term is presented as follows:

subject to the initial-boundary value conditions

From we obtain the local fractional iteration algorithm:

where the initial value is given as

Using , we have the first approximation:

In view of and , we get the second approximation:

Making use of and , the third approximate term reads as follows:

From and , the fourth approximate term can be written as follows:

Making the best of and , we can write the fifth approximate term as

Hence, we obtain the final term given as

In view of and , we suggest the exact solution of as

and its plot is shown in Figure 1.

Figure 1: The nondifferentiable solution of nonhomogeneous local fractional heat equation with nondifferentiable sink term for .

Example 2. We now consider the nonhomogeneous local fractional heat equation with the nondifferentiable source term:

subject to the initial-boundary value conditions

In view of , the local fractional iteration algorithm can be structured as follows:

Appling gives the first approximate term:

In view of and , the second approximate term reads as

Making use of and , we arrive at the third approximate term:

From and we give the fourth approximation:

In view of and , the fifth approximate term is presented as

Hence, we finally have

so that the exact solution of nonhomogeneous local fractional heat equation with nondifferentiable source term is

For the fractal dimension , the plot of the nondifferentiable solution of the nonhomogeneous local fractional heat equation with the nondifferentiable source term is shown in Figure 2.

Figure 2: The nondifferentiable solution of the nonhomogeneous local fractional heat equation with the nondifferentiable sink term for .

5. Conclusions

At the present work, the nonhomogeneous heat equations arising in the fractal heat flow were investigated. The local fractional Laplace variational iteration method was applied to obtain the nondifferentiable solutions for the nonhomogeneous local fractional heat equations with the nondifferentiable source and sink terms. Finally, the graphs of the obtained solutions are also shown.

Conflict of Interests

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

Acknowledgments

The work was supported by the Natural Science Foundation of Jiangsu Colleges and Universities (no. KK-12058) and Jiangsu Province R&D Institute of Marine Resource (no. JSIUMR 201210).

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