Abstract
The Fitzhugh–Nagumo equation is an important non-linear reaction–diffusion equation used to model the transmission of nerve impulses. This equation is used in biology as population genetics; the Fitzhugh–Nagumo equation is also frequently used in circuit theory. In this study, we give solutions to the fractional Fitzhugh–Nagumo (FN) equation, the fractional Newell–Whitehead–Segel (NWS) equation, and the fractional Zeldovich equation. We found the exact solutions of these equations by conformable derivatives. We have obtained the exact solutions within the time-fractional conformable derivative for these equations.
1 Introduction
Fractional differential equations (FDEs) are generalizations of known differential equations (ODEs). FPDEs are used effectively in many fields of science [1–4]. These equations are significant models to interpret plasma physics, relativistic physics, quantum mechanics, non-linear optics, etc. So, FPDE studies are getting more and more important. Recent developments and applications in fractional calculus have been discussed by many authors [5–9]. Many fractional models can be converted to a FODE, allowing us to use the power-series technique to find all open-series analytical solutions.
The fractional Fitzhugh–Nagumo equation is an important non-linear reaction–diffusion equation used to model the transmission of nerve impulses. This equation is used in biology as population genetics; the fractional Fitzhugh–Nagumo equation is also frequently used in circuit theory.
There are numerous effective ways to find solutions of PDEs. These methods are the -expansion [10], the sub-equation (11) and (12), the exp-function [13, 14], the first integral [15], the functional variable [16, 17], the modified simplest [18, 19], the Kudryashov [20, 21], the extended simple [22], and the extended tanh–coth methods [23, 24]. These techniques allow for the calculation of PDE solutions in a variety of different formats.
These equations we deal with in this study are effective equations that play a fundamental role in many phenomena such as plasma physics and optics. The non-linear phenomena of wave happen in different fields such as optical fiber, physics, and biology. It is necessary to gain the exact solutions of such models for the better understanding of non-linear wave phenomena.
In the second part, the conformable derivative is introduced. In the third part, the extended tanh–coth method is given. In the other sections, we found the exact solutions of the (1 + 1) dimensional time-fractional Fitzhugh–Nagumo (FN) equation, the Newell–Whitehead (NW) equation, and the Zeldovich equation via this method.
2 Conformable derivative
Definition 1The basic limit definition of this derivative is [25]:Some properties of this derivative are given in [26, 27].
Definition 2Let g: (0, ∞) → R be a function. The conformable derivative of g for order α is defined byfor all k > 0, α ∈ (0, 1).
If a function g: [0, ∞) → R is α − differentiable at t0 > 0, α ∈ (0, 1] and g is continuous at t0.
Let f and g be α − differentiable at a point t > 0, α ∈ (0, 1]:
If g is differentiable,Non-linear conformable partial differential equations (NCPDEs) with one independent variable are as follows:The non-linear fPDE (5) can be converted to a non-linear ODE using Eq. 6:
Let us now discuss how we will approach solving the Fitzhugh–Nagumo equation, the Newell–Whitehead equation, and the Zeldovich equation.
3 The extended tanh–coth method
The tanh–coth approach can be summed up as follows in [28]:
The wave variable into a PDEu (x, t) = U(ξ) is a traveling wave solution. A non-linear ODE can be converted from the non-linear PDE (9):
From the derivatives of the independent variable Y,The tanh method can be expressed as follows:The following is an extension of Eq. 13 [29]:
4 Solutions of the Fitzhugh–Nagumo equation
The FN equation is an important non-linear reaction–diffusion equation used to model the transmission of nerve impulses. Also used in biology as population genetics, this equation is also frequently used in circuit theory, where x is the space variable and t is a time variable.
Since the FN equation is (1 + 1)-dimensional and time-fractional, using the wave variable , Eq. 15 can be converted into a non-linear ODE (Eq. 16).where c and v are constants. By balancing (U)3 with U″ in Eq. 16, we obtain the following:
The solution form is as follows:Here a0 and a1, b1 are arbitrary constants. Eq. 19 is substituted into Eq. 16, and if the coefficients of Y are used to set the system of equations to zero, thenBy solving this system of equations, we get the following cases.
Case 1:Therefore, the exact solution is as follows:
Case 2:The exact solution is as follows:
Case 3:The exact solution is as follows:
Case 4:The exact solution is as follows:
Case 5:The exact solution is as follows:
Case 6:The exact solution is as follows:
Case 7:The exact solution is as follows:
Case 8:The exact solution is as follows:
Case 9:The exact solution is as follows:
5 Solutions of the Newell–Whitehead equation
If μ = −1 in Eq. 15, the NW equation is obtained by
Using the wave variable , Eq. 39 can be converted to Eq. 40.where c and v are constants. By balancing, we obtain
The solution form is as follows:Here, a0 and a1, b1 are arbitrary constants. Eq. 43 is substituted into Eq. 41, and if the coefficients of Y are used to set the system of algebraic equations to zero, thenBy solving this system of equations, we get the following cases.
Case 1:Therefore, the exact solution is as follows:
Case 2:The exact solution is as follows:
Case 3:The exact solution is as follows:
6 Solutions of the Zeldovich equation
If μ = 0 in Eq. 15, the Zeldovich equation is obtained:
Using the wave variable , Eq. 51 can be converted to Eq. 52.where c and v are constants. From Eq. 52, we obtain
The solution form is as follows:Here, a0 and a1, b1 are the arbitrary constants. Eq. 55 is substituted into Eq. 52, and if the coefficients of Y are used to set the system of equations to zero, thenBy solving this system of equations, we get the following cases.
Case 1:Therefore, the exact solution is as follows:
Case 2:The exact solution is as follows:
Case 3:The exact solution is as follows:
7 Conclusion
In this article, we present the extended tanh–coth method for solving non-linear space–time conformable PDEs. We found the exact and traveling wave solutions of some important space–time fPDEs via the extended tanh–coth method. We know many of the results obtained are new solutions that do not exist in the literature. The hyperbolic and trigonometric function solutions are significant to explain a variety of physical phenomena. This suggests that the extended tanh–coth method is more effective in finding the solutions of non-linear fPDEs. The 3D plots of the acquired solutions are presented by choosing appropriate values to the parameters in Figures 1–4. These are the advantages of the extended tanh–coth method. The offered method can be utilized to assist complicated models applicable to a wide variety of physical situations. We hope that the telecommunications industry and other such forms of waveguides will find this study to be beneficial.
FIGURE 1
Figure of u4 in (Eq. 4.14) with μ = 2, 0 < t < 1, and 0 < x < 10.
FIGURE 2
Figure of u5 in (Eq. 4.16) with a0 = 1/2, 0 < t < 10, and 0 < x < 10.
FIGURE 3
Figure of u7 in (Eq. 4.20) with a0 = 1/2, 0 < t < 10, and 0 < x < 10.
FIGURE 4
Figure of u9 in (Eq. 4.24) with μ = 2, 0 < t < 1, and 0 < x < 10.
Moreover, we come to the understanding that the newly obtained hyperbolic function and trigonometric function solutions in this study may help explain some complex physical aspects in non-linear physical sciences and are related to such physical properties. In future studies, these approaches can be easily used for other NFDEs, NFDE systems, fractional complex equations, fractional difference equations, etc.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.
Author contributions
ACC: Original Draft, Methodology, Validation; AB: Investigation, Supervision, Writing—Review and Editing; OAA: Conceptualization, Methodology, Writing; MA: Original Draft, Validation, Founding acquisition.
Funding
The work in this study was supported, in part, by the Open Access Program from the American University of Sharjah. This study represents the opinions of the authors and does not mean to represent the position or opinions of the American University of Sharjah
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
exact solutions, conformable derivative, the Fitzhugh-Nagumo equation, solitary wave solution, solitons
Citation
Cevikel AC, Bekir A, Abu Arqub O and Abukhaled M (2022) Solitary wave solutions of Fitzhugh–Nagumo-type equations with conformable derivatives. Front. Phys. 10:1028668. doi: 10.3389/fphy.2022.1028668
Received
26 August 2022
Accepted
06 October 2022
Published
26 October 2022
Volume
10 - 2022
Edited by
Yakup Yildirim, Near East University, Cyprus
Reviewed by
Emad Zahran, Benha University, Egypt
Amin Jajarmi, University of Bojnord, Iran
Updates
Copyright
© 2022 Cevikel, Bekir, Abu Arqub and Abukhaled.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Ahmet Bekir, bekirahmet@gmail.com
This article was submitted to Optics and Photonics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.