Analytical Solution of Nonlinear Fractional Gradient-Based System Using Fractional Power Series Method

A BSTRACT . This paper adapted a reliable treatment technique, called the fractional residual power series, to the fractional gradient- based system in solving a class of nonlinear programming model in Caputo’s sense. The gradient-based system has been constructed to transform the nonlinear programming problem with equality constraints to unconstrained optimization problem, and then the fractional residual power series method is implemented to obtain the essential behavior of underlying problem. The proposed methods have been applied effectively to produce optimal solution in rapidly convergent fractional series representations without linearization, or any limitations. To confirm the performance of the proposed methods, some optimization problems are tested. Further, numerical comparisons with other existing methods are also given. The results exhibit that the FRPS method is easy, simple, effective, and fully compatible with the complexity of such models.


INTRODUCTION
Over the past years, the topic of optimization has had the interest of many scholars in various applications of science and technology and associated with several categories of optimization problems. Besides, many effective methods have been developed to find the optimal solution to these problems. For more details see [1][2][3][4][5][6]. The gradient-based approach is one such method that is used to solve nonlinear programming (NLP) problems. Transforming the optimization problem into a system of ordinary differential equations (ODEs) is the basic idea of this method, which is equipped with ideal conditions to reach the optimal solutions to this problem [7][8][9][10][11][12][13]. Fractional calculus is a generalization of the derivatives and integrals of an arbitrary system. Recently, the subject of fractional calculus has received the attention of scientists and engineers because of its important applications in various fields, whether science or engineering [14][15][16][17][18][19]. Many real-life problems in various fields of applied science have been modeled using fractional differential equations (FDEs), which are generalizations of ODEs.
For describing the behavior of the unknowns of FDEs, many researchers usually implement some numerical or numerical analytical techniques instead. In this regard, some recent techniques are proposed for solving FDEs. Among them decomposition technique, symmetric perturbation technique, variable frequency technique, and partial differential transformation technique [20][21][22][23][24][25][26].
Further, more applications and promising approaches are utilized to treat the nonlinear fractional gradient-based systems of FDEs could be founded in the references [27][28][29][30]. Motivated by the existing techniques, the main contribution of this article is to transform equality constrained NLP problem to unconstrained optimization problem by identifying a penalty function, then construct a gradientbased system of FDEs. Besides the, the fractional residual power series (FRPS) technique is applied to provide us the accordance between the optimal solution of the NLP problem and the power series solution of the obtained FDEs system. FRPS technique is one of the modern numeric-analytic techniques was initially proposed in [31] to investigate the sequential solution of fuzzy differential equations of both first and second degree. It has been used to generate accurate approximate solutions in terms of fractional series formulas for several kinds for linear and non-linear FDEs, Partial FDEs and Fuzzy FDEs (see [32][33][34][35][36]). This scheme is used basically the residual-error function and employed the fractional differentiation in each stage in determining the coefficient of the suggested series expansion without linearity, division, or perturbation (see [37][38][39][40][41][42]). It does not require any converting while switching from the lower order to the higher order; as a result, the method can be applied directly to the given problem by choosing an initial guess approximation. FRPS is quick and easy calculation to find series solutions via utilizing software package. Also, different Taylor series method, FRPS needs easy computation state with high reliability and less time.
The organization of this paper was to present a brief presentation of some basic and necessary definitions and properties in fractional calculus in section 2, in addition to the fact that the central problem in this paper was presented in section 3. Section 4 presents the details of the application of the proposed technique and provides a clear and simple algorithm for the basic steps of this technique.
Clarifying the applicability and efficiency of the proposed technique by comparing the results derived from it for some numerical examples with the results of the fourth degree Rung-Kutta method, done in section 5. Section 6 is designed to provide some concluding observations.

PRELIMINARIES
In this section, we present the definition of the Riemann-Liouville fractional integral operator, the Caputo partial derivative, and some of their properties [43][44][45][46][47][48][49]. Throughout this paper, the symbol denotes the set of real numbers, the set of integers, and Γ(. ) is a gamma function. For more details, please see [45][46][47][48][49] and references therein. On the other hand, the operator has some basic properties such as, for any real number ,

OPTIMIZATION PROBLEM
The second part presents the details of the optimization problem to be studied in this paper. We consider the following NLP problem with equality constrains where ∈ ℝ is decision variable, ( ) is a vector-valued function of a real variable, and = ( 1 , 2 , … , ) : ℝ → ℝ ( ≤ ) are twice continuously differentiable function such that whose gradient ∇ ( ) has full rank. We assume that a feasible region of (2) is nonempty bounded set.
The penalty function can be defined as where ( ) is the penalty term defined on ℝ and has the following property One can be defined the penalty term ( ) as where > 0 is a constant.
A well-known penalty function for the problem (2) has been defined as where the penalty parameter satisfying the inequality 0 < < +1 for all , We assume that the unconstrained optimization problem (7) for each , has a solution and we denote it by . The main results that connect the solutions of the equality constrained NLP problem (2) and unconstrained problem (7) present in the following theorem.
Theorem 1. Let {x m } be a sequence generated by the penalty method of the unconstrained problem (7). Then any limit point of the sequence is a solution to the equality constrained NLP problem (2).
The author in [12] showed that the unconstrained optimization problem (7) can be described by the following gradient based dynamic system of ODEs where ∇ is the gradient vector with respect to the ∈ ℝ . We can describe the system (8) by an approach based on fractional gradient based dynamic system by the following system of FDEs The system (9) has an equilibrium point , if ∈ ℝ is satisfies the right-hand side of the system.

FRPS TECHNIQUE
This subsection devoted to applying the RPS method to derive analytic solution of the system of FDEs (10). We begin by propose the definition of fractional power series.
Algorithm 1. Algorithm of finding the coefficients of the k-th truncated series (16).
Step 4: Substitute the obtained values of back into Eq. (17).

NUMERICAL IMPLEMENTATION AND RESULTS
This section is designed to apply the proposed technique, FRPS, and evaluate its performance According to (6), the correspond penalty function for the problem (23) at = 2 is given by where the penalty variable ∈ ℝ + , → ∞. Hence, we get the following correspond system of FDEs where 0 < ≤ 1.
The efficiency of FRPS approach is introduced via establishing some numerical comparisons for the obtained results and the results obtained by Runge-Kutta approach and listed in Table 1 where the penalty variable ∈ ℝ + , → ∞. Depending on this penalty function, the correspond where 0 < ≤ 1. We adapt the FRPS technique to the FDEs system (31) with penalty variable = 300 at fractional order derivative = 0.9, we acquired solutions as shown in Table 2. Figure 2 present the obtained FRPS solutions 1 ( ) and 4 ( ) at various fractional derivative order .
Obviously, from Figure 2, the curves-FRPS approximate solutions consistent with each other and approach the exact curve with increasing fractional values to the integer-order value = 1.
The penalty function of this problem can be written as ), where 0 < ≤ 1.
The FRPS utilize to construct the solution of this system and we get the following numerical solutions of the NLP problem (32) as shown in Table 3. It is very clear that the results obtained for example 3 indicated that the proposed method is simple, and its performance is very effective comparing with Runge-Kutta method.