On numerical approximation of Atangana-Baleanu-Caputo fractional integro-differential equations under uncertainty in Hilbert Space

. Many dynamic systems can be modeled by fractional differential equations in which some external parameters occur under uncertainty. Although these parameters increase the complexity, they present more acceptable solutions. With the aid of Atangana-Baleanu-Caputo (ABC) fractional differential operator, an advanced numerical-analysis approach is considered and applied in this work to deal with different classes of fuzzy integrodifferential equations of fractional order fitted with uncertain constraints conditions. The fractional derivative of ABC is adopted under the generalized H-differentiability (g-HD) framework, which uses the Mittag-Leffler function as a nonlocal kernel to better describe the timescale of the fuzzy models. Towards this end, applications of reproducing kernel algorithm are extended to solve classes of linear and nonlinear fuzzy fractional ABC Volterra-Fredholm integrodifferential equations . Based on the characterization theorem, preconditions are established under the Lipschitz condition to characterize the fuzzy solution in a coupled equivalent system of crisp ABC integrodifferential equations. Parametric solutions of the ABC interval are provided in terms of rapidly convergent series in Sobolev spaces. Several examples of fuzzy ABC Volterra-Fredholm models are implemented in light of g-HD to demonstrate the feasibility and efficiency of the designed algorithm. Numerical and graphical representations of both classical Caputo and ABC fractional derivatives are presented to show the effect of the ABC derivative on the parametric solutions of the posed models. The achieved results reveal that the proposed method is systematic and suitable for dealing with the fuzzy fractional problems arising in physics, technology, and engineering in terms of the ABC fractional derivative


Introduction
Modeling processes of natural phenomena create perceptions and general impressions about the dynamic behavior of any physical system that may involve uncertain parameters that result from many factors such as measurement errors, estimates, expectations, and deficient data. In this direction, fuzzy set theory has been established to describe the uncertainty that appears in mathematical modeling. In 1965, the standard theory was originally introduced by Zadeh [1]. Afterwards, Dubois and Prade [2] proposed the fuzzy real numbers notions along with some of the essential characteristics as well. In a later separate work, Kandel and Byatt [3] introduced the concept of fuzzy differential equations. Anyhow, there are several suggestions for defining fuzzy derivative operators as well as studying fuzzy differential equations, including Seikkala, Goetschel-Voxman, Hukuhara (H-differentiability), Puri-Ralescu, and strongly generalized differentiable concepts [4][5][6][7]. The most popular approaches employing standard, strong, or generalized Hdifferentiability [8][9][10].
On the other hand, the theory of fractional calculus is an interesting topic, not only among mathematicians but also among physicists and engineers for its great importance applications in many fields of engineering and sciences [11][12][13][14][15][16][17]. It has been investigated extensively for describing memory and hereditary for various physical and engineering applications similar to rheology, continuum mechanics, entropy, electromagnetic problems, thermodynamics and so forth [18][19][20][21][22][23][24][25]. Further, in the literature, there are many definitions of fractional derivatives, such as the concepts of Caputo, Erdelyi-Kober, Riemann-Liouville, Graunwald-Letnikov, Riesz, and Caputo-Fabrizio [26][27][28][29][30][31][32][33][34][35]. The most common are the concepts of Riemann-Liouville and Caputo which bring some privacy. Nevertheless, they involve a singular kernel function that may adversely affect a realistic understanding of real-world problems. Indeed, Caputo-Fabrizio concept proposed in [19] includes an exponential kernel function to describe variants and structures of different scales, which cannot be well formulated by standard local derivatives possessing single kernels such as Riemann-Liouville and Caputo. In this orientation, Atangana and Baleanu proposed a novel concept as a generalization of the Caputo-Fabrizio derivative in light of the generalized Mittag-Leffler function to construct the non-singular as well as nonlocal kernel [22].
As a result of studying the fuzzy theory for fractional calculus, the term of fractional fuzzy differential equations (FFDEs) has been established in 2010 [36]. In [37], it has been proposed the fractional generalization by means of H-differentiable. Recently, the authors [38,39] defined a novel operator of fractional derivatives based on Atangana-Baleanu-Caputo (ABC) in view of fuzzy valued function with form of parametric interval, called ABC gH-differentiability. In this paper, we intend to study the effect of ABC gH-differentiability on the solution of different types of fuzzy fractional integrodifferential equations (FFIDEs). More specifically, we consider the underlying Fredholm-Volterra FFIDE: where ≥ , 0 < ≤ 1, is a fuzzy number, ( ) is a continuous fuzzy-valued function, ( ) is a continuous real-valued function with nonnegative or nonpositive values on [ , ], and + (•) denotes the ABC gH-derivative of order in view of continuous kernel functions ( , , ( )) and ( , , ( )).
By and large, there are no conventional techniques for finding exact solutions for FFIDEs. Therefore, there is an urgent need for advanced numerical methods to obtain accurate approximate solutions to these equations. In this analysis, we modify a numerical method based on the reproducing kernel theory to obtain approximate solutions of Eq. (1). The current method has many positive advantages. For instance, but not limited to, it is reliable and accurate numerical results can be achieved easily, it can be applied directly without any further assumptions on the structure of specific physical problems, it is not affected by cumulative calculation errors, and it is universal in nature and has a high capacity for solving various nonlinear mathematical issues. Therefore, the RKHS method has received enough attention in the last decade [40][41][42][43][44][45][46].
Motivated by the aforementioned discussion, this numerical research aims to design a novel iterative algorithm to obtain solutions to fuzzy integrodifferential equations in terms of the new ABC-fractional concept containing nonsingular and nonlocal kernel under gH-differentiability in addition to studying the effect of ABC-fractional derivative on these solutions. To begin with, several kernel functions are created to establish a complete orthogonal system in the Hilbert space. For this purpose, a linear bounded and invertible fractional operator is defined to extend analytical solutions over a dense and compact interval. Based on reproducing kernel property, the approximate solutions converge uniformly to the analytical solutions. Eventually, some numerical examples are presented to illustrate the reliability and efficiency of the suggested algorithm. This paper is organized as follows: In Section 2, some basic concepts related to fuzzy calculus and fractional calculus are presented. In Section 3, the FFIDE converted into a fractional system of integrodifferential equations. The procedures of RKHSM for solving the general form of both linear and nonlinear FFIDEs are discussed in Section 4. Some numerical examples are carried out in Section 5. This paper ends with a conclusion in section 6.

Preliminaries and mathematical concepts
In this section, some necessary definitions and mathematical preliminaries of fuzzy calculus and fractional calculus are introduced. To this end, the main concepts used in this analysis are presented, namely, the strongly generalized differentiability, ABC gH-differentiability, and Riemann integrability. Anyhow, a fuzzy number is a generalization of a real number in the sense that it does not refer to a single value but rather to a set of possible values, each with a weight between 0 and 1. This weight is called the membership function.
With the help of gH-difference (2), the parametric interval form of the gH-derivative for a function ( ) has only one of the following forms:  For integration of a fuzzy valued function, many approaches have been proposed such as Lebesgue integral [47] and fuzzy Riemann integral [51]. The Lebesgue integral ( ) within interval parametric form is more convenient, which is defined level wise by This analysis deals with ABC gH-derivative but before presenting its definition, it is worth recalling the Caputo and ABC fractional derivatives of crisp functions. Hereunder, the underlying notations will be used. [ , ] is given as where − 1 < ≤ , ∈ ℕ, > . Specifically, we have The CFD possesses a kernel with singularity that includes the memory effect, so this definition cannot describe the full effect of memory. Hereby, we present the Atangana-Baleanu definition of Caputo-type in which the singular kernel will be replaced by the Mittag-Leffler function.
Definition 2.7. [22] The integral associated with the ABC derivative of order is given as: For = 1, we obtain the classical integral as well as for = 0 , the ABC integral becomes the identity operator. Note that the ABC derivative of any constant is zero, i.e., C = 0 for any constant . Further, the composition relation between the ABC derivative and integral [52] so that . Then, is said to be Then, the fuzzy gH-Atangana-Baleanu-Caputo fractional differentiable fuzzy-valued function (ABC gH-differentiability) is defined by means of the underlying two cases: . Consequently, the interval parametric forms in the two cases can be deduced from Lebesgue integral in (3) as follows: , for = 1,2. Moreover, the integral associated with the ABC gH-derivative can be formulated in interval representation as It is worth to be mentioned that it was shown in [38] that for 0 ≤ ≤ 1,

ABC gH-differentiability formulation of FFIDEs
In this section, we adopt the interval ABC gH-derivative to handle a general form of Fredholm-Volterra FFIDE (1). Without loss of generality, we assume that the kernels are separable such that  (1) is Thus, the FFIDE (1) can be translated into one of the underlying systems: • In the case of • In the case of along with the following initial conditions 1 ( ) = 1 and 2 ( ) = 2 .
Algorithm 3.1. Let ( ) be the unique analytical solution of the FFIDE (1). To obtain an interval approach for the approximate fuzzy solution of FFIDE (1) in the sense of ABC gH-differentiability using RKHS technique, perform the underlying steps: Step 1: Take the -cut for both sides of Eq. (1) in light of assuming [( ) − ]-differentiability, = 1,2.

ABC FFIDEs in terms of RKHS algorithm
In this section, we present a quick review to some basic definitions and theorems concerning the reproducing kernel Theory. For this purpose, we recall the definition of direct sum of Hilbert spaces The property of reproducing any element of ℋ by the function К is called a reproducing property, in which the space ℋ of this case is called a reproducing Kernel Hilbert space (RKHS). Some of the desirable complete RKHSs [56] are constructed as follows.     For the boundedness of ϸ, let ∈ Υ[ , ], we have Therefore, since , = 1,2 is bounded, then is so.  ( ) ] 〉 Θ = 〈[ 11 12 21 22 ] The Nth-term approximate solution of Eq. (8) can be obtained by truncating the finite sum ( ) of the series solution (9) as follows.

Numeric Investigation of FFIDEs with ABC gH-differentiability
In this section, numerical solutions of various types FFIDEs with ABC gH-fractional derivative are investigated using RKHS algorithm. In specific, Volterra, Fredholm, and mixed Volterra-Fredholm FFIDEs are considered by taking different types of kernel functions, non-homogeneous terms, and fractional derivatives. The accuracy of our method is examined by comparing the exact, if exist, with RKHS approximate solutions. To see the effect of ABC gH-derivative on the behavior of the given FFIDEs, a comparison is also done between fuzzy approximate solutions under ABC gHdifferentiability with those of classical Caputo g-H-differentiability. In each example, the normalization coefficient is considered as ( ) = 1 − + Γ( ) . Our numerical results are carried out using Mathematica 10.     can be seen that all the plots are nearly compatible, analogous, and similar in behavior as well as fully consistent with each other, especially when considering the fuzzy ABC derivative of integer order = 1. Also, the solution is an interval at each instantaneous point of all levels of r-cut, which means that the solution is a fuzzy function at every point in the internal domain.
• For [(1) − ]-differentiability, the Volterra FFIDE (14) will be equivalent to the underlying crisp system of Volterra FIDEs: By means of [(1) − ]-differentiability, the Volterra FFIDE (17) will be equivalent to the underlying crisp system of Volterra FIDEs: whose exact solution is (t) = 3 . Using the RKHSM with =25, numerical results for Example 5.3 are given in Table 3 and Figures 15 and 16 in order to show the accuracy of the proposed method and to support the theoretical framework as well.
By means of [(1) − ]-differentiability, the Fredholm FFIDE (19) will be equivalent to the underlying nonlinear crisp system of Fredholm FIDEs: whose exact solution is (t) = ( , 2 − ) . Using the RKHSM with =15, the graphical results are reported in Figures 14 and 15.
whose exact solution for = 1 is ( ) = cosh( ). Using this concept, Table 4 summarizes the error in approximating using 30 iterations in view of the RKHSM, while Figure 16 shows the core of fuzzy approximate solution for different values of ABC gH-derivatives such that ∈ {1,0.95,0.9}.

Conclusions
In this work, a modified numerical algorithm has been profitably designed in light of RKHS method and employed to get approximate solutions of fuzzy fractional integrodifferential equations by means of Atangana-Baleanu-Caputo gH-differentiability. In this direction, characterization theorem was established for ABC-fractional order, in which the studied fuzzy fractional model was consequently transformed into a crisp system of fractional IVPs under fuzzy ABC calculus. The analytical solutions have been given in series form of the parametric interval of ABC in the space . The Nth-term approximate solutions and its derivatives were uniformly convergent to the analytical solutions and its derivatives, respectively. The convergent analysis and error estimation of the proposed method have been discussed as well. Several applications for both linear and nonlinear, Fredholm-Volterra FFIDEs have been presented to demonstrate the reliability and effectiveness of the RKHS method and to support the theoretical framework. With providing numerical examples, the accuracy of the analytical results has been illustrated. From the achieved results, it can be observed that the posed method yields accurate approximate solutions. Anyhow, the numerical results of ABC gH-differentiability have been compared with those of generalized Caputo derivative. Using ABC gH-differentiability, we conclude that the presented study can be effectively utilized as an extended planner in handling many kinds of fractional issues under uncertainty arising in engineering, physics, and natural sciences. Hopefully, the current analysis will be employed in the near future to study more uncertain fractional models by means of ABC gHdifferentiability of higher fractional order.