An Analytical View of Fractional-Order Fisher’s Type Equations within Caputo Operator

The present research article is related to the analytical investigation of some nonlinear fractional-order Fisher’s equations. The homotopy perturbation technique and Shehu transformation are implemented to discuss the fractional view analysis of Fisher’s equations. For a better understanding of the proposed procedure, some examples related to Fisher’s equations are presented. The identical behavior of the derived and actual solutions is observed. The solutions at diﬀerent fractional are calculated, which describe some useful dynamics of the given problems. The proposed technique can be modiﬁed to study the fractional view analysis of other problems in various areas of applied sciences.


Introduction
In mathematical science, the construction of exact and explicit solutions to nonlinear fractional-order partial differential equations (PDEs) is very significant and is one of the most exciting and especially active fields of study. It is well recognized that it is possible to divide all nonlinear PDEs into two parts: the nonintegrable ones and the integrable partial differential equations.
ere is an infinite number of exact solutions to the first form, i.e., the integrable equations. e most well-known problems among them are the sine-Gordon equation, Korteweg-de Vries equation, Boussinesq equations, Kawahara type equations, and nonlinear Schrodinger equation and the list can be expanded with other fundamental integrable problems, but it is not our purpose to give all the lists [1][2][3][4][5]. Nonlinear PDEs are considered to be in the class of nonintegrable partial differential equations with certain precise solutions or without precise solutions and will need special care to achieve their solutions because of the shape of the nonlinear differential equation and the pole of its solution. e Fitzhugh-Nagumo equation, Fisher equation, Burger-Huxley equation, and Ginzburg-Landau equation can be mentioned as the wellknown nonintegrable PDEs among them all [6][7][8][9][10][11][12][13].
Over the last few decades, considerable progress has been made in developing methods for obtaining precise solutions to nonlinear equations, but the progress accomplished is insufficient. Since, from our point of view, there is no single optimal way to achieve correct solutions to nonlinear differential equations of all forms. Based on the researchers' expertise and the sympathy for the method used, each method has its benefits and shortcomings. Also, all these techniques can be seen to be problem-dependent, namely, that certain techniques perform well on some concerns, but others do not. erefore, it is very important to apply certain well-known methods to nonlinear partial differential equations in the literature that are not solved with that method to look for potential new exact solutions or to check current solutions with different approaches [14][15][16][17].
Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation was first introduced by Fisher [18] and was later renamed Fisher equation. FEs have numerous applications in the fields of engineering and science [19][20][21][22]. e researchers investigated some important generalizations of this equation [23][24][25]. Numerous reaction-diffusion equations have wavefronts that show a vital part in explaining chemical, physical, and biological phenomena [26,27]. e reaction-diffusion systems can explain how changes in the concentration of one or more chemicals occur. One is the local chemical reactions that transform the substances into each other and the other is the diffusion, which allows the substances to spread through the air. e simplest equation for reaction-diffusion in one spatial dimension, where ψ(μ, I) shows single material concentration, P represents diffusion coefficients, and Q represents all local reactions. If R(ψ) � ψ(1 − ψ), we get FE which is used to define the biological populations dispersion. e Fisher-KPP advection equation is used to define population dynamics in advective environments [28]. e partial differential equation proposed by Fisher is nonlinear as Fisher proposed equation (2) as a model for gene selection, with ψ denoting the population density. e same equation also arises in the autocatalytic chemical reactions, nuclear reactor theory, flame propagation, neurophysiology, and Brownian motion process. e Fisher equation is considered to be an important equation because of its vast number of applications in the field of engineering. e homotopy perturbation technique was developed by He [29,30] in 1998. HPM provides the solution as a sum of the sequence having an infinite sum that converges rapidly to the exact results. HPM can be used to solve PDEs of higher dimensions and nonlinearity effectively.
In the present research article, effective utilization of the new developed technique, the homotopy perturbation method and Shehu transform, has been implemented to solve fractional FEs. e suggested technique is very effective for the solutions of other fractional PDEs because its required small computational work and higher degree accuracy. Moreover, the obtained results are in close resemblance with the actual solution of all fractional FEs.

Definition.
e integral of Shehu transformation is new and similar to other integral transformation which is described for exponential order functions. In set A, we take a function which is described by [33][34][35] A � ](η): e Shehu transformation which is defined by S(·) for a function ](η) is given as e Shehu transformation of a function ](η) is V(s, μ), and then ](η) is known as the inverse of V(s, μ) which is define as

Homotopy Perturbation Transform Method
To explain the fundamental ideas of this method, we get the following equation: where D α I � (z α /zI α ) is Caputo's derivative, M, N is the linear and nonlinear operator in μ, and h(μ, I) is the source function.
By taking Shehu transformation, we can write (11) as Now, using inverse Shehu transformation, we get where Now, if ρ is the parameter perturbation, we can write as where ρ is the perturbation parameter and ρ ∈ [0, 1]. e nonlinear term can be decomposed as where H n are He's polynomials of the form ψ 0 , ψ 1 , ψ 2 , . . . , ψ n , and can be determined as Using relations (15) and (16) in (2) and constructing the homotopy, we get On comparing coefficient of ρ on both sides, we obtain ρ 0 : ψ 0 (μ, I) � F(μ, I), e component ψ k (μ, I) can be calculated easily, which leads us to the convergent series rapidly. By taking ρ ⟶ 1, we obtain e obtained result is in the form of series and easily converges to exact solution of the problem.

Test Problems
To show the validity of the suggested technique, the following test problems are solved.
Applying Shehu transform to (21), we have Using inverse Shehu transformation, we get Applying the abovementioned homotopy perturbation technique as in (18)

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Comparing the coefficient of power ρ, we get ρ 0 : ψ 0 (μ, I) � β, , , Now, by taking ρ ⟶ 1, we obtain convergent series form solution as Putting α � 1, we get the same solution, Figure 1 compares the exact solution and approximate solution for the nonlinear fractional-order Fisher equation at α � 1. Figure 2 represents the graph of 2D of exact and analytical solutions and the second graph in Figure 2 shows the different fractional-order graphs of α.

Example. Consider the fractional-order Fisher equation is given by
with initial conditions Applying Shehu transform of (29), we have Using inverse Shehu transformation, we get Applying the abovementioned homotopy perturbation technique as in (18), we get ∞ k�0 ρ k ψ k (μ, I) � 1

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Now, by taking ρ ⟶ 1, we obtain convergent series form solution as ψ(μ, I) � ψ 0 + ψ 1 + ψ 2 + ψ 3 + · · · . ψ(μ, I) � Putting α � 1, we get the same solution Figure 4 compares the exact solution and approximate solution for the nonlinear fractional-order Fisher equation at α � 1. Figure 4 represents the graph of 2D of exact and analytical solutions and the second graph in Figure 4 shows the different fractional-order graphs of α.

Conclusion
In this paper, some computational works have been done to analyze Fisher's equations' fractional view analysis. For this purpose, the Shehu transformation is mixed with the homotopy perturbation method and derived a useful hybrid technique to handle the solution. e graphical representation of the solution of some illustrative examples is shown to be in closed contact. e fractional problem solution is convergent toward the integer-order solutions. Moreover, the accuracy of the proposed method is high and required less number of calculations. e suggested method can solve other fractional-order problems because of its simple and straight forward implementation.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Mathematical Problems in Engineering