New Exact Solutions of Kolmogorov Petrovskii Piskunov Equation, Fitzhugh Nagumo Equation, and Newell-Whitehead Equation

Chu, Yu-Ming; Javeed, Shumaila; Baleanu, Dumitru; Riaz, Sidra; Rezazadeh, Hadi

doi:https://doi.org/10.1155/2020/5098329

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 5098329 | https://doi.org/10.1155/2020/5098329

New Exact Solutions of Kolmogorov Petrovskii Piskunov Equation, Fitzhugh Nagumo Equation, and Newell-Whitehead Equation

Yu-Ming Chu,^1,2Shumaila Javeed ,³Dumitru Baleanu,^4,5Sidra Riaz,⁶and Hadi Rezazadeh⁷

Academic Editor: Zengtao Chen

Received13 Jun 2020

Revised28 Sept 2020

Accepted08 Oct 2020

Published09 Nov 2020

Abstract

This work presents the new exact solutions of nonlinear partial differential equations (PDEs). The solutions are acquired by using an effectual approach, the first integral method (FIM). The suggested technique is implemented to obtain the solutions of space-time Kolmogorov Petrovskii Piskunov (KPP) equation and its derived equations, namely, Fitzhugh Nagumo (FHN) equation and Newell-Whitehead (NW) equation. The considered models are significant in biology. The KPP equation describes genetic model for spread of dominant gene through population. The FHN equation is imperative in the study of intercellular trigger waves. Similarly, the NW equation is applied for chemical reactions, Faraday instability, and Rayleigh-Benard convection. The proposed technique FIM can be applied to find the exact solutions of PDEs.

1. Introduction

The nonlinearity in the world prevails thoroughly; thus, it is significant to develop nonlinear models including partial differential equations [1–4]. Nonlinear conformable PDEs attracted the interest of many researchers because of their vast applications in different fields, for example, in chemistry, acoustic, fluid dynamics, image processing, biology, physics, vibration, and control [5–7]. Nonlinear conformable PDEs have a great potential to apply in several fields; thus, researchers put noteworthy attention for their analytical and numerical solutions [8–12]. Different effective and reliable techniques are proposed like the homotopy analysis method [13], homotopy perturbation technique [14], extended hyperbolic tangent method [15, 16], hyperbolic function method [17], subequation method [18], and exponential rational function method [19] to get solutions.

Feng presented an effective technique to obtain travelling wave solutions of NPDEs, known as the FIM method [20–22]. FIM is based on the ring theory and commutative algebra. FIM provides first integral of explicit form having polynomials as coefficients by applying the division theorem. Contrary to other methods, the benefits of FIM are to produce exact and explicit solutions without complicated and lengthy calculations [23–25]. Despite several advantages, FIM can only be applied to integrable PDEs.

The focus of this paper is to find the exact solutions of conformable biological models. It includes KPP and its derived models, namely, FHN and NW. KPP is a general form of equation and we can obtain different equations from KPP, for example, FHN, NW, and Cahn Allen. The considered models are significant in biology. KPP describes the genetic model for spread of dominant gene through population. The graphical solutions of the KPP equation can be used for diallel analysis as diallel analysis requires graphical solutions of genes. In diallel analysis, graphical representation of genes is required and some further calculations enable researchers to have point estimation of recessive genes and dominant genes instead of providing an interval of estimation [26]. The FHN equation is used in the study of intercellular trigger waves. Trigger waves are pulses and oscillatory waves; these waves switch from one stable steady state to another [27]. Similarly, the NW equation is applied for Faraday instability, chemical reactions, Rayleigh-Benard convection, and biological systems [28]. Various techniques have been established for solving the KPP equation, for instance, the discrimination algorithm [29], the homotopy perturbation technique [30], the differential transform method [31], the (G/G)-expansion method [32], and the homotopy analysis method [33]. Generally, the solutions of KPP equations are based on series solutions or numerical solutions. In this work, an effective technique named as FIM was adopted to acquire the exact solutions of KPP, FHN, and NW equations. The work is novel as the exact solutions of considered models using FIM are not presented before in the literature.

This paper consists of the following sections. Conformable derivative is described in Section 2; the proposed technique FIM is discussed in Section 3; the solutions of conformable KPP, FHN, and NW equations are presented in Section 4, and Section 5 contains summary and further recommendations.

2. Preliminaries

2.1. Derivative: Conformable

Conformable derivative is defined by Khalil et al. [34, 35].

Definition 1. The conformable derivative for function of order is given aswhereas having . If function is -differentiable in , here and exists, then at 0, conformable derivative is represented as .

The conformable integral for function is given aswhere and .

Khalil et al. further proposed the succeeding theorem [34–36].

Theorem 2. Let the functions and at the point be -differentiable; for we have the following properties [34].(1) (2) (3) (constant functions)(4)(5)(6)Furthermore, if we have a differentiable function , then

The conformable derivative of any differential function at origin is zero; despite this flaw, several studies have been made on conformable derivative, as it explains higher order integration, sequential differentiation and integration, connection of differentiation and integration, property of linearity, derivative of constant function, quotient and product rule, chain rule, and power rule [34, 36–39]. Consequently, many researchers are working on the applicability of conformable derivative for real-world problems, such as Jacobi elliptical function expansion method used to solve conformable Boussinesq and combined Kdv-mKdv equation [40], conformable space-time fractional (2+1) dimensional dispersive long wave equation [41], conformable heat equation [42], and conformable perturbed nonlinear Schrodinger equation [43].

3. Methodology

Here, the methodology of FIM is presented.

Step 1. Conformable PDE is given as follows:

Step 2. Now using the following transformationSpecifically in case of conformable derivative, the next transformation is applied asThe transformation defined in equation (5) will convert conformable PDE in nonlinear ODE.where and transformed variable is denoted by .

Step 3. We will take other independent variables asAs a result, FIM will provide a system of ODEs (nonlinear) as

Step 4. We attain general solutions after integrating equation (8). There is no precise or sound technique to obtain first integrals in case of plane independent (autonomous) system, so it is difficult to get even one first integral. To determine the first integral, the division theorem is utilized. Hence, a first integral is derived (cf. equation (8)) with the help of the division theorem. In this way, nonlinear ordinary differential equations (ODEs) can be reduced into a first-order ODE system (integrable) through the division theorem. Afterwards, solving the obtained system (cf. equation (8)), the exact solutions can be acquired.

The theorem for complex domain and two variables is given as

Division Theorem. Consider polynomials and , in complex domain , where is irreducible. If at all zero points of , vanishes, then another polynomial exists in and the following equality holds:

4. Implementation of FIM: Conformable KPP Equation and Its Derived Equations

The exact solutions of KPP, FHN, and NW equations are presented in this section.

4.1. Conformable Space-Time Fractional KPP Equation

Andrey Kolmogorov, Ivan Petrovsky, and Nikolai Piskunov proposed a nonlinear PDE called the Kolmogorov Petrovsky Piskunov (KPP) equation to describe the genetic model for spread of dominant gene through population. Later, the KPP equation is applied in different natural sciences like in physics as combustion, in biology as propagation of nerve impulses, in chemical kinetics as propagation of concentration waves, and in plasma as evolution of set of duffing oscillators.

Consider conformable space-time fractional KPP equation defined as [32, 44]where , , and are constants and .

First, we use conformable derivative with the following transformation:where the transformation variable is . The transformation represented in equation (11) will provide the following conversions:

Here, and are constants. Then, we get ODE by using equation (12) in equation (10):

Now, we acquire a 2D system from equation (7) as

Afterwards, the division theorem will provide first integral. According to FIM, and are supposed to be nontrivial solutions of the system given (cf. equations (14) and (15)). Now, the division theorem provides us an irreducible polynomial in given as

where and . Now, we have a polynomial of form in such that

Using in equation (17) and equating coefficients for , then we have following equations:

Here, are polynomials in . As equation (18) shows has constant nature, hence and we can take . We conclude that can only be 0 or 1 by using and in equations (19) and (20) and after balancing the functions and degrees. Now, we can take ; therefore, equation (19) takes the following form:where is an integrating constant.

Afterwards, the substitutions of values of , in equation (20) provide a system of nonlinear algebraic equations by equating the power of . Thus, as a result, we obtain various values of constants given as follows.

Case 1. The following constants are acquired as follows:

Substituting equations (21) and (22) into equation (16), we get

Substitution of equation (23) into equation (14) provides the first solution of conformable fractional KPP equation.

Case 2. We get

Substituting equations (21) and (25) into equation (16), we obtain

Substitution of equation (26) into equation (14) provides the second solution of conformable fractional KPP equation.

Case 3. We have

Substituting equations (21) and (28) into equation (16), we get

Substitution of equation (29) into equation (14) provides the third solution of conformable fractional KPP equation.

Case 4. We obtain

Substituting equations (21) and (31) into equation (16), we obtain

Substitution of equation (32) into equation (14) provides the fourth solution of conformable fractional KPP equation.

Case 5. We get

Substituting equations (21) and (34) into equation (16), we obtain

Substitution of equation (35) into equation (14) provides the fifth solution of conformable fractional KPP equation.

Case 6. We have

Substituting equations (21) and (37) into equation (16), we obtain

Substitution of equation (38) into equation (14) provides the second solution of conformable fractional KPP equation.

The solutions are presented in Figure 1. For larger values of , the solutions attain more height which is depicted in Figure 2. Figure 3 shows the graphical presentation in 2D plot of genes for the KPP equation. The graphical solutions of the KPP equation can be used in diallel analysis as observed in [26].

4.2. Conformable Time-Fractional FHN Equation

Richard Fitzhugh proposed a model for transmission of impulses in nerve axon in 1961. Nagumo et al. made the identical circuit in succeeding years and presented the model of an excitable system. FHN is derived from KPP on substituting , [45, 46].

Consider conformable time-space fractional FHN equation aswhere and .

First, we use the conformable derivative with the following transformation:

where the transformation variable is . The transformation represented in equation (41) will provide the following conversions:

Here, is a constant. Then, we get an ODE by using equation (42) in equation (40):

Now, we obtain a 2D system from equation (7) as

Afterwards, the division theorem will give first integrals. According to FIM, and are supposed to be nontrivial solutions of the system given (cf. equations (44) and (45)). Now, the division theorem provides us an irreducible polynomial in aswhere and . Now, we have a polynomial of form in such that

Using in equation (47) and equating coefficients of , then we have the following equations:

Here, are polynomials in . As equation (48) shows has constant nature, thus and we can take . We conclude that can only be 0 or 1 by using and in equations (48) and (49) and after balancing the functions and degrees. Now, we can take ; therefore, equation (49) takes the following form:where is an integrating constant.

Afterwards, the substitutions of the values of , in equation (50) provide a system of nonlinear algebraic equations by equating the power of . Now, as a result, we have various constants given as follows.

Case 7. We get

Substituting equations (51) and (52) into equation (46), we get the following equation.

Substitution of equation (53) into equation (44) provides the first solution of conformable fractional FHN equation.

Case 8. We get

Substituting equations (51) and (55) into equation (46), we get the subsequent equation.

Substitution of equation (56) into equation (44) provides the second solution of conformable fractional FHN equation.

Case 9. We have

Substituting equations (51) and (58) into equation (46), we obtain the following equation.

Substitution of equation (59) into equation (44) provides the third solution of conformable fractional FHN equation.

Case 10. We obtain

Substituting equations (51) and (61) into equation (46), we get

Substitution of equation (62) into equation (44) provides the fourth solution of conformable fractional FHN equation.

Case 11. We get

Substituting equations (51) and (64) into equation (46), we obtain

Substitution of equation (65) into equation (44) provides the fifth solution of conformable fractional FHN equation.

Case 12. We have

Substituting equations (51) and (67) into equation (46), we get

Substitution of equation (68) into equation (44) provides the sixth solution of conformable fractional FHN equation.

The solutions are presented in Figure 4. For smaller value of , solutions attain more height which is depicted in Figure 5.

4.3. Conformable Space-Time Fractional NW Equation

NW has wide applications in mechanical and chemical engineering, ecology, and biology [34]. NW can be derived from KPP by substituting , , and .

Consider conformable space-time fractional NW equation aswhere and , , , and is a positive integer. For , , , and , equation (70) becomes

First, we use the conformable derivative with the following transformation:

where the transformation variable is . The transformation represented in equation (72) will provide the following conversions.

where is a constant. Then, we get ODE by using equation (73) in equation (71).

Thus, we obtain a 2D system from equation (7) as

Afterwards, the division theorem will give first integrals. According to FIM, and are supposed to be nontrivial solutions of the system given (cf. equations (75) and (76)). Hence, the division theorem provides us irreducible polynomial in given aswhere and . Now, we have a polynomial of form in such that

Using in equation (78) and equating coefficients of , then we have following equations:

Here, are polynomials in . As equation (79) shows has constant nature, thus and we can take . We conclude that can only be 0 or 1 by using and in equations (80) and (81) and after balancing the functions and degrees. Now, we can take ; therefore, equation (80) takes the following form:where is an integrating constant.

Afterwards, the substitutions of the values of , in equation (81) provide a system of nonlinear algebraic equations by equating coefficients of power of . Thus, we have various constants as given below.

Case 13. We acquire

Substituting equations (82) and (83) into equation (77), we get

Substitution of equation (84) into equation (75) provides the first solution of conformable fractional NW equation.

Case 14. We get

Substituting equations (82) and (86) into equation (77), we obtain

Substitution of equation (87) into equation (75) provides the second solution of conformable fractional NW equation.

Case 15. We have

Substituting equations (82) and (89) into equation (77), we get

Substitution of equation (90) into equation (75) provides the third solution of conformable fractional NW equation.

Case 16. We obtain

Substituting equations (82) and (92) into equation (77), we get

Substitution of equation (93) into equation (75) provides the fourth solution of conformable fractional NW equation.

Case 17. We get

Substituting equations (82) and (95) into equation (77), we obtain

Substitution of equation (96) into equation (75) provides the fifth solution of conformable fractional NW equation.

Case 18. We have

Substituting equations (82) and (98) into equation (77), we obtain

Substitution of equation (99) into equation (75) provides the sixth solution of conformable fractional NW equation.

The solutions are presented in Figure 6. In Figure 7, the results reveal that the peaks are getting sharper and sharper by reducing the value of .

5. Conclusion

The purpose of this paper was to find new exact solutions of some conformable biological models. The considered models were KPP, FHN, and NW. FIM was employed to obtain the solutions of fractional KPP, FHN, and NW equations. The proposed method was found brief and direct. The results indicate that FIM is one of the best techniques to calculate exact solutions of nonlinear fractional order problems appearing in biology, physics, and engineering.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Shumaila Javeed and Dumitru Baleanu were responsible for the conceptualization. Shumaila Javeed, Sidra Riaz, and Hadi Rezazadeh were responsible for the formal analysis. Shumaila Javeed and Sidra Riaz were responsible for the investigation. Shumaila Javeed and Sidra Riaz were responsible for the methodology. Dumitru Baleanu, Yu-Ming Chu, and Hadi Rezazadeh were responsible for the project administration. Dumitru Baleanu and Yu-Ming Chu were responsible for the resources. Hadi Rezazadeh was responsible for the software. Shumaila Javeed, Dumitru Baleanu, and Yu-Ming Chu were responsible for the supervision. Sidra Riaz, Yu-Ming Chu, and Hadi Rezazadeh were responsible for the validation. Shumaila Javeed and Sidra Riaz were responsible for writing the original draft. Shumaila Javeed and Yu-Ming Chu were responsible for writing, reviewing, and editing.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).

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Copyright © 2020 Yu-Ming Chu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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