Approximate and Iterative MethodsView this Special Issue
Local Fractional Laplace Variational Iteration Method for Solving Linear Partial Differential Equations with Local Fractional Derivative
The local fractional Laplace variational iteration method was applied to solve the linear local fractional partial differential equations. The local fractional Laplace variational iteration method is coupled by the local fractional variational iteration method and Laplace transform. The nondifferentiable approximate solutions are obtained and their graphs are also shown.
Fractional calculus  has successfully been used to study the mathematical and physical problems arising in science and engineering. Fractional differential equations are applied to describe the dynamical systems in physics and engineering (see [2, 3]). It is one of the hot topics for finding the solutions for the fractional differential equations for scientists and engineers. There are many analytical and numerical methods for solving them, such as the spectral Legendre-Gauss-Lobatto collocation method , the shifted Jacobi-Gauss-Lobatto collocation method , the variation iteration method , the heat-balance integral method , the Adomian decomposition method , the finite element method , and the finite difference method .
The above methods did not deal with some nondifferentiable problems arising in mathematics and physics (see [11–13]). Local fractional calculus (see [12–14] and the cited references) is the best choice to deal with them. Some methods for solving the local fractional differential equations were suggested, such as the Cantor-type cylindrical-coordinate method , the local fractional variational iteration method [16, 17], the local fractional decomposition method , the local fractional series expansion method , the local fractional Laplace transform method , and local fractional function decomposition method [21, 22]. More recently, the coupling schemes for local fractional variational iteration method and Laplace transform were suggested in . However, the results are very little. In this paper, our aim is to use the local fractional Laplace variational iteration method to solve the linear local fractional partial differential equations. The structure of the paper is suggested as follows. In Section 2 the basic theory of local fractional calculus and local fractional Laplace transform are introduced. Section 3 is devoted to the local fractional Laplace variational iteration method. In Section 4, the four examples for the local fractional partial differential equations are given. Finally, the conclusions are considered in Section 5.
2. Local Fractional Calculus and Local Fractional Laplace Transform
In this section, we present the basic theory of local fractional calculus and concepts of local fractional Laplace transform.
The local fractional derivative of of order is defined as [12–15] where The local fractional integral of of order in the interval is defined as [12–14, 16–23] where the partitions of the interval are , with , , , and , .
The local fractional series of nondifferentiable function used in this paper are presented as follows [12–14]: The properties of local fractional derivatives and integral of nondifferentiable functions are given by [12, 13] The local fractional Laplace transform is defined as [12, 20–22] where is local fractional continuous and .
3. Analysis of the Method
Let us consider the following nonlinear operator with local fractional derivative: where the linear local fractional differential operator denotes and is a source term of the nondifferential function.
Following the local fractional Laplace variational iteration method , we have the local fractional functional formula as follows: which leads to Using the local fractional Laplace transform, from (12), we get Taking the local fractional variation , we obtain which leads to where Hence, from (15) and (16) we get which yields Hence, we get the new iteration algorithm as follows: where the initial value is suggested as Therefore, we have the local fractional series solution of (10) so that The above process is called the local fractional variational iteration method.
4. The Nondifferentiable Solutions for Linear Local Fractional Differential Equations
In this section, we present the examples for linear local fractional differential equations of high order.
Example 1. The local fractional differential equation is presented as subject to the initial value From (19) and (23) we obtain where the initial value is given by Therefore, the successive approximations are Hence, we get and its graph is shown in Figure 1.
Example 2. We report the following local fractional partial differential equation: The initial value is given by In view of (19) and (29) the local fractional iteration algorithm can be written as follows: where the initial value is Making use of (31) and (32), the successive approximate solutions are shown as follows: Therefore, the nondifferentiable solution of (29) reads and its plot is presented in Figure 2.
Example 3. The following local fractional partial differential equation is considered and its initial value is Making use of (19) and (35) the local fractional iteration algorithm reads where the initial value is suggested as From (38) we have the successive approximations as follows: Hence, the approximate solution of (35) is given by and its graph is presented in Figure 3.
Local fractional calculus was successfully applied to deal with the nondifferentiable problems arising in mathematical physics. In this work we considered the coupling method of the local fractional variational iteration method and Laplace transform to solve the linear local fractional partial differential equations and their nondifferentiable solutions were obtained. The results are efficient implement of the local fractional Laplace variational iteration method to solve the partial differential equations with local fractional derivative.
Conflict of Interests
The authors declare that they have no competing interests in this paper.
This work was supported by National Scientific and Technological Support Projects (no. 2012BAE09B00), the National Natural Science Foundation of China (no. 51274270), and the National Natural Science Foundation of Hebei Province (no. E2013209215).
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
J. Klafter, S. C. Lim, and R. Metzler, Fractional Dynamics: Recent Advances, World Scientific, 2012.
A. H. Bhrawy and D. Baleanu, “A spectral Legendre-Gauss-LOBatto collocation method for a space-fractional advection diffusion equations with variable coefficients,” Reports on Mathematical Physics, vol. 72, no. 2, pp. 219–233, 2013.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Hristov, “An exercise with the He's variation iteration method to a fractional Bernoulli equation arising in a transient conduction with a non-linear boundary heat flux,” International Review of Chemical Engineering, vol. 4, no. 5, pp. 489–497, 2012.View at: Google Scholar
X.-J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong, 2011.
X.-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
G. A. Anastassiou and O. Duman, Advances in Applied Mathematics and Approximation Theory, Springer, New York, NY, USA, 2013.
A. M. Yang, Y. Z. Zhang, and X. L. Zhang, “The non-differentiable solution for local fractional tricomi equation arising in fractal transonic flow by local fractional variational iteration method,” Advances in Mathematical Physics, vol. 2014, Article ID 983254, 6 pages, 2014.View at: Publisher Site | Google Scholar
A. M. Yang, Z. S. Chen, H. M. Srivastava, and X.-J. Yang, “Application of the local fractional series expansion method and the variational iteration method to the Helmholtz equation involving local fractional derivative operators,” Abstract and Applied Analysis, vol. 2013, Article ID 259125, 6 pages, 2013.View at: Publisher Site | Google Scholar
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: Google Scholar