New Exact Traveling Wave Solutions of Fractional Time Coupled Nerve Fibers via Two New Approaches

In this paper, we obtain new soliton solutions of one of the most important equations in biology (fractional time coupled nerve ﬁbers) using two algorithm schemes, namely, exp (− ψ ( ξ )) expansion function method and ( θ ′ ( ξ ) / θ 2 ( ξ )) expansion methods. The equation and the solution methods have free parameters which help to make the obtained solutions are dynamics and more readable for dealing with fractional parameter and the initial and boundary value problem. As a result, various new exact soliton solutions for the considered model are derived which include the hyperbolic, rational, and trigonometric functions, and other solutions are obtained. In addition, the obtained results proved that the used methods give better performance compared with existing methods in the literature.


Introduction
Differential equations attained great importance with several applications in nature and live environment [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Differential equations were widely used in the modeling and simulation of the biological system long time ago, for example, population dynamics and spreading and transmission of viruses. Nerve conduction is one of the most important phenomena in biological systems. Several studies have attempted to provide an interpretation to the nature of the nerve conduction. ese studies started in the early last century in 1925 and seemed to indicate that local circuit currents were involved in the longitudinal propagation of activity [20][21][22][23].
is study presents a model of action potential propagation in bundles of myelinated nerve fibers. e nature of the conduction process on an isolated nerve axon is studied numerically and compared with the theoretical models [24]. Reutskiy et al. introduced a new model that combines the single-cable formulation of Goldman and Albus (1967) with a basic representation of the ephaptic interaction among the fibers. e conduction velocity (CV) behavior is investigated in the presence of various conductance parameters and temperatures [25].
ey allow an increase in the speed of a nerve impulse while decreasing the diameter of the nerve fiber. e first work concerning myelinated nerve fibers was developed by Rushton [49]. e primary content of the article is organized as follows. In section 2, the governing equations of the model are introduced. e properties of conformable fractional derivatives are given in Section 3. Two algorithms of the proposed analytical method, namely, exp(− ψ(ξ)) expansion function method and (θ′(ξ)/θ 2 (ξ)) expansion method for solving the reduced equation for fractional time coupled nerve fibers are introduced in Section 4. Finally, we briefly make a conclusion in Section 5.

Governing Model and Mathematical Analysis
To consider the equations for the coupled nerve fibers, consider their electrical analogy as in [50,51]: where n represents successive active nodes, M n represents the voltage across the membrane, N n represents the current flowing longitudinally through the fiber from node n to node n + 1, R i and R 0 are the inside and outside resistances, N C,n is the current supplying the capacity of the active node n, while N ion,n is the ionic current, comprising both sodium and potassium components. According to [51][52][53], the complex impedance of a capacitor reads. (2) en, the ionic current becomes e ionic current is given by the following relation: where M b and M a is the threshold voltage at sodium current and potential at the current returns to zero. By setting v (1) can be rewritten as where v 1 n � v n and v 2 n � w n . As long as δ n tends to x, equation (5) admits to

Revisitation of the Two Analytical Techniques
Let us consider the general form of nonlinear evolution equation of fractal order as By using the traveling wave variable where the prime denotes the differentiation with respect to ξ.
is method proposes that the solution of equation (9) can be written as and the constant α i will be evaluated later and ψ � ψ(ξ) verifies where λ and μ are constants and will be computed with the flow of the paper. e integer N is determined by balancing between the highest order derivative. e solutions of equation (12) read. Family 1. When μ ≠ 0 and Δ � λ 2 − 4μ > 0, then the hyperbolic function solution is Family 2. When μ ≠ 0 and Δ � λ 2 − 4μ < 0, then the trigonometric function solution is where Δ 1 � − λ 2 + 4μ.

e Extended
In view of the (θ ′ (ξ)/θ 2 (ξ)) expansion method, the quick gain of this method is the solution of where λ and μ are constants to be determined later. It is to be noted that the solution of equation (19) is given as follows.
Family 6. As long as λμ > 0, Family 7. If λμ < 0, then Family 8. In the limiting case λ ≠ 0 and μ � 0, we have Journal of Mathematics where ξ 1 and ξ 2 are constants to be determined later. Here, the term N in equation (19) can be computed taking into account the balancing between the highest order derivative and the nonlinear term in equation (10). Making use of equation (18) with equation (19) into equation (10), combining all the terms of the same power of (w ′ (ξ)/w 2 (ξ)), and performing some steps, we can get the values of α i , β i , w, and k. By inserting these values in equation (18) along with general solutions of equation (19), the solutions of equation (9) can be obtained directly.

Conclusion
In this paper, we introduced new solutions to one of the most important differential equations in biology. In this study, we have proposed a new model of myelinated nerve fibers described by a time fractional nonlinear evolution equation. Two algorithm schemes, namely, exp(− ψ(ξ)) expansion function method and (θ ′ (ξ)/θ 2 (ξ)) expansion method are used for constructing the new soliton solutions and other solutions such as hyperbolic function, trigonometric function, and rational function. To our knowledge, these new solutions have not been reported in former literature, and hence, they may be of significant importance for the explanation of some special physical phenomena. e results as in equations (29) and (34) are plotted in Figures 1-4. e obtained solutions here can be useful for applications in mathematical physics, engineering, and nonlinear optics. e achieved results here show the effectiveness and reliability of the proposed technique.

Data Availability
e data used to support the findings of this study are available within the manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest.