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

1 School of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing 210009, China 2 School of Mechanical Engineering, Huaihai Institute of Technology, Lianyungang 222005, China 3Department of Mathematics, University of Salerno, Via Giovanni Paolo II, Fisciano, 84084 Salerno, Italy 4 School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China 5 Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China 6 Electronic and Information Technology Department, Jiangmen Polytechnic, Jiangmen 529090, China

Recently, the local fractional calculus [20][21][22] was used to deal with the discontinuous problem for heat transfer in fractal media [23][24][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 2 Mathematical Problems in Engineering arising in heat flow with local fractional derivative are investigated. Finally, conclusions are shown in Section 5.

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 u be defined as [19] ∭u ( ) where the elements of the volume ΔΩ ( ) → 0 as → ∞ and the fractal dimension of the volume . The equality ( , , , ) is the temperature at the point ( , , ) ∈ Ω, 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 n , Δ ( ) → 0 as → ∞ for = (3/2) = 3 .
From (3) the local fractional Fourier law of the material in fractal media [19,23] was suggested as follows: where S ( ) is the fractal surface measure over Ω ( ) and 2 is the thermal conductivity of the fractal material.
From (15) the local fractional convolution of two functions is defined as [29][30][31][32] and we have From (19) we obtaiñ By the local fractional variation [23,27,29,30], we obtain such that From (24) we get̃{ 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 (28) we arrive at

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 (26) we obtain the local fractional iteration algorithm: where the initial value is given as Using (32), we have the first approximation: In view of (32) and (34), we get the second approximation: Making use of (32) and (35), the third approximate term reads as follows: From (32) and (36), the fourth approximate term can be written as follows: Making the best of (32) and (36), we can write the fifth approximate term as Hence, we obtain the final term given as In view of (28) and (29), we suggest the exact solution of (30) as and its plot is shown in Figure 1.
Mathematical Problems in Engineering

Example 2.
We now consider the nonhomogeneous local fractional heat equation with the nondifferentiable source term: In view of (26), the local fractional iteration algorithm can be structured as follows: Appling (43) gives the first approximate term: In view of (43) and (44), the second approximate term reads as̃{ Making use of (43) and (45), we arrive at the third approximate term: From (43) and (46) we give the fourth approximation: In view of (43) and (47), the fifth approximate term is presented as For the fractal dimension = ln 2/ ln 3, the plot of the nondifferentiable solution of the nonhomogeneous local fractional heat equation with the nondifferentiable source term is shown in Figure 2.

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.