Novel Investigation of Fractional-Order Cauchy-Reaction Diffusion Equation Involving Caputo-Fabrizio Operator

In this article, the new iterative transform technique and homotopy perturbation transform method are applied to calculate the fractional-order Cauchy-reaction di ﬀ usion equation solution. Yang transformation is mixed with the new iteration method and homotopy perturbation method in these methods. The fractional derivative is considered in the sense of Caputo-Fabrizio operator. The convection-di ﬀ usion models arise in physical phenomena in which energy, particles, or other physical properties are transferred within a physical process via two processes: di ﬀ usion and convection. Four problems are evaluated to demonstrate, show, and verify the present methods ’ e ﬃ ciency. The analytically obtained results by the present method suggest that the method is accurate and simple to implement.


Introduction
The convection-diffusion equation is a mixture of convection and diffusion equations and identifies physical processes where energy, particles, or other physical properties are transmitted inside a physical process due to two process steps: diffusion and convection. In standard form, the convection-diffusion model is written as follows: where D is the diffusivity, U is the variable term, such as thermal diffusivity for heat flow or mass diffusion coefficient for particle, and v ! is the average velocity that the volume is travelling. For instance, in convection, u might be the density of river in salt and then the flow velocity of water v ! . For example, in a calm lake, v ! would be the average velocity of bubbles rising to the surface due to buoyancy, and U would be the concentration of small bubbles. R defines "sinks" or "sources" of the quantity U. For a chemical species, R > 0 indicates that a chemical reaction is increasing the number of the species, while R > 0 indicates that a chemical reaction is decreasing the number of the species. If thermal energy is generated by friction, R > 0 may occur in heat transport. ∇ denotes gradient, while ∇· denotes divergence. Previously, different techniques have been applied to investigate these models such as Adomian's decomposition technique [1], variational iteration technique [2], Bessel collocation technique [3], and homotopy perturbation technique [4]. In recent decades, fractional derivatives have been used to interpret many physical problems mathematically, and these representations have produced excellent results in modelling real-world issues. Many basic definitions of fractional operators were given by Riesz, Riemann-Liouville, Hadamard, Weyl, Grunwald-Letnikov, Liouville-Caputo, Caputo-Fabrizio, and Atangana-Baleanu, among others [5][6][7][8]. Over the last few years, many nonlinear equations have been developed and widely used in nonlinear physical sciences like chemistry, biology, mathematics, and different branches of physics like plasma physics, condensed matter physics, fluid mechanics, field theory, and nonlinear optics. The exact outcome of nonlinear equations is crucial in determining the characteristics and behaviour of physical processes. Still, it is impossible to find exact results when dealing with linear equations. Many useful methods have been applied to investigate nonlinear fractional partial differential equations, for example, analytical solutions with the help of natural decomposition method of fractional-order heat and wave equations [9], fractional-order partial differential equations with proportional delay [10], fractional-order hyperbolic telegraph equation [11] and fractional-order diffusion equations [12], the variational iterative transform method [13], the homotopy perturbation transform method [14,15], the homotopy analysis transform method [16,17], reduced differential transform method [18,19], qhomotopy analysis transform method [20][21][22][23][24], the finite element technique [25], the finite difference technique [26], and so on [27][28][29][30].
Daftardar-Gejji and Jafari developed a new iterative method of analysis for solving nonlinear equations in 2006 [31,32]. It is the first application of Laplace transformation in iterative technique by Jafari et al. Iterative Laplace transformation method [33] was introduced as a simple method for estimating approximate effects of the fractional partial differential equation system. Iterative Laplace transformation method (NITM) is used to solve linear and nonlinear partial differential equations such as fractional-order Fornberg Whitham equations [34], time-fractional Zakharov Kuznetsov equation [35], and fractional-order Fokker Planck equations [36].
In 1999, He developed the homotopy perturbation method (HPM) [37], which combines the homotopy technique, and the standard perturbation method has been broadly utilized to both linear and nonlinear models [38][39][40]. The homotopy perturbation method is important because it eliminates the need for a small parameter in the model, eliminating the disadvantages of traditional perturbation techniques. The main goal of this paper is to use HPM to solve nonlinear fractional-order Cauchy-reaction diffusion equation using a newly introduced integral transformation known as the "Yang transform" [41]. The suggested technique is applied to analyse two well-known nonlinear partial differential equations. In the context of a quickly convergent series, we obtain a power series solution, and only a few iterations are required to obtain very efficient solutions. There is no need for a discretization technique or linearization for the nonlinear equations, and just a few few can yield a result that can be quickly estimated to utilize these methods.

Basic Definitions
We provide the fundamental definitions that will be used throughout the article. For the purpose of simplification, we write the exponential decay kernel as, KðI, ϱÞ = e ½−℘ðI−ϱ/1−℘Þ . Definition 1. The Caputo-Fabrizio derivative is given as follows [42]: Nð℘Þ is the normalization function with Nð0Þ = Nð1Þ = 1.
Definition 2. The fractional integral Caputo-Fabrizio is given as [42] CF I Definition 3. For Nð℘Þ = 1, the following result shows the Caputo-Fabrizio derivative of Laplace transformation [42]: Definition 4. The Yang transformation of ℙðIÞ is expressed as [42] Y Remark 5. Yang transformation of few useful functions is defined as below.
Proof. From Equation (5), we can achieve another type of the Yang transformation by putting I/v = ζ as Since L½ℙðIÞ = FðvÞ, this implies that Put I = ζ/v in (8), and we have Journal of Function Spaces Thus, from Equation (7), we achieve Also from Equations (5) and (8), we achieve The connections (10) and (11) represent the duality link between the Laplace and Yang transformation.

Lemma 7.
Let ℙðIÞ be a continuous function; then, the Caputo-Fabrizio derivative Yang transformation of ℙðIÞ is define by [43] Proof. The Caputo-Fabrizio fractional Laplace transformation is given by Also, we have that the connection among Laplace and Yang property, i.e., χðvÞ = Fð1/vÞ. To achieve the necessary result, we substitute v by 1/v in Equation (13), and we get The proof is completed.

Algorithm of the HPTM
The procedure of general nonlinear Caputo-Fabrizio fractional partial differential equations is through HPTM. Let us take a general nonlinear Caputo-Fabrizio partial differential equations with nonlinear function NðUðφ, IÞÞ and linear fractional LðUðφ, IÞÞ as [43] CFD ρ where the term gðφ, IÞ shows the source function. Using Yang transformation to Equation (16), one can obtain Implementing inverse Yang transformation, we obtain where the term Vðφ, IÞ shows the source function and with the initial condition. Now, we apply homoptopy perturbation method.
We decompose the nonlinear term NðVðφ, IÞÞ as where H i ðVÞ represents the He's polynomial and is calculated through the following formula: Substituting Equations (19) and (20) in Equation (18), we obtain We obtain the following terms by coefficients comparing of ρ in (22):

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As a result, the obtained solution of Equation (16) can be written as follows:

Error Analysis and Convergence
The following theorems are fundamental on the techniques address the original models [16] error analysis and convergence.
Proof. We make a sequence of ∑ ∞ i=0 V i ðφ, IÞ: To provide the correct outcome, we have to demonstrate that C i ðφ, IÞ forms a "Cauchy sequence." Take, for example, For i, n ∈ N, we acquire Since 0 < σ < 1, and V 0 ðφ, IÞ is bounded, let us take β forms a "Cauchy sequence" in H. It follows that the sequence fV i ðφ, IÞg ∞ i=0 is a convergent sequence with the limit lim i⟶∞ V i ðφ, IÞ = Vðφ, IÞ for ∃Vðφ, IÞ ∈ H . Hence, this ends the proof.
This ends the theorem's proof.

The General Procedure of NITM
The general solution of fractional-order partial differential equation is as follows: where N is nonlinear and M linear functions.
With the initial condition implementing the Yang transformation of Equation (30), we get Applying the Yang differentiation is given to Journal of Function Spaces Using inverse Yang transformation Equation (32), we get By iterative method, we get The nonlinear term N is identified as Putting Equations (35)- (37) in Equation (34), we have obtain the following solution: Lastly, Equations (30) and (31) provide the i-term solution in series form which is expressed as with initial and boundary conditions The methodology consists of applying Yang transformation first on both side in (40) and utilizing the differentiation property of Yang transformation, and we have Using Yang inverse transform, we get Now, we apply the new iterative transform method The series type solution is given as

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The approximate solution is achieved as Now applying the HPTM, we get where the polynomials represent the nonlinear functions are H i ðVÞ. For instance, the terms of He's polynomials are achieved through the recursive relationship H i ðVÞ = D 2 I V i − V i , ∀n ∈ N. Now, as the correspond power coefficients of p is comparison on both sides, the following solution is obtained as follows: Then, the homotopy perturbation method series form solution is defined as The analytical result of the above equation is defined as The exact result of the above equation is Figure 1 shows the analytical solution of two methods at different fractional-order ℘ = 1 and 0.8, and Figure 2 shows separate fractional-order at ℘ = 0:6 and 0.4 with close contact with each other. In Figure 3, the graph shows the different fractional-order ℘ of Example 10.

Example 11. Consider fractional-order Cauchy-reaction diffusion equation as [44]
with initial condition and the exact result is given as Now, we apply the new iterative transform method Journal of Function Spaces The series type solution is given as The approximate solution of the above equation is defined as Now by applying homotopy perturbation transform method, we get

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Comparing the coefficients of power p, we get The HPTM series solution is given as Now ℘ = 1; then, the actual result of Equation (52) is V ðφ, IÞ = e φ 2 +I . Figure 4 shows the analytical solution of two methods at different fractional-order ℘ = 1 and 0.8, and Figure 5 shows the separate fractional-order at ℘ = 0:6 and 0.4 with close contact with each other. In Figure 6, the graph shows the different fractional-order ℘ of Example 11.
Example 12. Consider fractional-order Cauchy-reaction diffusion equation [44] CF D with initial condition The exact result is By using the Yang transformation, we get Now, we apply the new iterative transform method The series type solution is given as 8

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The approximate solution of the above equation is defined as Now, using HPM, we get  Comparing the coefficients of power p, we get Proceeding in this path, the rest of the Vnðφ, IÞ for n ≥ 3 component can be completely recovered and the series solution can therefore be absolutely determined. Eventually, we approximate the numerical solution Vðφ, IÞ to the truncated series.
Now for ℘ = 1, the closed form of the above series is Figure 7 shows the analytical solution of two methods at different fractional-order ℘ = 1 and 0:8, and Figure 8 shows the separate fractional-order at ℘ = 0:6 and 0.4 with close contact with each other. In Figure 9, the graph shows the different fractional-order ℘ of Example 12.
Example 13. Consider fractional-order Cauchy-reaction diffusion equation as [44] CF D with initial condition The exact result is Now, we apply the new iterative transform method 10 Journal of Function Spaces The series type solution is given as Similarly, the remainder of the V i ðφ, IÞ components for n ≥ 4 can be completely achieved, thereby fully evaluating the series solutions. Finally, we estimate the approximate result Vðφ, IÞ by truncated sequence +⋯: ð80Þ Figure 10 shows the analytical solution of two methods at different fractional-order ℘ = 1, 0.8, 0.6, and 0.4 of Example 13. The special case for ℘ = 1, and the above problem close form is given as

Conclusion
The homotopy perturbation transform technique and the Iterative transform method are used in this article to obtain numerical solutions for the fractional-order Cauchy-reaction diffusion equation, which is broadly used in applied sciences as a problem for spatial effects. In physical models, the techniques produce a series of form results that converge quickly. The obtained results in this article are expected to be useful for further analysis of complicated nonlinear physical problems. The calculations for these techniques are very simple and straightforward. As a result, we can conclude that these techniques can be used to solve a variety of nonlinear fractional-order partial differential equation schemes.

Data Availability
The numerical data used to support the findings of this study are included in the article.