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.
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.
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).