Grundwald-Letnikov Operator and Its Role in Solving Fractional Differential Equations

. Leibnitz in 1663 introduced the derivative notation for the order of natural numbers, and then the idea of fractional derivatives appeared. Only a century later, this idea began to be realized with the discovery of the concepts of fractional derivatives by several mathematicians, including Riemann (1832), Grundwal, Fourier, and Caputo in 1969. The concepts in the definitions of fractional derivatives by Riemann-Liouville and Caputo are more frequently used than other definitions, this paper will discuss the Grunwald-Letnikov (GL) operator, which has been discovered in 1867. This concept is less popular when compared to the Riemann-Liouville and Caputo concepts, however, this concept is quite interesting because the concept of derivation is developed from the definition of ordinary derivatives. In this paper will be shown that the formulas for the fractional derivative using the GL concept are the same as the results obtained using the Riemann-Liouville and Caputo concepts. As a complement, we will give an example of solving a fractional differential equation using Modified Homotopy Perturbation Methods. This is an open acces article under the CC-BY license.

1882) put forward the concept of  order fractional integral as a generalization of the derivative proposed by Leibniz (1646-1716). Furthermore, Grundwald and Letnikov (1867), Fourier, and others including the last Caputo (1969) seemed to be competing to create and develop a definition of fractional derivative. The Riemann-Liouville and Caputo versions are more popular than the others, this is because they are more widely used by researchers, especially in the applied field.
Currently, the development of research on fractions is so rapid, both grouped in Fractional Integrals and Derivatives [1,2], and in the Calculus Fractional group [3]. Fractional models are also very diverse, including: Diffusion-Reaction Equation model and Fractional Gas Dynamics Equation model [4,10]. The most widely used fractional operators are Riemann-Liouville and Caputo operators, only a few use Grundwald-Letnikov as in [5,6,13]. To solve fractional differential equations, many methods are used, including Homotopy Perturbation Method [7], Modified Homotopy Perturbation Method [8,9], and Homotopy Perturbation Method using Sumudu Transformation [10]. Local Fractional [11,12], applied Neural Network [13] and viscosity analysis [14] are the last things that are used as references for this paper.
This paper, the Grundwald-Letnikow version of the fractional derivative operator is presented, with the aim of showing similarities with other versions, both in terms of the existence of a welldefined definition, determining the basic derivative formula, even in its application to solve a problem in fractional differential equation. The method used to determine the solution is the Modified Homotopy Perturbation Method.

Literatur Review
In this section, several definitions related with the subject will be presented. ( 1) ( ) ; 0 dengan (1) 1

Floor and Ceiling Function
1. If ∈ is between two integers. Ceiling function of x denoted by x   represents the smallest integer greater than or equal to x. 2. If ∈ is between two integers. Floor function from x denoted by x   represents the largest integer that is less than or equal to x.

Definition 2.5. Caputo Operator
Let is a real number, and −1 < ≤ where is natural number. Fractional derivative of f( ) with oder α is There is an example of how to determine the fractional derivative using the definition of the Riemann-Liouville operator. Example:

. Fractional Differential Equation
One of the general forms of a fractional differential equation of order  is with 0 <   1 , t > 0 , a , b the real constant, and the initial condition is u(0) = 0.

Method
Many methods can be used to solve this fractional differential equation, including the Modified Homotopy Pertubation Method. In this section, The Modified Homotopy Pertubation Method to solve Fractional Differential Equation model will be elaborated.
The Fractional Differential Equation Model is given below: where D  fractional derivative with order  , L is linear operator, N is nonlinear operator, and A(t) is a function of t . The initial condition is: (0) = , = 0, 1, 2, … , − 1 .
By substituting (8) to (7) and take p = 1, then solution function of (6) will be where ( ) is obtained from integral result of derivative function as follows

Results and Discussion
The results of the research that will be presented in this paper are the derivation of the Grundwald-Letnikov operator formula, the basic theorem on the general formula for fractional derivatives using the Grundwald-Letnikov operator, and examples of solving fractional differential equations using the Modified Homotopy Pertubation Method.

Grunwald-Letnikov
In contrast to Riemann-Liouville and Caputo who defined a fractional derivative through integration, Grunwald-Letnikov formulated a fractional derivative as a generalization of derivatives of the order of natural numbers.

Proof:
First, we know that the first derivative of the function Hence second derivative is So That theorem was proved. Furthermore, it is shown that the results of Grundwal-Letnikov are similar to those of Riemann-Liouville and Caputo on the basic formula of the derivative.