Effective Approach to Construct Series Solutions for Uncertain Fractional Differential Equations

Purpose: We construct the analytical approximate resiual power fuzzy series solutions of fuzzy conformable fractional differential equations in an -level depiction in the sense of strongly generalized -fuzzy conformable derivative in which of the all initial conditions are taken to be fuzzy numbers. Methodology: The certain fuzzy conformable fractional differential equation under strongly generalized -fuzzy derivative is converted to a crisp one as a family of differential inclusions and solved via resiual power method. The main drawback concerning the use of differential inclusions is that it does not contain a fuzzification of the differential operator; instead, the solution is not essentially a fuzzy valued function. Findings: (i) To show the efficiency of our proposed method: Several important and attractive test examples, which included the fractional conformable fuzzy integro-differential equation are discussed and solved in detail. (ii) To show the stability of approximate solutions to specific problems: some graphical results, numerical comparisons and tabulate data are created and discussed at different values of Value: Using the residual power series analysis methos is a powerful and easy-to-use analytic tool to solve initial problems on fuzzy conformable fractional differential equations and it successfully applied to solve real life problems such as the inductance–resistance–capacitance, RLC-series circuit.


Introduction
Most real-world optimisation problems often involve uncertainty or inaccuracy in the data due to measurement errors or some unexpected things. Fuzzy logic is a type of uncertainty that is very common in solving those kinds of problems and it is also used as an important mathematical tool to express the ambiguity and inaccuracy of human thinking [1][2][3][4]. Zadeh established the fuzzy set theory [1], fuzzy derivative [2] and also introduced the concept of the linguistic approach and its applications in artificial intelligence, decision-making process information retrieval, and economics [3]. However, Dubois and Prade [4] proposed the concepts of fuzzy differential and differential integration for fuzzy valued functions (F-VFs) and developed by many researchers such as Stefanini [5], Dubois, Prade [6][7][8], and Puri, and Ralescu [9][10]. Fuzzy fractional differential equations (F-FDEs) appeared as a generalisation of fuzzy differential equations (F-DEs) and they are playing a fundamental role in modelling many real-life fuzzy difficulties and various phenomena under uncertainty that arise many applications such as in quantum field theory and optics, doffing oscillator and population models, electronic, dynamical and control systems, artificial intelligence, industrial engineering, financial and management of banking, nature studies and technological processes . For example, Guo et al. [11] introduced the fuzzy population models; Agarwal et al. [12] presented the concept of F-FDE; Fard and Salehi [13]; Soolaki et al. [14] discussed the fuzzy fractional variational problems using Caputo and combined Caputo differentiability; Zhang et al. [15] proved the generalised necessary and sufficient optimality conditions for fuzzy fractional problems based on Atangana-Baleanu fractional derivative (FD) and generalised Hukuhara difference; Das and Roy [16] presented a new numerical method for solving F-FDEs using Adomian decomposition method in terms Riemann-Livoullie sense; Salahshour et al. [17] solved F-FDEs by fuzzy Laplace transforms; Ghaemi et al. [18] discussed the numerical solution (NS) of a fuzzy fractional kinetic equation in the term of fuzzy Caputo FD; Marc et al. [19] discussed the potential applications of fuzzy logic in financial and management of banking; Venkat et al. [20] used fuzzy logic in financial markets for decision making; Bede and Gal [21] introduced the strongly-generalised differentiability (SGD) for a F-VF; Hasan et el. [22] investigated the analytical and NSs of fractional fuzzy hybrid system in Hilbert space are devoted to model control via Atangana-Baleanu Caputo FD; Ahmadian et al. [23][24][25] used tau method for finding the NSs of a fuzzy fractional kinetic model, uncertain fractional viscoelastic model and linear F-FDEs, respectively; and Sh. Behzadi et al. [26] applied the Fuzzy Picard method for solving fuzzy quadratic Riccati and Fuzzy Painlev equations.
Generally, it is not straightforward to obtain exact solutions (ESs) for these kinds of problems because of the difficulties involved, so reliable numerical techniques are needed to deal with these types of problems. Recently, some methods were suggested for creating analytical NSs for uncertain F-FDEs in terms of SGD sense. These methods are the finite element, reproducing kernel, fuzzy Picard method, homotopy, and residual power series (R-PS) methods [21,[26][27][28][29][30][31][32][33].
The R-PS method developed by some authors [30][31] which is considered as effective optimisation technique to construct power series solutions (PSSs) of some F-DEs. Very recently, the R-PS technique was used for solving certain and uncertain F-FDEs [34][35][36][37][38][39][40][41] such as Tariq et al. [35] combined Laplace transform method with the R-PS method in new theory and view and used it to create PSSs for the target equations. Abu Arqub [30] used the R-PS method to obtain the PSSs of F-DEs under SGD; Alaroud et al. [36] presented a novel optimisation technique, the RPS for handling certain classes of F-FDEs of order 1 < ≤ 2 under SGD; Alshammari et al. [29] presented the analytic NSs of uncertain Riccati DEs using R-PS method; Abu Arqub and Al-Smadi [37] proposed the so-called fuzzy conformable fractional derivative (FC-FD) and integral and then used to obtain the solutions of certain fuzzy conformable fractional differential equations (FC-FDEs) under SGD and Alshammari et al. [38] proposed an accurate numeric-analytic algorithm on the R-PS to investigate the fuzzy NSs for a nonlinear fuzzy Duffing oscillator under SGD.
There are many definitions for FDs and the most two important of them are the Riemann-Liouville and the Caputo definitions which are used in many applications and real-natural phenomena. While in 2014, Khalil et al. [39] defined the conformable FD (C-FD) of an order α ∈ (0, 1] of f : [t 0 , ∞) → R at t 0 ≥ 0 as follows: and is differentiable, and the limits exists.
For more details about the basic rules and applications of C-FD, it can be found in the literature [39][40][41][42][43][44][45][46][47]. The main advantages of the C-FD can be summarised as follow: (i) It can be very easily computed compared with other previous fractional definitions. (ii) It satisfies all concepts of classical derivative while other fractional definitions fail to satisfy some of them. (iii) It can be computed for a non-differentiable function. (iv) It solves F-DEs, PF-DEs, and systems easily and efficiently.
(v) It modifies some important transforms such as Laplace, Sumudu, and Nature transforms, and they are used as efficacious tools for solving some singular F-DEs. (vi) Several applications have been remodeled using C-FD and it can be opened the door for various new applications. (vii) Several new comparison results can be established based on C-FD.
Given the above-mentioned features of the C-FD, we seek through this research to integrate this concept of the FD to the fuzzy equations for studying the extent of these features being reflected in the development of those equations and improving them in terms of realism and the form of the solution. In addition, we aim to investigate the applicability of the RPS method under the assumption of SGD to provide analytical-NSs for the FC-FDEs that of the following general form: subject to the fuzzy initial condition (F-IC): where T α is the C-FD of an order α, f is an analytic function,î : [a, b] × R F → R F is a continuous F-VF, in which R F stands to the group of fuzzy numbers. There are three main approaches to solve FC-FDEs with F-ICs. The first one is that if the initial value is only a fuzzy number, the solution will be a fuzzy function, where as a result, the derivatives must be regarded as fuzzy derivatives. To achieve this problem, the SGD for F-VFs must be used. The second approach is that the FC-FDE will be converted to a crisp one as a family of differential inclusions. The main drawback concerning the use of differential inclusions is that it does not contain a fuzzification of the differential operator; instead, the solution is not essentially a F-VF. The last approach is related to the crisp equation and the initial fuzzy values, which are in the solution. The weak point lies in having the solution in the fuzzy setting re-written, making the techniques of solution less friendly and more constrained with a lot of steps of computation to do. To achieve this point, we will replace the crisp equation and the initial fuzzy values in terms of real constants as well as arithmetic operations regarded as operations on fuzzy numbers in the final solution. The latest solutions approach, which focuses on searching the fuzzy set of real-valued functions (R-VFs), not F-VFs exemplified with these R-VFs, fulfils the said restrictions.
Furthermore, most of the four main interests bring us to fuzzy applications and R-PS method are: (i) They are convenient and remarkably powerful tools in solving numerous issues arising in physics, engineering, financial and management of banking and other sciences. (ii) They are useful in uncertain or estimated reasoning or discretionary thinking, especially for a framework with a scientific model that is difficult to derive. (iii) Useful in assessed qualities under deficient or questionable data, especially for control theory in optimality and security. (iv) (iv) Fuzzy controllers are widespread nonlinear controllers and each one of these examinations is fundamental in nature and more investigations should be possible. For the sake of idealism, the optimal fuzzy control field appears to be completely open.
This paper consists of five Sections. Some important definitions and concepts for the fractional fuzzy theory are given concisely in Section 2. In Section 3, we employ the R-PS method to construct the analytical approximate fuzzy PSSs for the initial value problem (IVP) as in Equations (2) and (3) in an r-level depiction in the sense of strongly generalised α-fuzzy conformable derivative (SG α-FCD). Some important and attractive test examples with illustrative graphs and tables are given in Section 4 in order to validate our proposed algorithm. Finally, the conclusion is presented in Section 5.

Conformable Fractional Fuzzy Theory
This section reviews and studies some fuzzy (fractional) theory that will be used in our investigations and findings throughout the paper.

Definition 2.2 ([18]):
If u ∈ R F and r ∈ [0, 1], then the r-level of u is called a crisp set and defined as: So, if u ∈ R F , then the r-level of u is a closed interval in R and defined by: where u 1r = u 1 (r) = min{s : s ∈ [u] r } and u 2r = u 2 (r) = max{s : s ∈ [u] r }. However, there are several common forms for the fuzzy number u, one of them is the triangular form which is defined by an ordered triple u = (μ 1 , μ 2 , μ 3 ) ∈ R 3 with μ 1 < μ 2 < μ 3 and whose r-level is [22]: (ii) H-difference: If there exists an element z ∈ R F such that u = v + z, then z is called Hdifference of u and v, denoted by u v, and defined as follows: (iii) Scalar multiplication: (iv) Multiplication: (v) Equality: Two fuzzy numbers u and v are equal if

Definition 2.4 ([18]):
The complete metric structure on R F is defined as d : Note that the metric space d in R F has many nice and interesting properties on addition operation and scalar multiplication and the reader can be found in the literature [18,36,[44][45][46][47][48][49][50].
which is called the r-level representation of a F-VF.

Definition 2.8 ([18,37,48-49]):
,î is called the SGD at t 0 if there exists an elementî (t 0 ) ∈ R F such that either: for all ε > 0 sufficiently near to 0 and the limits in a metric d.
for all ε > 0 sufficiently near to 0 and the limits in a metric d.

Definition 2.12 ([31]):
A series of the form: where 0 ≤ m − 1 < α ≤ m and t ≥ t 0 is called a fractional power series (F-PS) about t 0 , where t is a variable and c k 's are constants called the coefficients of the series.

Theorem 2.4: Suppose thatî(t) has an F-PS representation at t 0 of the form
where R is the radius of convergence of the F-PS. Then Proof: We can prove the theorem inductively. For m = 0, the formula is correct. With respect to m = 1 and according to Lemma 2.1 the formula is also true. Assume the formula is true for m = n; i.e To complete the proof, we need to show that the formula is true for m = n + 1. Thus,

Theorem 2.5 ([40]):
Suppose that i has a F-PS representation at t 0 of the form . ., then the coefficients c n in Equation (29) will take the form c n =ˆi (nα) (t 0 ) α n n! .

Construction the Residual Power Series Solution for the FCFDEs
In this section, we employ the R-PS method to construct analytical fuzzy NSs in the form of a F-PS for the IVP as in Equations (2) and (3) in an r-level depiction. To accomplish this, we switch the IVP in Equations (2) and (3) to crisp system of F-DEs. The crisp systems depend on the type of differentiability, whereî(t) is either α(1)-FCD or α(2)-F-CD. Without losing the generality, we construct the R-PSS to α(1)-FCD only, and in the same approach, we can construct the R-PSS to or α(2)-FCD. Now, assume thatî(t) is α(1)-FCD, then the IVP in Equations (2) and (3) can be reformulated to the following F-DEs system: with the F-ICs:ˆi where α ∈ (0, 1] and t ∈ [0, b]. The R-PS method supposes that the solution of the given DE has an F-PS. So, we assume that the solution of the system (30) and (31) has the following expansions: According to the F-ICs in Equation (31), it easy to see that b 0 =î 0,1r and c 0 =î 0,2r . Indeed, R-PS is a technique used to determine the coefficients of the F-PSS in a different approach than the traditional method. Thus, to determine the coefficients of the series in Equation (32), we need to define the so-called residual functions of the equations in Equation (32) as follows: , Indeed, we have the following facts for our approach via the R-PS method: (iii) T mα Res 1r (0) = T mα Res 2r (0) = 0, m = 0, 1, 2, . . .
Now, if we substitute the F-PS in Equation (32) into Equation (33) and use the Theorem 2.4, then we obtain the following expression of the residual functions: Since f 1r and f 2r are analytic functions of F-PS expansions, Equation (36) can be expressed in the following F-PS expansions: where g 1rk and g 2rk are multivariable functions of b 0 , b 1 , . . . , b k , c 0 , c 1 , . . . , c k generates according to f 1r and f 2r . Likewise, if we apply the C-FD T mα on both sides of Equation (37), then it can be expressed as an F-PS expansion as follows: According to Equation (38) and the fact in Equation (34), we define the so-called αmth-order DE as follows: Substitute t = 0 into Equation (39), we obtain the following iterative equations: Solving Equation (40) for b m+1 and c m+1 gives the following recurrence relations which determine the coefficients of the F-PS in Equation (32): Therefore, the exact solution (ES) of the system (30) and (31) in an F-PS form is given bŷ Furthermore, the mth-truncated of the F-PS in Eq (3.13) for an appropriate number of m gives the NS of the system (30) and (31) as follows: With the same previous approach, we can find a PSS to the system of F-DEs that corresponding to α(2)-FCD case. While in the next section, we consider both cases (α(1)-FCD and α(2)-FCD) in details for dealing the three important and interesting examples.

Illustrative Examples of FC-FDEs
In this chapter, we apply the R-PS method to three examples to illustrate the construction that was prepared in Section 3 and to confirm the effectiveness and efficiency of the proposed method used in constructing an F-PSS to FC-FDEs. The first example includes a linear FC-FDE, the second example deals with a nonlinear FC-FDE, where as the third example illustrates our method to construct the F-PSS to FC-integral equation.

Example 4.1: Given the following FC-FDE:
T αî (t) = −î(t) + sin(t α ), t ∈ [0, 1], 0 < α ≤ 1, subject to the F-IC:ˆi  (44) subject to the F-IC (45), we consider the following two cases: Case 1: The system of the ODEs corresponding to α(1)-FCD is subject to the F-ICs:ˆi To employ the RPS method to create the PSS to the system (46) and (47), we assume that the solution of this system can be expressed as F-PS expansion about the initial point t = 0 as follows:ˆi Using the F-ICs in Equation (47) give us b 0 = 24 25 + 1 25 r and c 0 = 101 100 − 1 100 r. Therefore, the solution of system (46) and (48) can be represented as: Apply the C-FD T α to Equation (49), we obtain: Define the residual functions of Equation (46) as follows: So, the αmth-order DEs of Equation (46) have the following forms: Through a few simple calculations, a new discretised version of Equation (53) can be obtained and given by: where It is obvious that, the αmth-derivative of the F-PS representation, Equation (54) is convergent at least at t = 0, for m = 0, 1, 2, . . .. Therefore, the substituting t = 0 into Equation (54) gives the following recurrence relation which determines the values of the coefficients b k and c k : If we collect and substitute these values of the coefficients back into Equation (49), then the ES of the system (46) and (47) has the general form, which is coinciding with the general expansion: If we rearrange the terms in Equation (57), then it can be written as follows: We can summarise the solutions in Equation (57) aŝ which are equivalent to: In fact, if α = 1, then the PSS to the system (46) and (47) will be as:

Case 2:
The system of the ODEs corresponding to α(2)-FCD is: subject to F-ICs:ˆi Similarly, assume that the F-PSSs of Equations (58) and (59) has the following form: Then the αm10th-order DEs corresponding to Equation (58) is: Operator T mα in Equation (64) gives a summarised form of the equation: where χ m (t) is defined as in Equation (55). The recurrence relation which determines the values of the coefficients b k and c k is: So, the ESs of Equations (61) and (62) have the general form: We can formulate the ES in Equation (67) aŝ Thus, the ES of the system (61) and (62) when α = 1 has the following expression: subject to the F-IC:ˆi where u = max(0, 1 − |ρ|), ρ ∈ R.

Case 1:
The system of the ODEs corresponding to α(1)-FCD is: subject to the F-ICs:ˆi Assume the F-PS representation of the solution to the Equations (72) and (73) is: Then the αmth-order DEs corresponding to Equation (72) are: where Therefore, by substitute t = 0 into Equation (75), the recurrence relations which determine the values of the coefficients b k and c k are: , m = 2, 3, 4, . . . , , m = 2, 3, 4, . . .
If we substitute these values in Equation (74), then the F-PSS of Equations (72) and (73) has the general form: The closed form of the solution in Equation (77) iŝ Thus, the ES to the Equations (70) and (71) in the sense of α(1)-FCD, can be expressed as follows:ˆi The closed form of the solution in Equation (86) is: Thus, the ES to the Equations (70) and (71) in the sense of α(2)-FCD, can be expressed as follows:ˆi In the next example, we apply the R-PS method to construct the PSS to a fractional conformable fuzzy integro-differential equation (FCFI-DE). The same procedure used in the previous examples will be used with a few minor differences.

Example 4.3:
Consider the inductance-resistance-capacitance, RLC-series circuit as in diagram below: Then the FCFI-DE is given by: subject to the F-IC:ˆi where R, L, and C are the resistance, inductance corresponding to the solenoid and capacitance, respectively, and V(t) = sin(t α ) represents the voltage function in the electric circuit..
For simplicity's sake and for numerical analysis, we assume that R = L = C =  (90) to ODEs systems that we exhibit in the following two cases: Case 1: The system of the FCFI-DEs corresponding to α(1)-FCDis: subject to the F-ICs:ˆi The ES in this case at α = 1 is: where Assume the F-PS representations of the solution for the Equations (91) and (92) are: So, the ES of Equations (91) and (92) has the general form: Case 2: The system of the FCFI-DEs corresponding to α(2)-FCD is: subject to the F-ICs:ˆi The ES in this case at α = 1 is: Our goal here is to illustrate some numerical results of the R-PSSs of Equations (89) and (90) to show the validity and efficiency of the proposed method. Anyhow, Tables 1 and  2 show the exact error of α(1)-FCD and α(2)-FCD 10th-approximate R-PSSs, respectively, of Equations (89) and (90) at α = 1 and various values of r and t. While Tables 3 and 4 show the relative error of α(1)-FCD and α(2)-FCD 10th-approximate R-PSSs, respectively, of Equations (89) and (90) at different values of α, r and t. The exact and relative errors are defined, respectively, as follows: Relative error : Example 4.3 was discussed in [37] by the reproducing kernel Hilbert space (RKHS) method. Tables 5 and 6 show the exact error of α(1)-FCD and α(2)-FCD solutions of Equations (89) and (90) at α = 1 and various values of r and t that were obtained by the RKHS and R-PS methods. The data shows that there is a slight improvement in error when using the R-PS method.

Conclusions and Discussions
In recent years, FDs have been used to model some natural phenomena using fuzzy equations. It has been observed that the order of the FD has a close relationship in controlling the shape of the solution so that it expresses the phenomenon more realistically. Our goal in this paper was to use C-FD in the fractional fuzzy equations instead of other previously used FDs, as the Caputo derivative. It was observed from the graphs that the solutions we obtained were smooth and simulating the ordinary derivative. Also, the mathematical calculations in finding the solutions were easier than previous equations in which other types of FDs were used. This is due to the advantages that the C-FD has over the other types. On the other hand, the RPS method was used in finding series solutions of initial problems on FC-FDEs in the sense of SGα-FCD due to their novelty and ease. Indeed, the choice of this method was successful, as an accurate approximate solution was found in general, and sometimes the ES was obtained as we presented in the first and second examples. The results show that the power series analysis method is a powerful and easy-to-use analytic tool to solve initial problems on FC-FDEs. How to apply this new approach for solving problems related to FC-PDEs still need further research.