Local Fractional Adomian Decomposition and Function Decomposition Methods for Laplace Equation within Local Fractional Operators
We perform a comparison between the local fractional Adomian decomposition and local fractional function decomposition methods applied to the Laplace equation. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.
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 .
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 , the fractional Fourier , the fractional Laplace transform , the harmonic wavelet [13, 14], the local fractional Fourier and Laplace transform , local fractional variational iteration [16–18], the local fractional decomposition , and the generalized local fractional Fourier transform  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.
2. Mathematical Fundamentals
Definition 1 (see [15–20, 23, 24]). The function is local fractional continuous at , if it is valid for with , for and . For , it is so called local fractional continuous on the interval , denoted by .
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 .
Local fractional derivative of high order is written in the form And local fractional partial derivative of high order is written in the form
Definition 3 (see [15–20, 23, 24]). A partition of the interval is denoted by , , and with and . Local fractional integral of in the interval is given by
If the functions are local fractional continuous, then the local fractional derivatives and integrals exist. Some properties of local fractional derivative and integrals are given in .
3. Analytical Methods
In order to illustrate two analytical methods, we investigate the nonlinear local fractional equation of order as follows: with constants and with boundary and initial conditions
3.1. Local Fractional Adomian Decomposition Method
Now, we decompose the unknown function as a sum of components defined by the series: The components are obtained by the recursive formula:
3.2. Local Fractional Function Decomposition Method
According to the decomposition of the local fractional function, with respect to the system , the following functions coefficients can be given by where Substituting (16) into (10) implies that Suppose that the Yang-Laplace transforms of functions and are and , respectively. Then, we obtain That is, Hence, we have Let Hence, we get Then, making use of (8) and (9) and rearranging integration sequence, we have the following several formulas about and .
If , then where .
Then, we get In case and , see .
4. 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 ) 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: subject to the fractal value conditions 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.
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of the paper.
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