Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

This paper aims to present a novel optimization technique, the residual power series (RPS), for handling certain classes of fuzzy fractional differential equations of order 1 < γ ≤ 2 under strongly generalized differentiability. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form.The RPS algorithm is significant and straightforward tool for creating a fractional power series solution without linearization, limitation on the problem’s nature, sort of classification, or perturbation. Some illustrative examples are provided to demonstrate the feasibility of the RPS scheme.The results obtained show that the scheme is simple and reliable and there is good agreement with exact solution.


Introduction
Fuzzy fractional differential equation is hot and important branch of mathematics. It has attracted much attention recently due to potential applications in artificial intelligence, industrial engineering, physics, chemistry, and other fields of science. Parameters and variables in many of the nature studies and technological processes that were designed utilizing the fractional differential equation (FDE) are specific and completely defined. Indeed, such information may be vague and uncertain because of experimentation and measurement errors that then lead to uncertain models, which cannot handle these studies. The process of analyzing the relative influence of uncertainty in inputs information to outputs led us to study solutions to the qualitative behavior of equations. Therefore, it is necessary to obtain some mathematical tools to understand the complex structure of uncertainty models [1][2][3][4][5]. On the other hand, the theory of fractional calculus, which is a generalization of classical calculus, deals with the discussion of the integrals and derivatives of noninteger order, has a long history, and dates back to the seventeenth century [6][7][8][9][10]. Different forms of fractional operators are introduced to study FDEs such as Riemann-Liouville, Grunwald-Letnikov, and Caputo. Out of these forms, the Caputo concept is an appropriate tool for modeling practical situations due to its countless benefits as it allows the process to be performed based on initial and boundary conditions as is traditional and its derivative is zero for constant [11][12][13][14][15][16][17].
The residual power series (RPS) method developed in [18] is considered as an effective optimization technique to determine and define the power series solution's values of coefficients of first-and second-order fuzzy differential equations [19][20][21][22]. Furthermore, the RPS is characterized as an applicable and easy technique to create power series solutions for strongly linear and nonlinear equations without being linearized, discretized, or exposed to perturbation [23][24][25][26][27]. Unlike the classical power series method, the RPS neither requires comparing the corresponding coefficients nor is a recursion relation needed as well. Besides that, it calculates the power series coefficients through chain of equations of 2 International Journal of Differential Equations one or more variables and offers convergence of a series solution whose terms approach quickly, especially when the exact solution is polynomial.
The remainder of this paper is organized as follows. In Section 2, essential facts and results related to the fuzzy fractional calculus will be shown. In Section 3, the concept of Caputo's H-differentiability will be presented together with some closely related results. In Section 4, basic idea of the RPS method will be presented to solve the fuzzy FDEs of order 1 < ≤ 2. In Section 5, numerical application will be performed to show capability, potentiality, and simplicity of the method. Conclusions will be given in Section 6.

Preliminaries
In this section, necessary definitions and results relating to fuzzy fractional calculus are presented. For the fuzzy derivative concept, the strongly generalized differentiability will be adopted, which is considered H-differentiability modification.
A fuzzy set V in a nonempty set is described by its membership function V : Definition 1 ([28]). Suppose that V is a fuzzy subset of R. Then, V is called a fuzzy number such that V is upper semicontinuous membership function of bounded support, normal, and convex.
If V is a fuzzy number, then is called the -level representation or the parametric form of a fuzzy number V. Theorem 2 ([29]). Suppose that V 1 , V 2 : [0, 1] → R satisfy the following conditions: (1) V 1 is a bounded nondecreasing function.

Definition 3 ([29]
). Let V, ∈ R F . If there exists an element P ∈ R F such that V = + P, then we say that P is the Hukuhara difference (H-difference) of V and , denoted by V ⊖ .
The sign ⊖ stands always for Hukuhara difference. Thus, it should be noted that
) exist, for each >0 sufficiently tends to 0 and where the limit here is taken in the complete metric space (R F , ).
The next characterization theorem shows a way to convert the FFDEs into a system of ordinary fractional differential equations (OFDEs), ignoring the fuzzy setting approach.

Theorem 13 ([34]). Consider the below fuzzy fractional IVPs
subject to where (iii) there is a constant (say) ℓ > 0 such that and Therefore, there are two systems of OFDEs that are equivalent to FFDEs (4) and (5) as follows:

Formulation of Fuzzy Fractional IVPs of Order 1 < ≤ 2
Consider the below fuzzy fractional differential equation subject to fuzzy initial conditions where , ∈ R F , : [ , ] × R F → R F is a linear or nonlinear continuous fuzzy-valued function, ( ) is a continuous real valued function with nonnegative values on [ , ], and ( ) is unknown analytical fuzzy function to be determined. We assume that the fuzzy fractional IVPs (10) and (11) have unique smooth solution on the domain of interest.
Next, some theorems and definitions which are used later in this paper are presented.
The aim of the next algorithm is to perform a strategy to solve the FFIVPs (10) and (11) in terms of its -cut representation form. Indeed, there are four cases that depend on type of differentiability.

Case (II). If ( ) is Caputo
Step 1: Solve the required system.

Description of Fractional RPS Method
In this section, the RPS scheme is presented for constructing an analytical solution of FFIVPs (10) and (11) through substituting the expansion of fractional power series (FPS) among the truncated residual functions. In view of that, the resultant equation helps us to derive a recursion formula for the coefficients' computation, where the coefficients can be computed recursively through the recurrent fractional differentiating of the truncated residual function.

Conclusion
In this paper, the RPS algorithm is successfully developed, investigated, and applied to solve the fuzzy differential equation of fractional order 1 < ≤ 2 with fuzzy initial constraints under the fuzzy concept of Caputo Hdifferentiability. The fuzziness is represented using upper semicontinuous membership function of bounded support, convex, and normalized fuzzy numbers based on its single parametric form. The behavior of approximate solution for different values of fractional order is discussed quantitatively as well as graphically. The numerical results in this paper demonstrate the efficiency of the algorithm. We conclude that the proposed scheme is highly accurate in solving widely array of fuzzy fractional issues.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
International Journal of Differential Equations

Conflicts of Interest
The authors declare that they have no conflicts of interest.