A new analytical approach to the (1+1)-dimensional conformable Fisher equation

In this paper, we use an effective method which is the rational sine-Gordon expansion method to present new wave simulations of a governing model. We consider the (1+1)-dimensional conformable Fisher equation which is used to describe the interactive relation between diffusion and reaction. Various types of solutions such as multi-soliton, kink, and anti-kink wave soliton solutions are obtained. Finally, the physical behaviours of the obtained solutions are shown by 3D, 2D, and contour surfaces


Introduction
Mathematical modelling of physical systems leads to linear and nonlinear differential equations in physics, engineering, and other fields.Understanding physical processes described by nonlinear equations necessitates finding exact solutions.Aside from that, exact solutions can be used to calculate specific critical physical quantities analytically and for simulation [1].There are some methods to calculate analytically including the Hirotas' direct method [2], the tanh method [3], the extended tanh-function method [4], the multiple exp-function method [5], the transformed rational function method [6], the first integral method [7], the modified simplest equation method [8], the improved Bernoulli sub-ODE method [9], the Sine-Gordon expansion method [10,11,12], the modified exponential method [13].Besides these analytical methods, there are many efficient numerical techniques have been submitted to the literature by mathematicians.For example, the q-homotopy analysis transform technique [14], the trapezoidal base homotopy perturbation method [15], and others [16,17,18].The paper aims to find exact soliton solutions of the conformable (1+1)-dimensional Fisher equation [19] by using the rational sine-Gordon expansion method.The (1+1)-dimensional Fisher equation describes the process of diffusion and reaction interacting.Fisher presented this equation as a model for mutant gene propagation, with u(x, t) denoting the density of favourable mutations, α 2 as diffusion factor [20].This equation is used in chemical kinetics and population dynamics and is also used to solve problems like the nonlinear evolution of a population in one-dimensional habitual space or the neutron population in a nuclear reaction [21].method, the generalized Kudryashov method and the extended (G /G)-expansion method, and gained the bright-like, dark-like and singular-like solitary wave solutions [19].Triki and Wazwaz have used the trial equation method to a generalized Fisher equation, and gained some new wave solutions [22].As a different approach, Matinfar et al. [23] focused on the numerical solution.In this context, they obtained solutions compatible with the exact solution by applying the generalized two-dimensional differential transform method.In this article, we will apply the rational sine-Gordon expansion method to the (1+1)-dimensional Fisher equation with conformable derivative to construct wave solutions.The conformable derivative operator overcomes some limitations of other fractional operators (Caputo, Riemann-Liouville, Caputo-Fabrizio and etc.) and provides basic properties of classical calculus such as the quotient rule, the chain rule, the product of two functions, Rolle's and mean value theorems.The application of the conformable derivative is simpler and very effective.
The rest of the paper is organized as follows: In Section 2, we describe the conformable derivative and its fundamental properties.In Section 3, the basic steps of the rational sine-Gordon expansion method, which is the novelty of the paper are presented.In Section 4, we apply the proposed method to the (1+1)-dimensional Fisher equation.Several conclusions are given in the last section.

Preliminary remarks on the conformable derivative
Definition 1 Given a function h : [0, ∞) → R. Then the conformable derivative of h order γ is defined by Theorem 1 Let Lγ be the derivative operator with order γ ∈ (0, 1] and h, k be γ-differentiable at a point t > 0. Then we have the following properties [24,25]: 3 Fundamental structure of the RSGEM Before giving the rational sine-Gordon expansion method (RSGEM) [26, 27, 28], we will explain the sine-Gordon expansion method (SGEM).Let us suppose the sine-Gordon equation where 2), gives the nonlinear ordinary differential equation (NODE) as, where Φ = Φ(ξ), µ is the amplitude and c is the velocity of the travelling wave, respectively.We find as follows after full simplification of Eq. (3); where C is the constant of integration.Replacing in Eq. ( 4), gives; Setting A = 1 in Eq. ( 5), gives ω = sin(ω).( 6)

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Solving Eq. ( 6) by variables separable, we get the two significant properties of trigonometric functions as follows; where p = 0 is the integration constant.Let's consider the nonlinear partial differential equation (NPDE) of the form below, for which the solution is searched; We apply the wave transformation, ϕ = ϕ(x, t) = Φ(ξ), ξ = kx + w t γ γ into Eq.( 9), it gives the following equation, where N is a nonlinear ordinary equation (NODE) that has partial derivatives of Φ depending on ξ.The SGEM, the solution of Eq. ( 9) is considered in the following form Eq. ( 11) can be rearranged considering Eqs. ( 7) and ( 8) as follows; As we know, rational functions are more general functions than polynomial functions.We can obtain significantly general forms of wave solutions which are including polynomial function solutions by this way.The different point of the method is the solution functions have two auxiliary functions, viz.sech(ξ), tanh(ξ) We consider the following solution form which is also written as where a i , b i , c i , d i , a 0 , b 0 are constants that will be determined later.a i , b i , c i , d i values are not all zero at the same time.The value of M is determined using the balance principle between the highest power nonlinear term and the highest derivative in NODE.
After equating the coefficients of sin i (ω) cos j (ω) to zero, we find a set of algebraic equations.a i , b i , c i , d i , a 0 , b 0 values are found in solving the set of algebraic equations by Wolfram Mathematica 12.At the end, we substitute these values into Eq.( 13) and get the new travelling wave solutions of Eq. ( 9).

Application of RSGEM
The (1+1)-dimensional conformable Fisher equation is given as where γ is the order of the conformable derivative between 0 < γ ≤ 1.
We use the wave transformation as given below, where k, w are constants that will be determined.Getting partial derivatives of U(ξ) the function with respect to x, t, we find a non-linear ordinary differential equation as According to the homogeneous balance principle, we obtain a relationship between U and U 3 in Eq. ( 17), M = 1.For M = 1, Eq. ( 13) turns to the below form.
We put Eq. ( 18) and its first and second-order derivatives into Eq.( 17) and can obtain a system of algebraic equations.By using Wolfram Mathematica 12, a 0 , a 1 , b 0 , b 1 , c 1 , d 1 and the other parameters can be found.Finally, we put these coefficients into Eq.( 13) and obtain new travelling wave solutions of Eq. ( 1). Case-1 Putting the above coefficients into Eq.( 13), yields where Considering the suitable values of parameters, we can find wave simulations for Eq. ( 19) as following Figures 1  and 2:  Putting the above coefficients into Eq.( 13), yields

Case-2
When we consider the suitable values of parameters, we can find wave simulations for Eq. ( 20) as following figures: Case-3 Putting these coefficients into Eq.( 13), yields When we consider the suitable values of parameters, we can find wave simulations for Eq. ( 21) as following figures:

Case-4
Putting these values into Eq.( 13), yields When we consider the suitable values of parameters, we can find wave simulations for Eq. ( 22) as following figures: -20 -10 10 20 x u4(x,t)

Case-5
Putting these values into Eq.( 13), yields When we consider the suitable values of parameters, we can find wave simulations for Eq. ( 23) as following figures:

Conclusion
In this paper, the rational sine-Gordon expansion method has been applied to the (1+1)-dimensional conformable Fisher equation.
We have obtained some new wave solutions including hyperbolic and trigonometric functions.Figure 1 shows multi-soliton solution surfaces both imaginary and real parts of Eq. ( 19). Figure 3, Figure 5, and Figure 9 show the anti-kink soliton surfaces for Eq. ( 20), Eq. ( 21) and Eq. ( 23), respectively.Figure 7 shows the kink soliton surface for Eq. ( 22).Kink-type solitons are travelling wave solutions that climb up or climb down from one phase to another, and kink soliton reaches a constant at infinity.The mentioned model is used for modelling the relationship between the rate of inflation and both real and nominal interest rates, population dynamics in nonlinear media, and logistic population growth models, as well [20,29,30].Fisher's model has been investigated by a numerical technique which is the q-homotopy analysis transform method (q-HATM) in [31].They considered the time-fractional Fisher's model in Caputo's sense.Besides, they assumed special values of the coefficients in the model.The main advantage of the proposed method is the derived solutions include many other analytical techniques.According to new results and all figures, it has been observed that this method is a powerful tool for obtaining analytical solutions of nonlinear partial differential equations such as governing models.We hope that the provided solutions may be useful for scientists in mathematical biology, neurophysiology, chemical reactions, and economy.