Fractional nonlinear Volterra–Fredholm integral equations involving Atangana–Baleanu fractional derivative: framelet applications

In this work, we propose a framelet method based on B-spline functions for solving nonlinear Volterra–Fredholm integro-differential equations and by involving Atangana–Baleanu fractional derivative, which can provide a reliable numerical approximation. The framelet systems are generated using the set of B-splines with high vanishing moments. We provide some numerical and graphical evidences to show the efficiency of the proposed method. The obtained numerical results of the proposed method compared with those obtained from CAS wavelets show a great agreement with the exact solution. We confirm that the method achieves accurate, efficient, and robust measurement.


Introduction
Recently, many scientists have applied fractional derivatives with different types of definitions, such as Atangana-Baleanu fractional integral [1], Caputo fractional derivative [2], and Caputo-Fabrizio fractional derivative [3], to many real-world problems and pointed out the powerfulness of using such noninteger-order and nonlocal kernels to numerically solve different types of integral equations and to describe the dynamics and properties of these problems; see, for example, .
One of these problems is studying numerical solution of the nonlinear Volterra-Fredholm integral equations by involving the well-known Atangana-Baleanu fractional derivative. Note that nonlinear Volterra-Fredholm integral equations appear in many applications in different disciplines such as neural networks [38], the pulses of sound reflections [39], and mathematical physics such as Lane-Emden-type equations [40], and many more can be found, for example, in [41] and references therein. On the other hand, finding exact solutions of such equations is usually difficult and sometimes even impossi-ble. Therefore developing numerical algorithms to simulate exact solutions with this new involvement of noninteger order is crucial.
There are many numerical methods for solving Fredholm and Voltera integral equations, such as the Galerkin method, collocation method, Taylor series, Legendre wavelets, Taylor and recently Chebyshev polynomials, homotopy perturbation method, power series method, and expansion method [42][43][44][45][46][47][48]. Besides these contributions, other authors have also applied wavelet bases and gained much attention during the last decade. However, these bases are typically nonredundant, and thus corruption or loss of transform coefficients can be serious. Motivated by the above, our main goal in this work is to propose a new efficient method based on framelet systems to numerically solve fractional nonlinear Volterra and Fredholm integral equations by involving the Atangana-Baleanu fractional order derivative. Framelet theory is a relatively emerging area in mathematical analysis and known as redundant systems. The redundancy of the framelet system requires that a given function would be represented by a different structure as a convergent sum. These expansions have recently emerged as another effective tool and popular through the use in numerous applications. One of the major advantages of a redundant system is that it is implemented by a frame fast transform, which provides us with better recovery and higher accuracy order. Indeed, one of our main contributions in this work is simulation of the solution of a given fractional nonlinear Volterra and Fredholm integral equation based on these framelet expansions (redundant setting). This means that the right representation is critical if we intend to effectively perform our solution.
A framelet system contains a set of functions called generators. We construct them using some known and effective principles such as the unitary and oblique extension principles (of course, including their generalizations) and based on nonnegative functions called Bsplines. This provides us with simple and better reconstruction of the coefficients to obtain the corresponding unknown elements of the space L 2 (R) and also gives us better accuracy order and relatively small errors. In practice, framelet-based methods have been applied to provide accurate and efficient numerical schemes for solving several types of integral and differential equations; see, for example, [49][50][51][52][53][54][55][56][57][58][59].
We consider the following form of fractional nonlinear Volterra-Fredholm integral equation (FV-FIE): with initial conditions (ICs) u (p) (0) = d p , p = 0, 1, 2, . . . , m -1, and λ ∈ (n, n + 1], n ∈ N, (1.2) where D λ u is the Atangana-Baleanu fractional-order derivative given by where M(λ) is a normalization function satisfying M(0) = M(1) = 1, and E λ is the Mittag-Leffler function. The integral operator corresponding to this definition is given by We refer the reader to [1,13,60,61] for more details and properties of the fractional derivative. The paper is organized as follows. In Sect. 2, we provide some preliminaries and basics of frames with necessary theory needed for the construction of the framelet systems. In Sect. 3, we establish a matrix formulation of the proposed method based on the constructed framelet systems and using the collocation technique. Numerical examples with numerical comparison and graphical illustration are presented in Sect. 4 to validate our main expansion technique.

Framelet expansion method
The purpose in this section is providing an approximate solution of the FV-FIE given in Equations (1.1)-(1.2) in the form of truncated expansions of a given framelet system.
A set of functions is called a frame for L 2 (R) if there exists positive numbers A, B such that for all functions v ∈ L 2 (R). The set is called tight frame (or framelet) if it is possible to have A = B. Note that, according to inequality (2.1), for a function g ∈ L 2 (R), we obviously obtain the following associated framelet representation: The framelets are constructed using B-spline functions. The B-splines B M of order M, where M ∈ N, are recursively defined by the equation where B 1 (x) is the indicator function over the interval [0, 1).
The Fourier transform of an integrable function f , denoted byf , is defined bŷ and the discrete Fourier series of a sequence a(k), denoted byâ, is given bŷ B-splines are nonnegative refinable functions in the sense that where p(ω) is a polynomial of trigonometric functions with p(0) = 1, andâ is a 2π -periodic function in the frequency domain, called the low mask of B M .
The framelet system X( ) is constructed via the oblique extension principle (OEP) [49] and it has the form and satisfies the following equations: whereâ 0 ,â , = 1, . . . , r, are the low and high masks of u = B M , respectively. The OEP deals with the notion of constructing a framelet system using a refinable function φ where for some trigonometric functionˆ (ω), 1}. Then the system X( ) defined in Equation (2.4) forms a framelet system for L 2 (R). The representation in Equation (2.2) is truncated by the series W n such that where c ,j,k = g, u ,j,k . Let us present some examples of framelet systems via the OEP setting.
Therefore the set X( 2 ) where 2 = {ψ 1 , ψ 2 } forms a framelet system for ∈ L 2 (R). The graph of the generators of the corresponding framelet system is plotted in Fig. 1 (the right subfigure).

Matrix formulation using framelets
In this section, we provide a general framework of the proposed algorithm based on the collocation division of the domain.
where I λ is the Riemann-Liouville fractional-integral operator defined by Equation (1.3). With a few algebra, Equation (3.1) can be simplified to Hence, Now, based on a dyadic division points of the domain of the framelet system used, say {ξ q , q ∈ }, plugging these points into this equation, we have (ξ qt) 1-λ ds = g(ξ q ).
By approximating the integrals in this equation based on the composite trapezoidal rule, we get a generated system of equations, which can be solved to obtain the unknown coefficients c ,j,k , as in Equation (2.6), in order to have an approximated solution of order m.

Numerical applications
In this section, we consider some examples to test the proposed algorithm. The absolute errors are given by For comparison, we provide some numerical results for Example 4.1 based on Cos and Sin (CAS) wavelets defined by and based on the collocation method, where the translation parameter is given by The truncated expansion using such bases is given by      Table 2 Numerical results of Example 4.1 using the framelet systems X( 1 ) and X( 2 ) for m = 3 and  The exact solution is In Figs. 6 and 7, we present some graphical illustrations to compare the exact and approximate solutions and error bounds.
The exact solution is u(x) = e xx -1. The graphical illustrations are given in Figs. 8 and 9.  The exact solution for this equation when λ = 6/5 is u(x) = x 2 . The numerical results of this example are presented in Table 4, and the graphical illustrations of the exact, approximate, and error results are depicted in Figs. 10 and 11.

Conclusion
In this work, we presented an efficient method based on framelet systems to solve nonlinear Volterra-Fredholm integral and integro-differential equations by involving the Atangana-Baleanu fractional derivative. We used some constructed framelet systems generated using the B-spline functions with high vanishing moments to numerically solve the related equations. We converted the considered problem given in Equation (1.1) to a matrix system after employing the Atangana-Baleanu fractional derivative definition, Figure 11 Error graphs of Example 4.4 for m = 3 and λ = 1.2 using framelet systems X( 1 ) and X( 2 ), respectively where the resulted integrals are approximated using the composite trapezoidal rule. The obtained results indicate that the method produces high accuracy order and reliable results with only a few terms of the truncated framelet partial sums with a proper discretization. We have also supported our results by some graphical illustrations of the exact and approximate solutions and provided error bounds of the solved examples.