On Convergence Analysis and Analytical Solutions of the Conformable Fractional Fitzhugh–Nagumo Model Using the Conformable Sumudu Decomposition Method

: The current article studied a nonlinear transmission of the nerve impulse model, the Fitzhugh–Nagumo (FN) model, in the conformable fractional form with an efﬁcient analytical approach based on a combination of conformable Sumudu transform and the Adomian decomposition method. Convergence analysis and error analysis were also carried out based on the Banach ﬁxed point theory. We also provided some examples to support our results. The results obtained revealed that the presented approach is very fantastic, effective, reliable, and is an easy method to handle speciﬁc problems in various ﬁelds of applied sciences and engineering. The Mathematica software carried out all the computations and graphics in this paper.


Introduction
Numerous nonlinear fractional models have paramount importance in applied science and engineering like fluid mechanics, geophysical fluid mechanics, fluid mechanics, thermodynamic, plasma physics, relaxation vibrations, heat transfer, and optics [1][2][3][4]. For instance, The fractional Fitzhugh-Nagumo model (FN) is an important nonlinear model for describing the transmission of thermal energy in thermodynamics, circuit theory, biology, and in the area of population genetics [5][6][7].
During the last few years, the conformable fractional derivative and integral have received much attention, and many applications have been remodeled using their definitions. Moreover, they have many exciting advantages that make them more comfortable and more flexible than the definitions of other fractional derivatives, especially Caputo and Reimann Liouville derivatives. Among these advantages, the conformable fractional derivative (CFD) satisfies all ordinary calculus concepts such as product, quotient, Rolle's theorem, and mean-value theorem, chain rules. A non-differentiable function can be µ-differentiable in terms of conformable sense [19][20][21]. FN has been studied very extensively and, there is pervasive literature available of the solutions of FN differential equations of fractional order, where the fractional derivatives are in terms of Caputo or Reimann-Liouville. Nevertheless, there is very little or no work available on solving the FN involving conformable fractional derivatives. Motivated by those mentioned above, we feel compelled to solve the FN model in the form of conformable space.
In this paper, we develop the conformable Sumudu decomposition method (CSDM) application to study the conformable fractional FN equation. The CSDM is a modified algorithm based on the combination of the Adomian [22] decomposition scheme and conformable Sumudu transform method [23]. The remaining part of this article is structured as follows: In the next section, we present some basic definitions of the conformable fractional derivatives and the conformable Sumudu transform. In Section 3, the main idea of the proposed method is described. The convergence of the solution is discussed and proved in Section 4. In Section 5, we devote ourselves to applying the (CSDM) for conformable fractional FN equations. In Section 6, we discuss the numerical results and illustrate the accuracy and efficiency of the CSDM. Conclusions are outlined in Section 7.

Example 2.
Let , n ∈ R and µ ∈ (0, 1], then the CST for of specific functions is calculated by: In particular,

Analysis of (CSDM)
Herein, we demonstrate the proposed approach by considering the general form of the nonlinear conformable fractional equation.
with initial condition Taking the CST S µ t , of Equation (6), we have using the differentiation property of the (CST), we obtain Transforming the inverse CST both sides of Equation (9), we get Now, Adomian solution is and, we decompose the nonlinear term according to the following series of the Adomian polynomials where, Substituting Equations (11) and (12) in (10), we get comparing both sides of (13), we get

Convergence Analysis
In this segment, we discuss the sufficient condition that guarantees the CFN equation's unique solution, and we present the proposed method's error analysis.
Proof. Let = (C[0, T], . ) be the Banach space of all continuous functions , we define a mapping G : → as follows , Now suppose R and N are also Lips- where ε 1 and, ε 3 are Lipschitz constants φ and,φ are different functions.
For 0 < ϑ < 1, the mapping G is a contraction. Thus, According to the Banach fixed point theorem for contraction, (14) has a unique solution.
In the next theorem, we discuss the convergence of the solution. Proof. Let F n = ∑ n j=0 φ j x λ λ , t µ µ , be the nth partial sum. Using a new formulation of Adomian polynomial we get Let n = m + 1, we have F m+1 − F m ≤ ϑ m F 1 − F 0 , by using the triangle inequality we have F m+1 − F m ≤ ϑ m−1 − 1 φ 1 but since 0 < ϑ < 1, 0 < 1 − ϑ < 1, therefore, which proves the theorem.

Numerical Examples
Example 3. Consider the following CFN of the form: with the IC The exact solution of Equation (16) is given by The Adomian polynomials for the nonlinear term −φ 3 can be computed as follows Using Equation (14), we have In the same pattern, we compute the following terms Thus, the 5th-order approximate solution of Equation (18) is given by

Results and Discussion
This segment discusses the proposed method's precision and applicability by comparing the approximate and exact solutions using graphs and tables. Figures 1 and 2, depict the behaviors of the exact solutions of Examples 3 and 4, when µ = λ = 1. We can observe that the solution φ x λ λ , t µ µ increases quickly when we increase x and t.

Conclusions
This study has efficiently implemented the conformable Summdu transform and Adomian decomposition method to obtain an approximate solution of the conformable fractional Fitzhugh-Nagumo model. The CSDM gives us a solution in an infinite series with small error and high convergence. Furthermore, the convergence and the error analysis of the proposed method were stated and proven. Two examples were employed in order to illustrate the preciseness and effectiveness of the employed method. To provide better understanding of the characteristics of the solutions, the solution graphs were plotted in Figures 1-8, by considering different values of parameters, x, and t within the interval [0, 1]. Moreover, we have discussed the behavior of the solution φ x λ λ , t µ µ when (λ = ν = 1) and approximate solutions when λ and ν taking different fractional values. The obtained solutions were in full agreement as compared with exact solutions. Finally, the exact solutions and approximate solutions were plotted, and we can see the agreement among the solutions. The results lead us to say that the proposed method is reliable, accurate, and much understandable compared to other methods. Hence, it is concluded that this method can also be applied to solve other fractional non-linear differential equations involving conformable fractional derivatives.

Conflicts of Interest:
The authors declare no conflict of interest.