{'fraction_non_alphanumeric': 0.042586175451395224, 'fraction_numerical': 0.013952215940178734, 'mean_word_length': 4.391838741396263, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
New numerical scheme of Atangana-Baleanu fractional integral: an application to groundwater flow within leaky aquifer

Many physical problems can be described using the integral equations known as Volterra equations. There exists quite a number of analytical methods that can handle these equations for linear cases. For non-linear cases, a numerical scheme is needed to obtain an approximation. Recently a new concept of fractional differentiation was introduced using the Mittag-Leffler function as kernel, and the associate fractional integral was also presented. Up to this point there is no numerical approximation of this new integral in the literature. Therefore, to accommodate researchers working in the field of numerical analysis, we propose in this paper a new numerical scheme for the new fractional integral. To test the accuracy of the new numerical scheme, we first revisit the groundwater model within a leaky aquifer by reverting the time classical derivative with the Atangana-Baleanu fractional derivative in Caputo sense. The new model is solved numerically by using this new scheme.


Introduction
The concept of fractional differentiation with non-singular and non-local kernel has been suggested recently and is becoming a hot topic in the field of fractional calculus. The concept was tested in many fields including chaotic behaviour, epidemiology, thermal science, hydrology and mechanical engineering [1,2,3]. The numerical approximation of this differentiation was also proposed in [3]. In the recent decade, the integral equations were revealed to be great mathematical tools to model many real world problems in several fields of science, Technology and Engineering. In many research papers under some conditions (see e.g. [5,6,7] and references therein), it was proven that there is equivalence between a given differential equation and its integral equation associate. the subsurface water flowing within a confined aquifer model with the new derivative with fractional order in time component. This paper is organized as follows: In Section 2, for the convenience of the reader, we recall some definitions and properties of fractional Calculus within the scope of Atangana-Baleanu. In Section 3 a numerical approach of Atangana-Baleanu derivative with fractional order is introduced. In Section 4, as application of the new numerical approximation of Atangana-Baleanu fractional integral. We study analytically and numerically the model of groundwater flow within a leaky aquifer based upon the Mittag-Leffler function. Finally, Section 5 is dedicated to our perspectives and conclusions.

Somes definitions of the Atangana-Baleanu fractional derivative and integral
We recall the definitions of the new derivative with non singular kernel and integral introduced by Atangana and Baleanu in the senses of Caputo and Riemann-Liouville derivatives [1,4].
Let (a, b) ⊂ R and let u be a function of the Hilbert space L 2 (a, b). We define by u ′ the derivative of u as distribution on (a, b).
Definition 2.2. Let α ∈ (0, 1) and a function u ∈ H 1 (a, b), b > a. The Atangana-Baleanu fractional derivative in Caputo sense of order α of u with a based point a is defined as where B(α) has the same properties as in Caputo and Fabrizio case, and is defined as is the Mittag-Leffler function, defined in terms of a series as the following entire function , α > 0, λ < ∞, and β > 0, The Mittag-Leffler functions appear in the solution of linear and nonlinear fractional differential equations [8]. The above definition is very helpful to discuss real world problems and will also have a great advantage when using the Laplace transform with initial condition. Now let us recall the definition of the Atangana-Baleanu fractional derivative in the Riemann-Liouville sense. Definition 2.3. Let α ∈ (0, 1) and a function u ∈ H 1 (a, b), b > a. The Atangana-Baleanu fractional derivative in the Riemann-Liouville sense of order α of u is defined as Notice that, when the function u is constant, we get zero.
Definition 2.4. The Atangana-Baleanu fractional integral of order α with base point a is defined as

Numerical approach of the Atangana-Baleanu integral with fractional order
In this section, we will start by given the discretization for Riemmann-Liouville fractional integral [7,9]. Next, following the same idea as [6,11], we introduce a numerical scheme to discretize the temporal fractional integral, and give the corresponding error analysis which will be used to give the numerical solution of the Atangana-Baleanu fractional integral and also to obtain the solution of the modified groundwater flow within a leaky aquifer.
Let f ∈ C 2 (a, b). Then b a f (x)dx could be discretized as follows as On the other hand With the Riemann-Liouville fractional integral we have the following.
Using the same approach as in [10], the error is given as From the Taylor series at the point ξ we can approximate

Now taking the norm, we get
since the function f is differentiable.Hence Using the second approach of the discretization (3.2), we obtain the following approximation (3.9) If the function f is differentiable such that its Atangana-Baleanu fractional integral exists then Then for k = 0, 1, 2, 3, . . . , n the Atangana-Baleanu integral can be decomposed as follows Let us now assume that f is two times differentiable on [0, T ]. Then, we obtain With the above relation in hand, we can conclude that Here we have Therefore, we have the following relationship (3.12) Finally the equation above can be reformulated as  = (a, b) be an open and bounded subset of R n (n ≥ 1), with boundary ∂Ω. For a given α ∈ (0, 1), and a function ϕ(r, t) ∈ H 1 (Ω) × [0, T ], which represents the head, we seek ϕ such that the flow of water within the leaky aquifer is governed by where ̟ = KDc and β 2 = Sc̟ −2 , S denotes the coefficient of storage, K denotes the conductivity.
The problem (4.1) of groundwater flow within the leaky aquifer will be approached analytically and solved numerically using the implicit scheme.

4.2.
Existence and uniqueness of the solution of the problem. In the following, we discuss the existence and uniqueness of solutions of the direct problem.
for the initial datum ϕ c ∈ L ∞ (Ω), ϕ c > 0, and where Given ϕ c ∈ L ∞ (Ω), ϕ c > 0, a solution of (4.2) is a positive function ϕ ∈ H 1 (Ω) ×[0, T ] such that if we apply the fractional integral defined in (2.4) to (4.2) it yields We want to use the contraction mapping theorem, so for this purpose we need to build a closed set ε of H 1 (Ω) × [0, T ] such that the nonlinear operator g be a contraction which maps ε into itself.
So we first show that g is a contraction mapping.
We have by the triangular inequality As the derivative operator satisfy the Lipschitz conditions in H 1 , hence there exist two positive constants c 1 and c 2 such that Then g is a contraction mapping. Proof. We shall follow the idea of [12]. Since the nonlinear operator g is locally Lipschitz, forφ c = ϕ c H 1 there exists Cφ c such that 0 < Cφ c < ∞ and endowed with the norm ϕ ε = sup Now we consider the following associated problem of (4.4) defined on ε, For all ϕ ∈ ε and t ≥ 0 we have ϕ(s, r)) H 1 ds.
But since g is locally Lipschitz for all s ∈ [0, T 1 ], it follows that This shows that ψ is a contraction mapping in ε. Thus ψ has a fixed point which is a solution to (4.2). Now let us show that the problem (4.2) has an unique solution.

4.3.
Exact solution of the flow within the leaky aquifer based upon Atangana-Baleanu fractional derivative. In the model here discussed for water flow in the leaky aquifer, the head ϕ(r, t), which appears in (4.1), is assumed to be governed by the one-dimensional time-fractional differential equation involving the Atangana-Baleanu fractional derivative.
Applying Laplace transform to (4.1), the fundamental solutionφ(r, p), p ∈ N results to be: where L t :=φ denotes the Laplace transform. Replacing each term by its value, we get Hence the following differential equation in the form holds (4.6) Since q is positive, the exact solution of the differential equation (4.6) is given in terms of Bessel function of the first kind, J 0 and modified kind, K 0 as where A and B are the constants and J 0 , K 0 respectively given as Using the boundary condition in (4.7), we obtain B = 0, then the solution is reduced to Due to the difficulties to obtain the inverse Laplace transform of (4.8), we therefore propose to obtain the approximate solution of (4.2) by using the proposed numerical approximation of the Atangana-Baleanu integral.

4.4.
Numerical analysis of the ground water flow within the leaky aquifer based upon Atangana-Baleanu fractional derivative. To achieve this, we revert the fractional differential equation to the fractional integral equation using the link between the Atangana-Baleanu derivative and the Atangana-Baleanu integral to obtain (4.9) ϕ(r, t) − ϕ(r c , 0) = AB 0 I α t g r, t, ϕ(r, t) . Using the implicit finite differences method, a discrete approximation of g r, t, ϕ(r, t) given by (4.3) can be obtained as follows In order to use the numerical approximation proposed in this work, discrete solution of (4.2) is then given as follows Therefore the numerical approximation can be given as follows In order obtain the plots of the solution given by (4.13), we shall consider 350 equidistant nodes in [0, 1]. For the value α = 1/2, the approximate solution found by using the numerical method of the Atangana-Baleanu integral proposed is obtained. Figure 1 shows the approximate solution for the different time step k = 0, k = 20 and k = 50 respectively. . Approximate solution of (4.13) for α = 1/2, k = 0 -blueand k = 50-orange-(graph in the left). Approximate solution of (4.13) for α = 9/10, k = 0 -blue-and k = 50-orange-(graph in the right).

Conclusions
The new scheme of the fractional integral in the sense Atangana-Baleanu has been proposed in this work. The error analysis of the novel scheme was successfully presented and the error obtained shows that the scheme is highly accurate. A new model of groundwater flowing within a leaky aquifer was suggested using the concept of fractional differentiation based on the generalised Mittag-Leffler function in order to fully introduce into mathematical formulation the complexities of the physical problem as the flow taking place in a very heterogeneous medium. The Mittag-Leffler operator provide more natural observed fact than the more used power law. The new model was analysed, as the uniqueness and the existence of the solution was investigated with care. To further access the accuracy of the proposed numerical scheme, we solved the new model numerically using this suggested scheme. Some simulations have been presented for different values of fractional order. We strongly believe that this numerical scheme will be applied in many fields of science, technology and engineering for those problems based on the new fractional calculus.