## Mathematical Modeling of Heat and Mass Transfer in Energy Science and Engineering 2014

View this Special IssueResearch Article | Open Access

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

**Academic Editor:**Jun Liu

#### 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).

#### References

- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier Science, Amsterdam, The Netherlands, 2006. View at: MathSciNet - B. J. West, M. Bologna, and P. Grigolini,
*Physics of Fractal Operators*, Institute for Nonlinear Science, Springer, 2003. View at: Publisher Site | MathSciNet - J. Klafter, S. C. Lim, and R. Metzler,
*Fractional Dynamics: Recent Advances*, World Scientific, 2012. View at: MathSciNet - D. Baleanu, J. A. T. Machado, and A. C. Luo,
*Fractional Dynamics and Control*, Springer, New York, NY, USA, 2012. - A. B. Alkhasov, R. P. Meilanov, and M. R. Shabanova, “Heat conduction equation in fractional-order derivatives,”
*Journal of Engineering Physics and Thermophysics*, vol. 84, no. 2, pp. 332–341, 2011. View at: Publisher Site | Google Scholar - J. M. Angulo, M. D. Ruiz-Medina, V. V. Anh, and W. Grecksch, “Fractional diffusion and fractional heat equation,”
*Advances in Applied Probability*, vol. 32, no. 4, pp. 1077–1099, 2000. View at: Publisher Site | Google Scholar | MathSciNet - Y. Z. Povstenko, “Fractional heat conduction equation and associated thermal stress,”
*Journal of Thermal Stresses*, vol. 28, no. 1, pp. 83–102, 2004. View at: Publisher Site | Google Scholar | MathSciNet - H. M. Youssef, “Theory of fractional order generalized thermoelasticity,”
*Journal of Heat Transfer*, vol. 132, no. 6, 7 pages, 2010. View at: Publisher Site | Google Scholar - M. A. Ezzat and A. S. El-Karamany, “Fractional order theory of a perfect conducting thermoelastic medium,”
*Canadian Journal of Physics*, vol. 89, no. 3, pp. 311–318, 2011. View at: Publisher Site | Google Scholar - M. A. Ezzat, “State space approach to thermoelectric fluid with fractional order heat transfer,”
*Heat and Mass Transfer/Waerme- und Stoffuebertragung*, vol. 48, no. 1, pp. 71–82, 2012. View at: Publisher Site | Google Scholar - H. H. Sherief, A. M. A. El-Sayed, and A. M. Abd El-Latief, “Fractional order theory of thermoelasticity,”
*International Journal of Solids and Structures*, vol. 47, no. 2, pp. 269–275, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. Vázquez, J. J. Trujillo, and M. P. Velasco, “Fractional heat equation and the second law of thermodynamics,”
*Fractional Calculus and Applied Analysis*, vol. 14, no. 3, pp. 334–342, 2011. View at: Publisher Site | Google Scholar | MathSciNet - J. Hristov, “An inverse stefan problem relevant to boilover: heat balance integral solutions and analysis,”
*Thermal Science*, vol. 11, no. 2, pp. 141–160, 2007. View at: Publisher Site | Google Scholar - J. Hristov, “A note on the integral approach to non-linear heat conduction with Jeffrey’s fading memory,”
*Thermal Science*, vol. 17, no. 3, pp. 733–737, 2013. View at: Publisher Site | Google Scholar - K. Davey and R. Prosser, “Analytical solutions for heat transfer on fractal and pre-fractal domains,”
*Applied Mathematical Modelling*, vol. 37, no. 1-2, pp. 554–569, 2013. View at: Publisher Site | Google Scholar | MathSciNet - M. Ostoja-Starzewski, “Towards thermoelasticity of fractal media,”
*Journal of Thermal Stresses*, vol. 30, no. 9-10, pp. 889–896, 2007. View at: Publisher Site | Google Scholar - H. Qi and X. Jiang, “Solutions of the space-time fractional Cattaneo diffusion equation,”
*Physica A: Statistical Mechanics and its Applications*, vol. 390, no. 11, pp. 1876–1883, 2011. View at: Publisher Site | Google Scholar | MathSciNet - A. H. Bhrawy and M. A. Alghamdi, “A Legendre tau-spectral method for solving time fractional heat equation with nonlocal conditions,”
*The Scientific World Journal*, vol. 2014, Article ID 706296, 7 pages, 2014. View at: Publisher Site | Google Scholar - A. Atangana and A. Kılıçman, “The use of Sumudu transform for solving certain nonlinear fractional heat-like equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 737481, 12 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - X. J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis,”
*Boundary Value Problems*, vol. 2013, article 131, 2013. View at: Publisher Site | Google Scholar - X. J. Yang, D. Baleanu, and W. P. Zhong, “Approximate solutions for diffusion equations on cantor space-time,”
*Proceedings of the Romanian Academy A*, vol. 14, no. 2, pp. 127–133, 2013. View at: Google Scholar | MathSciNet - X. Yang, D. Baleanu, and J. A. Tenreiro Machado, “Systems of Navier-Stokes equations on Cantor sets,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 769724, 8 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - X. J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science, New York, NY, USA, 2012. - X. J. Yang, H. M. Srivastava, J. H. He, and D. Baleanu, “Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives,”
*Physics Letters A*, vol. 377, no. 28, pp. 1696–1700, 2013. View at: Publisher Site | Google Scholar | MathSciNet - M. Hu, D. Baleanu, and X. Yang, “One-phase problems for discontinuous heat transfer in fractal media,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 358473, 3 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - A. M. Yang, C. Cattani, C. Zhang, G. N. Xie, and X. J. Yang, “Local fractional Fourier series solutions for non-homogeneous heat equations arising in fractal heat flow with local fractional derivative,”
*Advances in Mechanical Engineering*, vol. 2014, Article ID 514639, 5 pages, 2014. View at: Publisher Site | Google Scholar - X. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,”
*Thermal Science*, vol. 17, no. 2, pp. 625–628, 2013. View at: Publisher Site | Google Scholar - A. Yang, C. Cattani, H. Jafari, and X. Yang, “Analytical solutions of the one-dimensional heat equations arising in fractal transient conduction with local fractional derivative,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 462535, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - C. F. Liu, S. S. Kong, and S. J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,”
*Thermal Science*, vol. 17, no. 3, pp. 715–721, 2013. View at: Google Scholar - A. M. Yang, J. Li, H. M. Srivastava, G. N. Xie, and X. J. Yang, “The local fractional Laplace variational iteration method for solving linear partial differential equations with local fractional derivatives,”
*Discrete Dynamics in Nature and Society*, vol. 2014, Article ID 365981, 2014. View at: Publisher Site | Google Scholar - C. G. Zhao, A. M. Yang, H. Jafari, and A. Haghbin, “The Yang-Laplace transform for solving the IVPs with local fractional derivative,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 386459, 5 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - Y. Z. Zhang, A. M. Yang, and Y. Long, “Initial boundary value problem for fractal heat equation in the semi-infinite region by Yang-Laplace transform,”
*Thermal Science*, vol. 18, no. 2, pp. 677–681, 2014. View at: Publisher Site | Google Scholar

#### Copyright

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.