Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott

A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transformation method to obtain approximate solutions for the time-fractional Gray-Scott model. This method gives a more realistic series of solutions that converge rapidly to the exact solution. We can ensure convergence by solving the series resultant. We study the convergence analysis of fractional homotopy analysis transformation method by determining the interval of convergence employing the 
 
 
 
 ℏ
 
 
 u
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 v
 
 
 
 -curves and the average residual error. We also test the accuracy and the efficiency of this method by comparing our results numerically with the exact solution. Moreover, the effect of the fractionally obtained derivatives on the reaction-diffusion is analyzed. The fractional homotopy analysis transformation method algorithm can be easily applied for singular and nonsingular fractional derivative with partial differential equations, where a few terms of series solution are good enough to give an accurate solution.


Introduction
Differential equations play a significant role within the field of finance, engineering, physics, and biology. Therefore, these applications can be modelled through differential equations [1,2]. Reaction-diffusion system (RDS) is known as a set of partial differential equations, which correspond to many physical phenomena. RDS can be applied in physics, biology, chemistry, epidemiology, etc. (see, for example, [3,4]).
An RDS can be represented by the Gray-Scott model (GSM). The classical (integer derivative) GSM has been studied by several numerical techniques [5,6]. Moreover, the existence and stability of the solution to this model in one dimension are discussed in [7]. In recent years, solutions to the fractional (noninteger) GSM have been spread at the same rate with the classical (integer derivative) GSM [8,9].
Three are many definitions of FC, such as the Riemann-Liouville and the Liouville-Caputo [22]. Recently, Caputo and Fabrizio (CF) proposed a new concept of fractional differentiation using the exponential decay as the kernel instead of the power law [23,24]. Thereafter, Atangana and Baleanu (AB) developed a new concept of differentiation with nonsingular [25,26], based on the general Mittag-Leffler function. These two concepts with fractional order in Riemann-Liouville and Liouville-Caputo sense have a nonlocal kernel.
Despite the difficulty of finding exact solutions in FC's case, the numerical and approximate technique to obtain approximate solutions is needed. Several methods have been applied for solving fractional differential equations, such as the fractional natural decomposition method [27,28], q -homotopy analysis transform method [28][29][30], and Adams Bashforth and the Fourier spectral methods [31]. Khan et al. [32] and Kumar et al. [33,34] coupled the homotopy analysis method (HAM) [35][36][37] with the Laplace transform to solve a nonlinear differential equation. This method is called the fractional homotopy analysis transform method (FHATM). The main advantage of this method is its ability to combine two powerful methods to obtain a rapidly convergent series for fractional differential equations. The FHATM provides us with a convenient way to control the convergence of the series solution.
In this paper, RDS can be represented by GSM. In order to find an approximate solution to the proposed model, the FHATM is applied. To the best of our knowledge, this paper is the first one that introduced the approximate analytic solution for the time-fractional Gray-Scott system using a nonsingular fractional derivative.

Preliminaries and Notations
2.1. The Model. We consider the reaction-diffusion system for the cubic autocatalysis. This system contains two chemical species U and V , whose concentration is referred by variables u and v, respectively. Cubic autocatalysis is given by two reactions, which occur at a different rate: where P is some inert product of reaction. Following [38], when quantity depends on one spatial coordinate ðξÞ, the GSM in one space dimension is equivalent to the following two equations: The left-hand side of the above equations represents the change in concentration of U (upper equation) and the concentration of V (lower equation) over time. Moreover, Δu and Δv represent the Laplacian operator on 1-D. The second term in both equations (the concentration of U times the square of the concentration of V ) represents the reaction term. As shown by the minus uv 2 in u (upper equation) and the positive uv 2 in v (lower equation), the decrease in u equals the increase in v. This term shows that U is converted to V . As a result, this amount uv 2 is subtracted from the first equation and added to the second equation. The third term in the upper equation represents the replenishment term, while the third term in the lower equation represents the diminished term. The chemical U is added to a given rate (+A, scaled by (1 − u), so u does not exceed 1). On the other hand, the chemical V is removed to a given removal rate (−B), scaled by the concentration of V , so v does not go below zero. As a result, 1 would be the maximum value for u, and 0 would be the minimum value of v. In the context of this model, A ≤ B. The Gray-Scot model's parameters and functions (2) and (3) are given in Table 1.
In this study, we extend the classical GS model to the following time-fractional Gray-Scott model (TFGSM) of the orders δ and η. Let uðξ, ρÞ = u and vðξ, ρÞ = v; then with initial conditions  [22], the Caputo-Fabrizio fractional derivative [23], and the Atangana-Baleanu fractional derivative [26] in the Caputo sense are defined, respectively, as where ρ > 0 and FðγÞ > 0 is a normalization function satisfying where Fð0Þ = Fð1Þ = 1 and E γ ð:Þ denotes the Mittag-Leffler function, defined by The Liouville-Caputo fractional integral [22], the Caputo-Fabrizio fractional integral [39], and the Atangana-Baleanu fractional integral [40] in the Caputo sense are defined, respectively, as follows: Here, when γ equals zero, the initial function is recovered, and when γ equals unity, the classical ordinary integral is obtained.
The Laplace transformation of the Liouville-Caputo fractional derivative [22], the Caputo-Fabrizio fractional derivative [23], and the Atangana-Baleanu fractional derivative [26] in the Caputo sense are given, respectively, as follows: 2.3. Homotopy Series. The following properties can be found in [41]. Let φ 1 and φ 2 be a homotopy series of a homotopy parameter q given by Then, the nth-order homotopy derivative is given as which holds the following:

Homotopy and Laplace Transform for FHATM
Applying the Laplace transformation on Equations (4) and (5), using the Laplace transformation formula of LC, CFC, and ABC, and then simplifying these equations, we obtain where Y 1,δ ð:Þ and Y 1,η ð:Þ are defined in Table 2. It is difficult to evaluate the Laplace transformation of unknown solutions u and v specifically when combined in a nonlinear form.
We define the homotopy maps as follows:

Journal of Function Spaces
Journal of Function Spaces where The rest of the parameters and functions are defined in Table 3.
By requiring the left-hand side of Equations (16) and (17) to be zero, we construct the so-called zeroth-order deformation equation subject to the initial conditionŝ There are three cases of solutions depending on the parameter q ∈ ½0, 1: (a) If q = 0 (we are on the linear operator), wherê (b) If q = 1 (we are on the nonlinear operator), wherê (c) If q varies from zero to one, the solution of the Equations (4) and (5) vary from the initial guesses u 0 ðξ, ρÞ and v 0 ðξ, ρÞ to the exact solutions uðξ, ρÞ and vðξ, ρÞ.
Expandingûðξ, ρ ; qÞ andvðξ, ρ ; qÞ by the Taylor series with respect to the embedding parameter q, we obtain where If u 0 ðξ, ρÞ, v 0 ðξ, ρÞ, the auxiliary parameter ℏ u,v , and the auxiliary linear operator L are properly chosen, then according to [36], the series (25) and (26) converges at q = 1, and we haveû which must be one of the solutions of Equations (4) and (5). Let us define the vectors that deduce the mth-order deformation equations from the zeroth-deformation Equations (20) and (21), given as follows:  (16) and (17).

Meaning Condition q
The embedding parameter The auxiliary linear operator The nonlinear operator See Equations (18) and (19) 5

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Upon differentiating the zeroth-deformation in Equations (20) and (21) m times with respect to the embedding parameter q, setting q = 0, and finally dividing them by m!, we have the so-called mth-order deformation equations as follows: Applying the inverse Laplace transform to Equations (31) and (32), we obtain Here, Consequently, the solutions of the mth-order deformation equation are given as Consider the initial guesses u ð:Þ 0 ðξ, ρÞ = uðξ, 0Þ and u ð:Þ 0 ðξ, ρÞ = vðξ, 0Þ; then using Equations (35) and (36), the first two terms are given as where Y s ð:Þ, s = 2, 3, 4 is defined in where the superscript (.) is replaced by (LC), (CFC), and (ABC). 6.×10 -9

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8.×10 -9 Figure 1: The ℏ u,v curves obtained from the 3rd order of the FHATM solutions using the ABC, CFC, and LC when δ and η tend to 1 and ξ = 10:         Figure 2 and Table 5 show the average residual error for the LC, CFC, and ABC operators. These show SE n,u ðℏ u Þ and SE n,v ðℏ v Þ for 3 terms obtained using FHATM. We set into Equations (45) and (46) the parameter values given in Table 4. Using the command "Minimize" in Mathematica, we plotted the residual error against ℏ u,v to get the optimal values ℏ * u,v . From Table 5, it is seen that the FHATM for LC, CFC, and ABC operators converges rapidly. Note that only three iterations are considered here. Therefore, the accuracy of the results can be improved by considering more terms, where the error converges to zero. Figure 3 and Table 6 show the comparison of 3 terms in the FHATM solution for the LC, CFC, and ABC operators with the exact solution in Equations (43) and (44). Table 6 presents the absolute error of the FHATM solution using  Journal of Function Spaces parameter values given in Table 4. We noted from this table that the FHATM solution for LC, CFC, and ABC operators is in excellent agreement with the exact solutions. Moreover, Figure 3 shows the comparison between the exact solution and approximate solution obtained by 3 terms of FHATM for the LC, CFC, and ABC operators for parameter values listed in Table 4. We observe from Figure 3 that the solution obtained by FHATM increases rapidly to the exact solution following the increase in δ and η. Those tables and figures demonstrate the efficacy of the presented algorithm for solving the time-fractional Gray-Scott equation.

Conclusion
In this paper, the Gray-Scott equation was extended to the time-fractional Gray-Scott equation of Liouville-Caputo (LC), Caputo-Fabrizio-Caputo (CFC), and Atangana-Baleanu-Caputo (ABC) type. The fractional homotopy analysis transform technique is used to derive analytic solutions for TFGSE. This method gives the solutions in a series form that converges rapidly in nonlinear time-fractional GS equation. The interval of the convergence by ℏ u,v curves in Figure 1 and the optimal value of ℏ u,v were found by least square error as given Figure 2. Also, the solutions obtained were compared with the exact solution, which were in excellent agreement. The effect of the fractional derivative on the concentration of U increases when δ decreases while the concentration of V is decreases. Moreover, the results obtained using FHATM agree well with the numerical result presented in [5], and the absolute error less than 3 × 10 −6 as given in Table 6. In conclusion, the FHATM method is a powerful method to handle fractional operators of LC, CFC, and ABC type, generating highly accurate data.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no competing interests.