Fuzzy transform-based approximation method for solving fractional semi-explicit differential-algebraic equations

We present an efficient numerical method to approximate the solution of a system of fractional-order linear semi-explicit differential-algebraic equations with variable coefficients. The method is based on the use of the direct and inverse fuzzy transforms (F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}$$\end{document}-transforms). By employing this method, we obtain an analytical approximate solution to the main problem in terms of flexible basic functions. The non-local property of fuzzy transforms helps us to have an efficient method for problems involving non-singular kernels. The error analysis and convergence evaluation of the method are demonstrated in detail. We give some examples to illustrate the significant features of the method.


Introduction
A fractional differential equation (FDE) is a generalization of an ordinary one with an operator of a non-integer (fractional) order. The most important property of fractional operators is that they consider the entire history of the phenomena. Thus, they are excellent tools to describe the memory and hereditary properties of phenomena and processes; and the mathematical models of real-world problems are fractionalorder systems in general. For example, Mainardi in Mainardi (1997) has given some applications of fractional derivatives in continuum and statistical mechanics. For more applications, we refer the reader to Yang (2019).
In many cases, when modeling real-world physical problems with hereditary effects, the states of the physical systems have in some ways constraints, for instance, by conservation laws such as Kirchhoff's laws in electrical networks, or by position constraints such as the movement of mass points on a surface. Then, the corresponding mathematical models contain algebraic equations to describe these constraints, and the fractional differential equations that describe the dynamics of the system. Such systems, comprising of both fractional differential and algebraic equations, are called fractional differential-algebraic equations (FDAEs). In fact, differential algebraic equations (DAEs) and FDAEs are natural ways to model dynamical systems subject to constraints and hereditary effects(see Campbell et al. 2019). They have been studied by researchers in Damarla and Kundu (2015), Ghanbari et al. (2018), Wang and Chen (2014), Westerlund and Ekstam (1994).
Most of FDAEs do not have exact solutions, and then, numerical techniques must be used to get approximate solutions for these types of equations; however, the numerical treatment of FDAEs may be more complicated than the numerical treatment of classical DAEs. By the authors' knowledge, there are only a few numerical methods for solving FDAEs; for instance, high-order Legendre collocation method (Ghanbari et al. 2018), the generalized triangular function operational matrices method (Damarla and Kundu 2015), the variational iteration method (VIM) and the Adomian decomposition method (ADM) ( Ibis and Bayram 2011), and homotopy analysis method (Zurigat et al. 2010). So introducing an efficient numerical method for solving a system of FADEs is the subject of this paper. Two issues are central to construct an efficient method to approximate solutions of FDAEs: the singularity of the kernel of fractional derivatives, and the dimension of the algebraic system. The non-local property of fuzzy transforms and the flexibility of basic functions with local compact supports lead us to have an efficient method for problems involving nonsingular kernels and systems with arbitrary dimension.
In this study, we are interested in using the technique of fuzzy transforms (F-transforms) to give a simple structure and accurate approximate solution to the initial value problem for linear semi-explicit differential-algebraic equations with fractional-order and variable coefficients.
The F-transform that was firstly introduced by Perfilieva in 2006 received significant attention because of its strong connection with real-world problems such as the construction of approximate models, filtering, solving differential equations, application in signal processing, decompression of images, and data compression Perfilieva 2006Perfilieva , 2007Novák et al. 2008;Perfilieva and Daňková 2008;Di Martino et al. 2008;Tomasiello 2017;Hurtik and Tomasiello 2019;Tomasiello 2021). The approximation property of F-transforms and the effect of the shapes of basic functions on the approximation quality were described in Perfilieva (2006); Alikhani et al. (2017). One interesting feature of F-transform is its significant performance in noisy problems in which the inputs of the problem possess some disturbing noise. It has been demonstrated that the F-transform acts as a filter and removes effectively the noises. Another interesting feature of F-transform is that in contrary to the traditional methods, which result in discrete solutions in the grid points, it gives continuous and even differentiable solutions. The approximate solution is as smooth as the basic functions A k (x). It means that if we need a smooth approximation, then we have to utilize the smooth basic functions. The most important properties of the F-transforms technique can be summarized as follow: • the error bound depends only on the modulus of continuity of the solution; • the method is flexible in implementation; • it gives sufficiently smooth piecewise best approximation in small support; • it does not require any starting point or auxiliary function for starting. • since the support of basic functions is compact, the computational cost decreases. • it is as accurate as the most of existing numerical methods. • it can be generalized to the F m -transform-based method which is more accurate. • it can be applied to problems with non-smooth coefficients because of the non-local property of the transform.
The structure of this contribution is the following: a brief review of F-transforms is given in Sect. 2. The new technique is introduced in Sect. 3. The solvability of the corresponding algebraic system is investigated in Sect. 4. Illustrative examples are given in the final section.

Review of the fuzzy transforms
The method of fuzzy transforms is a well-known soft computing method applied to many practical problems. In this section, we recall some definitions and results from the literature that will be used throughout the paper.
Definition 1 Let [a, b] be an interval on R, n ≥ 2, and let t 1 , . . . , t n be fixed nodes within [a, b], such that a = t 1 < . . . < t n = b, t 0 = t 1 , t n+1 = t n . We say that the fuzzy sets A 1 , . . . , A n , identified with their membership functions A 1 (t), . . . , A n (t) defined on [a, b], form a fuzzy partition of [a, b] if they verify the following conditions for k = 1, . . . , n, and for all k = 1, . . . , n − 1 strictly decreases on [t k , t k+1 ].
The membership functions A 1 , . . . , A n are called basic functions.
Remark 1 A fuzzy partition A 1 , . . . , A n , n ≥ 2, is called h−uniform, if the nodes t 1 , . . . , t n are h−equidistant, i.e., t k = a + h(k − 1), k = 1, . . . , n, where h = b−a n−1 , and the following two additional properties are verified: 1. for all k = 2, . . . , n−1 and for all t The uniform fuzzy partitions constructed by the triangular and sinusoidal membership functions are the famous fuzzy partitions for a given interval [a, b] (see Perfilieva 2006).
Definition 2 (Direct F-transform). Let A 1 , . . . , A n be basic functions, which form a fuzzy partition of [a, b], and f be any continuous function on [a, b].
F n ( f )] of real numbers with the components given by is called the F-transform of f with respect to A 1 , . . . , A n , and is denoted by F( f ).
Definition 3 Let f : [a, b] → R be a given function, and let Then, for each k = 1, . . . , n, by applying the trapezoidal rule with the nodes t k−1 , t k , t k+1 to numeric integration of The following results hold true for the F-transform of f (see Perfilieva et al. 2011): (A) Let f be a given continuous function on [a, b]. Then, the kth component of the F-transform of f minimizes the function a, b], and for all α, β ∈ R, Definition 4 (F-transform of a vector-valued function). Let A 1 , ..., A n be basic functions, which form a fuzzy partition of [a, b], and g : Definition 5 Let m ∈ N and g : [a, b] → R m be a given vector-valued function, and is called the inverse F-transform of g.
We are now ready to give two theorems that play crucial roles in our discussion.
be the error of jth component of g. By Theorem 13.14 in Bede (2013) Theorem 2 (see Zeinali et al. 2018 . . , F n (g)] are the F-transforms of h s and g, respectively, then

The method of fuzzy transforms to FDAEs
In this section, we present a numerical method based on fuzzy transforms to the following initial value problem of FDAEs We also suppose that the consistency condition is verified. For the convenience of notations and without loss of generality, we concentrate on the case b = 1. Here, the operator D α 0 denotes the fractional operator of order α in the Caputo sense defined as is the Riemann-Liouville-type fractional integral operator of order α and (α) denotes the Gamma function (see Podlubny 1998;Diethelm 2010). We will use the following relations We are going to employ a new method based on the Ftransform to approximate the solution of (1). Since H (t) is invertible, without loss of generality we assume that Let n ∈ N, n ≥ 2 and A k , k = 1, . . . , n be a uniform fuzzy partition of interval [0,1] with the step size h = 1 n−1 , t 1 = 0, t n = 1. We first apply the operator I α to both sides of (4). Then, using (2) and linearity of I α , we obtain where K 1 (s) = −F(s)S(s), K 2 (s) = −F(s)G(s). By applying F-transform on both sides of (5) and Theorem 2, we deduce for k = 1, . . . , n and inserting for k = 2, . . . , n − 1, and we can rewrite the equations above in the following compact matrix form where I is the n × n identity block matrix with block I m , the m × m identity matrix; X i , B i , R i are vectors with m components, and M i j are m × m matrices with

E(s) + K 2 (s) P 1 (s)r n (s)ds,
By solving (10), the unknown vectorsX 1 , . . . ,X n are obtained. The solutions of this system,X 1 , . . . ,X n , are the approximate values of X 1 , . . . , X n , which are the components of F−transform of x(t). Thus, the approximate solution of Problem (1) is given by the inverse F-transform, i.e., (11)

Solvability and convergence
In this section, we first show that under some sufficient conditions, System (10) is solvable. Then, we investigate the convergence of the approximate solution to the exact solution of Problem (1). For solvability of System (10), by applying the geometric series theorem, it suffices to show that M < (α) (see Appendix Atkinson 2009). We prove this assertion and go into details regarding the discussion of case m = 1. The general case is proved exactly by the similar method, and the main difference is using a norm instead of the absolute value. For the block matrix M, the norm M is considered as max j k+1 j=1 M k j , where M k, j refers to any matrix norm.
Theorem 3 Let fuzzy sets A k , k = 1, . . . , n form a fuzzy partition to [0, 1] for n ∈ N, and M be the matrix given in System (10) Proof We have to compute the norm of the coefficient matrix M. To do this, we first notice that for k = 2, . . . , n with M n,n+1 = 0, we have For j ≥ 2 and −1 ≤ u ≤ 1, using the mean value theorem there exists −1 < δ < 1 such that From (12), (13) and substituting h = 1 n , we have By a similar calculation, we can prove that where C 1 is a constant independent from n. By (12) and (15), we conclude for k = 2, . . . , n. In a similar way, we can show that there exists a constant C 2 such that Thus, Finally, for sufficiently small C we conclude that C max{C 1 ,C 2 } (α) < 1 which implies that (I − 1 (α) M) is invertible.
The theorem above guarantees that System (10) has a unique solution. At this stage, we intend to prove the convergence of the proposed method for Problem (1).
Theorem 4 Assume the hypothesis of Theorem 3. Letx n (t) = n k=1X k A k (t), whereX k are the solutions of System (10) and x(t) be the exact solution of the initial value problem (1). Then, be the inverse fuzzy transform of x(t), t ∈ [0, 1] andx n (t) be as in (11). Then, From Theorem 1, we have On the other hand, where [X 1 , . . . , X n ] and [X 1 , . . . ,X n ] are the solutions of the systems respectively. Hence, We notice that (I − 1 (α) M) −1 exists. Thus, and consequently, where It is easy to show thatĆ is a constant independent of n. In a similar argument, we can obtain the same upper bound for R 1 and R n . Thus, (17) and (18) result Since lim n→∞ ω(x, 1 n ) = 0, by recalling Theorem 3, we deduce the assertion.
For n = 11(h = 0.1), n = 21(h = 0.02), n = 51(h = 0.01) and different values of α, we report the maximum values of absolute errors in Table 1. The plots of these errors are shown in Figs. 1 and 2 for α = √ 2 5 , n = 11 and n = 51. Example 2 Consider the fractional-order differential-alge braic equations where Q(t) = (α+3) 2 t 2 , S(t) = 0 and the exact solution is {x(t), y(t)} = t 2+α , −t 3 2 +α . By the same manner as we described in Example 1, we solve the system to find the approximationsX 1 , . . . ,X n to X 1 , . . . , X n . Then, we use the inverse F-transform to find the approximation to the exact solution x(t).

Conclusion
In this paper, we proposed a numerical method based on fuzzy transforms for solving the fractional-order linear semiexplicit differential-algebraic equations. We discussed the convergence analysis of the method and investigated the efficiency of the method by some illustrative examples. The implication of the method is fast and can be applied to Problem (1) with arbitrary α ∈ (0, 1)(rational or irrational one) and m ∈ N. However, some methods are restricted to only rational orders(see Ghanbari et al. 2018) with m = 1. The method can be generalized to the problems with higher order. Due to the fact that the fuzzy transforms is defined as an integral operator, the methods based on F-transforms are effective methods for problems with singular terms as we illustrated in Example 3.