Local Fractional Adomian Decomposition and Function Decomposition Methods for Laplace Equation within Local Fractional Operators

Many problems of physics and engineering are expressed by ordinary and partial differential equations, which are termed boundary value problems. We can mention, for example, the wave, the Laplace, the Klein-Gordon, the Schrodinger, the Advection, the Burgers, the Boussinesq, and the Fisher equations, and others [1]. Several analytical and numerical techniques were successfully applied to deal with differential equations, fractional differential equations, and local fractional differential equations [1–10].The techniques include the heat-balance integral [11], the fractional Fourier [12], the fractional Laplace transform [12], the harmonic wavelet [13, 14], the local fractional Fourier and Laplace transform [15], local fractional variational iteration [16–18], the local fractional decomposition [19], and the generalized local fractional Fourier transform [20] methods. In this paper, we investigate the application of local fractional Adomian decomposition method and local fractional function decomposition method for solving the local fractional Laplace equation [21, 22] with the different fractal conditions. This paper is organized as follows. In Section 2, the basic mathematical tools are reviewed. Section 3 presents briefly the local fractional Adomian decomposition method and the local fractional function decomposition method. Section 4 presents solutions to the local fractional Laplace equation with differential fractal conditions.


Introduction
Many problems of physics and engineering are expressed by ordinary and partial differential equations, which are termed boundary value problems.We can mention, for example, the wave, the Laplace, the Klein-Gordon, the Schrodinger, the Advection, the Burgers, the Boussinesq, and the Fisher equations, and others [1].
In this paper, we investigate the application of local fractional Adomian decomposition method and local fractional function decomposition method for solving the local fractional Laplace equation [21,22] with the different fractal conditions.This paper is organized as follows.In Section 2, the basic mathematical tools are reviewed.Section 3 presents briefly the local fractional Adomian decomposition method and the local fractional function decomposition method.Section 4 presents solutions to the local fractional Laplace equation with differential fractal conditions.
We notice that there are existence conditions of local fractional continuities that operating functions are right-hand and left-hand local fractional continuities.Meanwhile, the right-hand local fractional continuity is equal to its left-hand local fractional continuity.For more details, see [20].

Analytical Methods
In order to illustrate two analytical methods, we investigate the nonlinear local fractional equation of order 2 as follows: with constants  1 ,  2 ,  3 , 0 <  ≤ 1 and with boundary and initial conditions

Local Fractional Adomian Decomposition Method.
We rewrite (10) in the following form: Advances in Mathematical Physics 3 Applying the inverse operator  (−2)  to both sides of ( 12) yields where the term (, ) is to be determined from the fractal initial conditions.Now, we decompose the unknown function (, ) as a sum of components defined by the series: The components   (, ) are obtained by the recursive formula:

Local Fractional Function Decomposition Method.
According to the decomposition of the local fractional function, with respect to the system {sin    (/)  }, the following functions coefficients can be given by where Substituting ( 16) into (10) implies that Suppose that the Yang-Laplace transforms of functions V  () and   () are   () and   (), respectively.Then, we obtain That is, Hence, we have Hence, we get Then, making use of ( 8) and ( 9) and rearranging integration sequence, we have the following several formulas about V 1, () and V 2, ().

Solutions of Local Fractional Laplace Equation in Fractal Time-Space
In this section, two examples for Laplace equation are presented in order to demonstrate the simplicity and the efficiency of the above methods.
The local fractional Laplace equation (see [21]) is one of the important differential equations with local fractional derivatives.In the following, we consider solutions to local fractional Laplace equations in fractal time-space.
Example 7. Consider the following local fractional Laplace equation: subject to the fractal value conditions According to formula (15), we have where Hence, from (29) we obtain where Making use of (31), we present Proceeding in this manner, we get Thus, the final solution reads as follows: Now, we solve Example 7 by using the local fractional function decomposition method.
Example 8. We consider the following local fractional Laplace equation: Advances in Mathematical Physics subject to the fractal value conditions  (, 0) = 0,    (, 0)   = −  (  ) . (44) Now we can structure the same local fractional iteration procedure (15).Hence, we have Finally, we can obtain the local fractional series solution as follows: Thus, the final solution reads as follows: Now, we solve Example 8 by using the local fractional function decomposition method.

Conclusions
In this work solving the Laplace equations using the local fractional function decomposition method with local fractional operators is discussed in detail.Two examples of applications of the local fractional Adomian decomposition method and local fractional function decomposition method to the local fractional Laplace equations are investigated in detail.The reliable obtained results are complementary with the ones presented in the literature.

Figure 2 :
Figure 2: The plot of solution to local fractional Laplace equation with fractal dimension  = ln 2/ ln 3.