The behavior of solution function of the fractional differential equations using modified homotopy perturbation method

The discourse regarding fractional calculus, in particular those related to fractional differential equation, is still continue to attract researcher attention. Previous studies have elaborated on the variation of fractional differential equation models. This study aims to uncover the problem of convergence of the solution function sequence related to the order of fractional differential equation. Firstly, this study presents how to find a solution for the model by using a Modified Homotopy Perturbation Method as the improvement of Homotopy Perturbation Method. Furthermore, the solution function with the sequence of fractional order is drawn by Maple. Using the geometrical analysis, the result of this study shows that if fractional order sequence is convergent to α, then sequence of its solution function will be convergent to a solution function of fractional differential equation with order α.


Introduction
The literature of the fractional calculus problem has emerged since the discovery of the conventional derivative concept with a natural numbered order by Leibnitz. However, recently the discourse seems to be stagnant with unclear reasons [1,2]. Although some mathematicians such as Caputo, Riemann-Liouville, and Grundwald-Letnikov discovered the formula for determining fractional derivatives, the development of fractional calculus is still limited. This is probably due to the many non-equivalent definitions of fractional derivatives [3,4]. Conditions like that until the middle of the 20th century.
In the 21st century, the topic about fractional differential equation model has become popular in the literature of fractional derivatives. Fractional differential equation model is a differential equation that contains the derivative of a fractional order. Similar to the differential equation of natural number order, the type of equation is divided into linear and nonlinear fractional differential equations. One of the nonlinear fractional equation is Riccati fractional differential equation. The development of fractional differential equation has emerged particularly related to the problem of convergence of the solution function sequence [5,6]. It turns out that many models in interdisciplinary cases could be easier to be expressed in the form of fractional differential equation model, such as in the field of finance [7]. Moreover, the problem of potential energy and fluid mechanics such as viscosity and surface tension are the other examples of cases that could be solved by fractional derivatives [8,9].
Various methods have been developed to find solutions of fractional differential equations, including Homotopy Analysis Method (HAM) which was firstly devised by Shijun [10] and Homotopy Perturbation Method (HPM) introduced by He [11]. Those methods combine perturbation and homotopy technique in order to eliminate small parameter in the equations which solve the limitation of the existing perturbation method. Moreover, those approaches are modified by Odibat and Momani [12] into Modified Homotopy Perturbation Method (MHPM) to create a more efficient solution approach which move within rapid iteration process to achieve higher speed of convergence in order to get faster solution function [13,14].
This paper discusses the Modified Homotopy Perturbation method which is used to find the solution of a fractional differential equation and to analyze the convergence and behavior of the solution function sequences. The results show that the sequence of order of the fractional differential equation converging to a number, will cause the sequence of solution functions to converge to the solution function of the fractional differential equation with order is that number.

Method
In this section, we discuss the basic concepts of fractional derivatives followed by fractional differential equations and the methods that are used to find solutions.
Different from Riemann-Liouville and Grunwald-Letnikov, based on Caputo (1969), definition of fractional derivative as follows: Let is a real number, and −1 < ≤ where is natural number. Fractional derivative of orde of f (t) to x is: From the three formulas above, specifically obtained that if ( ) = then -th derivative of ( ), or derivative of ( ) with fractional order  as follows: The results of the Riemann-Liouvile formula will be different from the results of the Caputo formula if f(t) is a constant function, where the result of ( ) by Caputo is zero, while according to Liouville is not zero.

Fractional Differential Equation
The general form of the fractional differential equation model with order  is: where D  is operator of fractional derivative with order  , L is linear operator, N is nonlinear operator, and A(t) is a function of t. The specific form of model (1) is known as Riccati fractional differential model, presented as follow: with 0 <   1 , t > 0 , a , b the real constant, and the initial condition is u(0) = 0.
In case b = 0, model (2) will become a fluid relaxation problem model, in particular for A(t) is an exponent function, the equation will become a viscosity model [9].

Method to finding Solution
In this section, the algorithm of Homotopy Pertubation Methods combining perturbation and homotopy technique will be presented. Given a nonlinear fractional differential equation as follows: with the initial condition: where is linear differential operator, is nonlinear differential operator, f( ) is a known analytic function, B is a boundary, n is the unit outward normal and is the boundary of the domain .
In this Homotopy Perturbation Method, homotopy is defined as where ∈ Ω and ∈ [0,1] are the attached parameters, 0 is initial approach value which fulfil the initial condition. From equations (4) and (5) The adjustment process of from 0 to 1 is equal to the change of v ( , p) from 0( ) into ( ). This process is known as deformation while L( ) − ( 0) and L( ) + N( ) − ( ) is called as homotopic.
Hence the solution of (4) and (5) in form of p-power series is By substituting (9) to (10) and take p = 1, then solution function of (6) will be: where ( ) is obtained from integral result of derivative function as follows on equation (13).

Result and Discussion
In this section the method to find a Fractional Differential Equation Model solution will be provided using the Modified Homotopy Perturbation Method as described above, and continued by analyzing the convergence of the solution function sequence.

Solution Function
Again, we see the general form of Nonlinear Fractional Differential Equation If we take a = -1 , b = 1 , and A(t) = 2t , the equation become with initial conditions u(0) = 0.
Based on Modified Homotopy Perturbation Method, homotopy equation (10) became So, the solution of this equation is = 0 + 1 + 2 2 + 3 3 + ⋯ . By substituting the basic assumptions and initial conditions for the homotopy equation obtained Therefore, solution function of equation (14):

Convergence Analysis
In this section, convergence analysis of solution function sequence will be elaborated in order to capture the behavioral change of solution function graph based on the change of fractional order. It is developed according to two examples in the previous section, which are solution of linear and nonlinear fractional differential equations.
Initially, it is given a fractional order of a number sequence (n ) = ( +1  Figure 1 presents the graph of the solution function of nonlinear fractional differential equation with order n -( ) + 2 ( ) = 2 as in (14) with solution of ( ) in (15).

Figure 1. Graphs of solution function of nonlinear FDE (14) with n = +1
Hence based on Figure 1, it can be seen that when n is closer to 1, the solution function graph will be closer to nonlinear FDE solution function graph with 1 order.
Similarly, figure 2 presents graph for the second case of Linear FDE in (16) with the solution of (17). Hence, it can be seen that the solution functions sequence moves according to the sequence of fractional orders, starting from red graph to black graph as a convergence function.

Conclusion
Based on what has been discussed above, the conclusion is that the Modified Homotopy Perturbation Method guarantees the existence of a solution of Fractional Differential Equation model. The second conclusion is that if the fractional order sequence is converges to , then the sequence of solution functions is also converges to solution fuction of Fractional Differential Equation with order  .