Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations

In this study, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method. For this purpose, we derive a V-O fractional integration operational matrix (OM) for CWs and use it in our method. In the established scheme, we approximate the unknown functions by CWs with unknown coefficients and reduce the problem to an algebraic system. In this way, we simplify the computation of nonlinear terms by obtaining some new results for CWs. Finally, we demonstrate the applicability of the presented algorithm by solving a few numerical examples.


Introduction
Fractional calculus is a useful extension of the classical calculus by allowing derivatives and integrals of arbitrary orders. It arose from a famous scientific discussion between Leibniz and L'Hopital in 1695 and was developed by other scientists like Laplace, Abel, Euler, Riemann, and Liouville [1]. In recent years, fractional calculus has become a popular topic for researchers in mathematics, physics, and engineering because the fractional differential (integral) equations govern the behavior of many physical systems with more precision [2]. We remind that the main advantage of using fractional differential (integral) equations for modeling applied problems is their nonlocal property [3], i.e., in a fractional dynamical system, the next state depends on all the previous situations so far [3].
Another interesting extension to fractional order calculus is considering the fractional order to be a known time-dependent function α(t) [4]. This generalization is called variable-order (V-O) fractional calculus. This subject finds enormous applications in science and engineering because the nonlocal property of fractional calculus becomes more evident [4]. Usually, the V-O fractional functional equations are difficult to solve, analytically. So, finding the exact solutions for these problems is impossible in most cases. Therefore, it is very important to propose approximation/numerical procedures to find numerical solutions for these problems. Some recent studies and numerical methods for V-O fractional functional equations can be observed in [5][6][7][8][9][10].
During last decades, orthogonal functions have been applied extensively for solving different classes of problems. The major reason for this is that solving the main problem turns into solving a simple algebraic system [11][12][13]. We remind that the Chebyshev polynomials can be effectively utilized for approximating any sufficiently differentiable function. In this case, the approximation error is rapidly converted to zero [14]. This property usually is named "the spectral accuracy".
Wavelets, compared to other functions, favor many advantages which allow the investigation of problems which the conventional numerical methods cannot handle [14]. During last years, different categories of fractional problems have been solved by eliciting the OM of classical fractional integration for well-known orthogonal wavelets, e.g., [15][16][17]. CWs are a particular type of orthonormal wavelets which satisfy orthogonality and spectrality, which are the properties of the Chebyshev polynomials in addition to the properties of wavelets. These useful features have led to the widespread use of CWs in solving fractional differential equations (FDEs). In [18], a category of fractional systems of singular integrodifferential equations was solved using CWs. Multi-order FDEs were solved in [19] by proposing a CWs-based numerical method. Also, nonlinear fractional integro-differential equations in large intervals were solved by Heydari et al. in [20] by CWs. A class of nonlinear FDEs was solved by applying CWs in [21]. The generalized Burgers-Huxley equation was solved by applying a CWs-based collocation method in [22]. In [23], CWs were utilized for solving FDEs with nonsingular kernel.
The major aim of the current study is to treat the following nonlinear V-O fractional system by proposing the CWs Galerkin method:  1) is a generalization of the classical fractional system introduced in [36]. So, the method of this paper can also be used for the fractional system investigated in [36].
To implement the proposed method, a new OM for CWs is elicited as follows: Eventually, we draw a conclusion in the last section.

Mathematical preliminaries
Here, some preliminaries which are necessary for this study are reviewed.

V-O fractional integral
In the development of the fractional calculus theory with V-O, many definitions have appeared. In this section, we give the most popular definition of V-O fractional integral.

Lemma 2.3 ([39]) If X is an arbitrarym-column vector and (t) is the HFs vector in
. . , d, be arbitrary constant vectors, and F : R d+1 → R be any continuous function. Then we define

Lemma 2.7 If V T (t) and U T i (t) are the approximations of v(t) = t and u i (t) by HFs, then
for any continuous function F : R d+1 → R.

Chebyshev wavelets
Herein, CWs and some of their properties, which will be used in the sequel, are reviewed.

CWs and function expansion
CWs are defined over [0, 1] as follows [18]: Here, T m (t) denotes the Chebyshev polynomials which are recursively defined over [-1, 1] as follows [41]: Let w n (t) be a weight function defined as The CWs can be utilized to expand any function u(t) on [0, 1] as where c nm = u(t), ψ nm (t) w n (t) . This function can be approximated as where the symbol T denotes transposition and (t) and C arem = 2 k M column vectors. Relation (3.5) can be simplified as follows: where ψ i (t) = ψ nm (t), c i = c nm , and i = Mn + m + 1. This results in Likely, the CWs can be used to expand any two-variable functions u(x, t) ∈ L 2 w n,n

Novel results for CWs
Here, some technical results are extracted for CWs which are applicable in the following.

Lemma 3.1 Assume (t) and (t) are respectively the CWs and the HFs vectors in (3.7)
and (2.8). Then we have where Q is the CWs matrix of sizem ×m with Proof The function ψ i (t) (ith component of (t)) can be approximated via HFs as follows: in which Q i is the ith row of the matrix Q. Then the proof is concluded.

Corollary 3.2 If X is anm-column vector and (t) is the CWs vector in
where X = Qdiag(Q T X)Q -1 , and the matrix X is the product OM for the CWs of sizem ×m.
Proof The proof is easy by applying Lemmas 2.3 and 3.1.

Corollary 3.3 If (t) is the CWs vector in (3.7) and A is anm-order matrix, then
Proof The proof is immediate from Lemmas 2.4 and 3.1.

Corollary 3.4
Assume that V T (t) and U T i (t) are the approximations of v(t) = t and u i (t) via CWs, respectively, for i = 1, 2, . . . , d. Then we have for any continuous function F : Proof The proof is clear by Lemmas 2.7 and 3.1.
where Q andP (α) are respectively defined in (2.14) and (3.9), and P (α) is the V-O fractional integration OM of CWs.
Proof By Theorem 2.5 and Lemma 3.1, we get Hence, by applying Lemma 3.1, the desired result is obtained.

Existence and uniqueness
The uniqueness and the existence of a solution for fractional system (1.1) is studied in this section. Since norms in R d are equivalent, we use the sup-norm · which, for any with the above norm constitutes a Banach space. It is also worth mentioning that we have

Definition 4.1 (Lipschitz continuous function [42]) Let
These functions are Lipschitz continuous if there exist ρ, σ ∈ R + such that (4.4) Remark 2 Fractional system (1.1) can be rewritten as AU = U, where the operator A : and t ∈ [0, 1] is defined as follows: Since ρ + σ K 1 α(t) < 1, using the contraction mapping principle, the desired result is obtained.
Remark 3 The Lipschitz hypotheses are spontaneously satisfied when the vectors F and K adopt as follows: Here, F ij and G ij : [0, 1] → R are continuous maps. Hence, the system of nonlinear integral equations (1.1) can be defined as and Theorem 4.2 admits a unique solution.

The established wavelet method
Herein, a new computational method using CWs is established for solving the VO fractional system (1.1). We utilize the results yielded in Sect. 4 for transforming fractional system (1.1) to an algebraic system by expressing the functions u i (t),K i (t, s) for i = 1, 2, . . . , d and v(t) = t via CWs as follows: and where U i are unknown vectors, K i are coefficient matrices for kernels K i for i = 1, 2, . . . , d, and V is the coefficient vector for v(t). By substituting (5.1) and (5.2) into (1.1), we have where G i arem ×m matrices given as for i = 1, 2, . . . , d. By substituting (5.5) into (5.4) and employing the fractional integration matrix for CWs, we get By Corollary 3.3, we have where K T i = B T i Q -1 and B i arem-column vectors given as for i = 1, 2, . . . , d. So, the residual functions R i (t) for fractional system (1.1) can be written as follows: Similar to the typical Galerkin method [41],md nonlinear algebraic equations are generated as follows: where the index j is computed as j = Mn + m + 1 and ψ j (t) = ψ nm (t).
Remark 4 Note that a set ofmd nonlinear algebraic equations is generated by Eq. (5.11) as follows: Finally, we get a wavelet solution for the original system (1.1) using (5.1) by solving the above system for the vectors U i , i = 1, 2, . . . , d.
Remark 5 After simplification we can compute the vectors B i in (5.9) as follows:

Analysis of convergence
The convergence analysis of CWs is surveyed in this section.
Remark 6 The weighted inner product in the Sobolev space Hm w (a, b) is given as   (6.5) form ≤ N + 1.
Remark 9 The Sobolev space Hm w (a, b) constructs a hierarchy of Hilbert spaces, because (a, b).
Then we have Proof It is the case that By considering t = b-a 2 x + b+a 2 and dt = b-a 2 dx as change of variables, we get By (6.6), we obtain v -P N v L 2 w (-1,1) ≤CN -m |v| Hm ;N w (-1,1) , (6.12) where By considering t = b-a 2 x + b+a 2 and b-a 2 dx = dt as change of variables, we obtain v (j) 2 (6.14) Hence, (6.4) and (6.14) yield and (6.10)-(6.15) yield The following equality holds for the infinity norm: By considering t = b-a 2 x + b+a 2 as change of variables, we obtain Considering (6.7) we have Finally, (6.16)-(6.21) result in which completes the proof.

Conclusion
In this paper, we established a Chebyshev wavelets (CWs) Galerkin technique for a class of systems of variable-order (V-O) fractional integral equations. First, we derived a fractional integral operational matrix (OM) for CWs which was then employed to find approximate solutions for the problem. Also, the hat functions were reviewed and used to elicit a method for forming this novel matrix. We also obtained the expansion of the unknown functions in terms of CWs with undetermined coefficients. Then, by employing some properties of CWs and their fractional integration OM, we reduced the fractional system to an algebraic system. Also, the existence of a unique solution for the system under consideration was proved. Furthermore, the reliability of the presented scheme is studied on some numerical examples which show the accuracy of the established technique.