# An Effective New Iterative Method to Solve Conformable Cauchy Reaction-Diffusion Equation via the Shehu Transform

Authors: Shahram Rezapour; Muhammad Imran Liaqat; Sina Etemad
Journal: Journal of Mathematics
 (2022)

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## Abstract

For the first time, we establish a new procedure by using the conformable Shehu transform (CST) and an iteration method for solving fractional-order Cauchy reaction-diffusion equations (CRDEs) in the sense of conformable derivative (CD). We call this recommended method the conformable Shehu transform iterative method (CSTIM). To evaluate the efficacy and consistency of CSTIM for conformable partial differential equations (PDEs), the absolute errors of four CRDEs are reviewed graphically and numerically. Furthermore, graphical significances are correspondingly predicted for several values of fractional-order derivatives. The results and examples establish that our new method is unpretentious, accurate, valid, and capable. CSTIM does not necessarily use He’s polynomials and Adomian polynomials when solving nonlinear problems, so it has a strong advantage over the homotopy analysis and Adomian decomposition methods. The convergence and absolute error analysis of the series solutions is also offered.


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## Body

## 1. Introduction

Fractional calculus is well used for proceedings with any order of derivatives or integrals. It is the preferment of integer derivatives and integrals. There are innumerable categories of definitions for fractional-order derivatives. Although the most frequent fractional derivatives, for instance, Riemann–Liouville, Caputo derivative, and Caputo–Fabrizio derivative, are utilized by numerous researchers for studying numerical solutions of distributed order differential equations [1], mathematical modeling of COVID-19 [2], analysis of cancer tumor [3], analytical studies on the physical phenomena [4], and analysis of different diseases [5, 6] and viruses [7], but there are several dissertations in the literature that state that these operators have a bit of a limitation [8, 9].In 2014 [10], Khalil et al. offered a novel operator, so-called conformable derivative (CD), which contains several of the axioms that the existing operators do not satisfy [8, 9]. The CD is relatively similar to the derivative in the classical limit method, and it is rather uncomplicated to manage. Consequently, it has been acknowledged so swiftly and has been the subject of numerous dissertations for scientists [11–13].In the disciplines of science and engineering, we find natural and physical events that, when characterized by mathematical models, happen to be differential equations (DEs). For instance, the equations of Kaup–Kupershmidt (KK), Burgers–Huxley equation, and the deflection of a beam are characterized by DEs [14–16]. Consequently, the solutions of DEs are required. Many DEs arising in applications are so intricate that it is sometimes impractical to have close-form solutions. Numerical methods provide a powerful alternative tool for solving the DEs under the given initial conditions. For example, in the past couple of years, various methods have been presented to solve DEs containing the Elzaki residual power series method [17], the Laplace decomposition method [18], the fractional natural decomposition approach [19], the variational iteration technique [20], the operational matrix scheme [21], the homotopy analysis approach [22], the Sumudu decomposition method [23], the Lie-group integrator based on GL(4, R) [24], and the residual power series method [25].The reaction-diffusion equations (RDEs) explain numerous nonlinear structures in physics, chemistry, ecology, and biology [26–28]. The RDE is described in the following procedure:(1)∂ξν,τ∂τ=Δξ+φξ,∇ξ;Ω,τ.The expression ofΔξ is diffusion term and φξ,∇ξ;Ω,τ is the reaction function.In this study, we discussed the conformable CRDEs of one-dimensional systems, which is described as follows:(2)Tτλξν,τ=σDννξν,τ+γν,τξν,τ,0<λ≤1,where Tτλ is the CD.Ifλ=1, then, it converts into a classical RDE; consequently, ξν,τ is the concentration, γν,τ is the reaction parameter, and σ>0 is the diffusion coefficient, by means of the preliminary and boundaries circumstances:(3)ξν,0=θν,ν∈ℝ,(4)ξ0,τ=θ1τ,Dνξ0,τ=θ2τ,τ∈ℝ+.Equations (2) and (3) are named as the characteristic Cauchy prototypal in the domain Ω=ℝ×ℝ+, and equations (2) and (4) are termed as the noncharacteristic Cauchy equation in the domain Ω=ℝ+×ℝ.In this study, we recognized an effectual CSTIM, which is the collaboration of the ST in the sense of CD and the iterative method familiarized by Daftardar-Gejji and Jafari [29] for gaining approximate and closed-form solutions to CRDEs. The superiority of this new method, which we planned, is that it makes the calculation effortless and extremely precise, allowing us to approximate the definite result.The subsequent organization of this work is structured as follows. In Section2, we recall some straightforward definitions and consequences dealing with the CD and ST. The fundamental recommendation beyond the CSTIM with convergence and absolute error analysis for conformable CRDEs is demonstrated in Section 3. In Section 4, we demonstrated numerical examples of CRDEs to exemplify the competency, potential, and straightforwardness of the new method. Lastly, in Section 5, the consequences are gathered in the conclusions.


## 2. Fundamental Concepts

In this section, we review some necessary definitions, theorems, and mathematical preliminaries concerning CD and ST that will be used in this work.Definition 1 (see [30]).
For given a functionξ:0,∞⟶ℝ, the CD of ξ of order λ is formulated as(5)Tτλξτ=limε⟶0ξτ+ετ1−λ−ξτε,for τ>0, and λ∈0,1. If ξ is ξ-differentiable in some 0,ϒ, ϒ>0, and limτ⟶0+Tλξτ occur subsequently, it becomes(6)Tλξ0=limτ⟶0+Tλξτ.Definition 2 (see [31]).
Shehu transform (ST) is a new integral transformation which is defined for function of exponential order. We take a function in the setη defined by(7)η=ξτ:∃Ξ1,Ξ2>0s.t.ξτ<E^eτ/Ξi,ifτ∈0,∞.
The ST which is represented byϱ⋅ for a function ξτ is defined as(8)ϱξτ=Y^ϖ,ω=∫0∞ξτe−ϖτ/ωdτ,τ>0.
The ST of a functionξτ is Y^ϖ,ω. Then, ξτ is called the inverse of Y^ϖ,ω which is expressed as ϱ−1Y^ϖ,ω=ξτ, for τ≥0, ϱ−1 is called the inverse ST.Definition 3 (see [32]).
Let0<λ≤1 and ξ:0,∞⟶ℝ be a real value function. Then, the conformable Shehu transform (CST) of order λ is defined by(9)ϱλξ=Y^ϖ,ω=∫0∞eϖτλ/ωλξτλ−1dτ.Definition 4 (see [33]).
The Mittag–Leffler function is defined as follows:(10)Eλτ=∑α=0∞ταΓαλ+1,λ∈ℂ,Reλ>0.Theorem 1 (see [34]).
Let0<λ≤1 and ξ1 and ξ2 be λ-differentiable at a point τ>0. Then,(1)
Tτλχ1ξ1+χ2ξ2=χ1Tτλξ1+χ2Tτλξ2,∀χ1,χ2∈ℝ.\(2)
Tτλτχ=χτχ−λ,∀χ∈ℝ(3)
Tτλc=0,c∈ℝ(4)
Tτλξ1ξ2=ξ1Tτλξ2+ξ2Tτλξ1(5)
Tτλξ1/ξ2=ξ2Tτλξ1−ξ1Tτλξ2/ξ22
In addition, ifξ is differentiable, then Tτλξτ=τ1−λdξ/dττ.Theorem 2 (see [32]).
Letξ:0,∞⟶ℝ be a real valued map and 0<λ≤1; then, we have(11)ϱλTτλξν,τ=ϖωϱλξν,τ−ξν,0.Theorem 3 (see [32]).
Letm,w and c∈ℝ and 0<λ≤1. We have the following:(1)
ϱλc=cω/ϖ(2)
ϱλτw=Y^ϖ,ω=λw/λω/ϖw/λ+1Γ1+w/λ(3)
ϱλsinmτλ/λ=Y^ϖ,ω=mω2/ϖ2+m2ω2,ϖ/ω>0(4)
ϱλcosmτλ/λ=Y^ϖ,ω=ϖω/ϖ2+m2ω2,ϖ/ω>0(5)
ϱλsinhmτλ/λ=Y^ϖ,ω=mω2/ϖ2−m2ω2,ϖ/ω>m(6)
ϱλcoshmτλ/λ=Y^ϖ,ω=ϖω/ϖ2−m2ω2,ϖ/ω>m
In Section3, we present a novel computational methodology for solving conformable CRDEs utilizing the CST and iteration method. The convergence and absolute error analysis of the series solutions obtained through the mentioned method are also implemented.


## 3. Methodology of the CSTIM with the Convergence and Error Analysis

In this section, we illustrate the basic idea of CSTIM to solve conformable CRDEs.


### 3.1. Methodology of the CSTIM

We contemplate the following equation in the common operator system by means of the preliminary condition to demonstrate the straightforward idea of the CSTIM for the CRDEs:(12)Tτλξν,τ+ϑξν,τ+ℑξν,τ=Φν,τ,ξν,0=θν,0<λ≤1,where Tτλ is the CD, ϑ is a linear operator, ℑ is nonlinear operator, and Φν,τ is continuous function. Employing CST at both sides of equation (12), we obtain as follows:(13)ϱλTτλξν,τ+ϑξν,τ+ℑξν,τ=ϱλΦν,τ.By consuming the linear axiom of CST, we get as follows:(14)ϱλTτλξν,τ+ϱλϑξν,τ+ϱλℑξν,τ=ϱλΦν,τ.Utilizing the following definition of CST(15)ϱλTτλξν,τ=ϖωϱλξν,τ−ξν,0,and using equations (15) in (14),(16)ϖωϱλξν,τ−ξν,0+ϱλϑξν,τ+ϱλℑξν,τ=ϱλΦν,τ.By simplifying equation (16), we obtain as follows:(17)ϱλξν,τ=ωϖξν,0+ϱλΦν,τ−ωϖϱλϑξν,τ−ωϖϱλℑξν,τ.By utilizing the inverse CST on both sides of equation (17), we obtain as(18)ξν,τ=ϱλ−1ωϖξν,0+ϱλΦν,τ−ϱλ−1ωϖϱλϑξν,τ−ϱλ−1ωϖϱλℑξν,τ.Next, assume the following:(19)Θν,τ=ϱλ−1ωϖξν,0+ϱλΦν,τ,Ψξν,τ=−ϱλ−1ωϖϱλϑξν,τ,ℵξν,τ=−ϱλ−1ωϖϱλℑξν,τ.As a result, equation (18) can be written as(20)ξν,τ=Θν,τ+Ψξν,τ+ℵξν,τ,where Θν,τ is a recognized function and ℵ and Ψ are specified nonlinear and linear operators of ξν,τ correspondingly. The outcome of equation (20) can be represented in the expansion form:(21)ξν,τ=∑α=0∞ξαν,τ.Also, we take(22)Ψ∑α=0∞ξαν,τ=∑α=0∞Ψξαν,τ.The nonlinear operatorℵ is written as [29](23)ℵ∑α=0∞ξαν,τ=ℵξ0ν,τ+∑α=0∞ℵ∑β=0αξβν,τ−ℵ∑β=0α−1ξβν,τ.Therefore, equation (20) can be characterized as this arrangement:(24)∑α=0∞ξαν,τ=Θν,τ+∑α=0∞Ψξαν,τ+ℵξ0ν,τ+∑α=0∞ℵ∑β=0αξβν,τ−ℵ∑β=0α−1ξβν,τ.Defining the recurrence relation by using equation (24), we set(25)ξ0ν,τ=Θν,τ,ξ1ν,τ=Ψξ0ν,τ+ℵξ0ν,τ,ξk+1ν,τ=Ψξkν,τ+ℵξ0ν,τ+ξ1ν,τ+⋯+ξkν,τ−ℵξ0ν,τ+ξ1ν,τ+⋯+ξk−1ν,τ.So, we have the following:(26)ξ1ν,τ+ξ2ν,τ+⋯+ξk+1ν,τ=Ψξ0ν,τ+ξ1ν,τ+⋯+ξkν,τ+ℵξ0ν,τ+ξ1ν,τ+⋯+ξkν,τ.Namely,(27)∑α=0∞ξαν,τ=Θν,τ+Ψ∑α=0∞ξαν,τ+ℵ∑α=0∞ξαν,τ.Thekth approximate solution of equation (12) is given by(28)ξkν,τ=ξ0ν,τ+ξ1ν,τ+⋯+ξk−1ν,τ.


### 3.2. Convergence and Absolute Error Analysis of CSTIM

In this section, we illustrate the convergence and absolute error analysis of CSTIM for the conformable CRDEs.The following theorem explains and governs the condition for the convergence of the expansion solution.Theorem 4.
Letϒ be a Banach space; then, the expansion result of ξν,τ is convergent if ∃ζ∈0,1 s.t. ξαν,τ≤ζξα−1ν,τ, ∀α∈ℕ.Proof.
Consider(29)Skν,τ=ξ0ν,τ+ξ1ν,τ+ξ2ν,τ+⋯+ξkν,τ.
It is essential to validate that the series ofkth partial sums Sk is a Cauchy series in the Banach space. For this, we have(30)Sk+1ν,τ−Skν,τ≤ξk+1ν,τ≤ζξkν,τ≤ζ2ξk−1ν,τ≤⋯≤ζk+1ξ0ν,τ.
For everyk,ℓ∈ℕ and k≤ℓ and by using equation (30) and the triangle inequality, we obtain(31)Skν,τ−Sℓν,τ≤Sℓ+1ν,τ−Sℓν,τ+Sℓ+2ν,τ−Sℓ+1ν,τ+⋯+Skν,τ−Sk+1ν,τ≤Sℓ+1ν,τ−Sℓν,τ+Sℓ+2ν,τ−Sℓ+1ν,τ‖+⋯+‖Skν,τ−Sk+1ν,τ≤ζℓ+1ξ0ν,τ+ζℓ+2ξ0ν,τ+⋯+ζkξ0ν,τ≤ζℓ+11+ζ+ζ2+⋯+ζk−ℓ−1ξ0ν,τ≤ζℓ+11−ζk−ℓ1−ζξ0ν,τ.
Since0<ζ≤1, then 1−ζk−ℓ≤1. Therefore, from inequality (31), we have(32)Skν,τ−Sℓν,τ≤ζℓ+11−ζξ0ν,τ.
So,ξ0ν,τ is bounded. Hence, we have(33)limx⟶k,ℓ⟶∞∞Skν,τ−Sℓν,τ=0.
Therefore,Sk is a Cauchy series in Banach space, so the expansion solution of equation (12) is convergent.
We propose an absolute error study of the planned scheme in the successive theorem.Theorem 5.
Letξν,τ be the approximate solution of the truncated finite series ξν,τ=∑α=0δξαν,τ. Suppose that it is reasonable to acquire a real number Ξ∈0,1, s.t. ξα+1ν,τ≤Ξξαν,τ, ∀α∈ℕ. Moreover, the greatest absolute error is(34)ξν,τ−∑α=0δξαν,τ≤Ξδ+11−Ξξ0ν,τ.Proof.
Let the series∑α=0δξαν,τ be finite. Then,(35)ξν,τ−∑α=0δξαν,τ=∑α+1∞ξαν,τ≤∑δ+1∞ξαν,τ≤∑δ+1∞Ξδξ0ν,τ≤Ξδ+11+Ξ+Ξ2+Ξ3+⋯ξ0ν,τ≤Ξδ+11−Ξξ0ν,τ,and the proof is completed.
The applicability of the recommended method is examined in Section4.


## 3.1. Methodology of the CSTIM

We contemplate the following equation in the common operator system by means of the preliminary condition to demonstrate the straightforward idea of the CSTIM for the CRDEs:(12)Tτλξν,τ+ϑξν,τ+ℑξν,τ=Φν,τ,ξν,0=θν,0<λ≤1,where Tτλ is the CD, ϑ is a linear operator, ℑ is nonlinear operator, and Φν,τ is continuous function. Employing CST at both sides of equation (12), we obtain as follows:(13)ϱλTτλξν,τ+ϑξν,τ+ℑξν,τ=ϱλΦν,τ.By consuming the linear axiom of CST, we get as follows:(14)ϱλTτλξν,τ+ϱλϑξν,τ+ϱλℑξν,τ=ϱλΦν,τ.Utilizing the following definition of CST(15)ϱλTτλξν,τ=ϖωϱλξν,τ−ξν,0,and using equations (15) in (14),(16)ϖωϱλξν,τ−ξν,0+ϱλϑξν,τ+ϱλℑξν,τ=ϱλΦν,τ.By simplifying equation (16), we obtain as follows:(17)ϱλξν,τ=ωϖξν,0+ϱλΦν,τ−ωϖϱλϑξν,τ−ωϖϱλℑξν,τ.By utilizing the inverse CST on both sides of equation (17), we obtain as(18)ξν,τ=ϱλ−1ωϖξν,0+ϱλΦν,τ−ϱλ−1ωϖϱλϑξν,τ−ϱλ−1ωϖϱλℑξν,τ.Next, assume the following:(19)Θν,τ=ϱλ−1ωϖξν,0+ϱλΦν,τ,Ψξν,τ=−ϱλ−1ωϖϱλϑξν,τ,ℵξν,τ=−ϱλ−1ωϖϱλℑξν,τ.As a result, equation (18) can be written as(20)ξν,τ=Θν,τ+Ψξν,τ+ℵξν,τ,where Θν,τ is a recognized function and ℵ and Ψ are specified nonlinear and linear operators of ξν,τ correspondingly. The outcome of equation (20) can be represented in the expansion form:(21)ξν,τ=∑α=0∞ξαν,τ.Also, we take(22)Ψ∑α=0∞ξαν,τ=∑α=0∞Ψξαν,τ.The nonlinear operatorℵ is written as [29](23)ℵ∑α=0∞ξαν,τ=ℵξ0ν,τ+∑α=0∞ℵ∑β=0αξβν,τ−ℵ∑β=0α−1ξβν,τ.Therefore, equation (20) can be characterized as this arrangement:(24)∑α=0∞ξαν,τ=Θν,τ+∑α=0∞Ψξαν,τ+ℵξ0ν,τ+∑α=0∞ℵ∑β=0αξβν,τ−ℵ∑β=0α−1ξβν,τ.Defining the recurrence relation by using equation (24), we set(25)ξ0ν,τ=Θν,τ,ξ1ν,τ=Ψξ0ν,τ+ℵξ0ν,τ,ξk+1ν,τ=Ψξkν,τ+ℵξ0ν,τ+ξ1ν,τ+⋯+ξkν,τ−ℵξ0ν,τ+ξ1ν,τ+⋯+ξk−1ν,τ.So, we have the following:(26)ξ1ν,τ+ξ2ν,τ+⋯+ξk+1ν,τ=Ψξ0ν,τ+ξ1ν,τ+⋯+ξkν,τ+ℵξ0ν,τ+ξ1ν,τ+⋯+ξkν,τ.Namely,(27)∑α=0∞ξαν,τ=Θν,τ+Ψ∑α=0∞ξαν,τ+ℵ∑α=0∞ξαν,τ.Thekth approximate solution of equation (12) is given by(28)ξkν,τ=ξ0ν,τ+ξ1ν,τ+⋯+ξk−1ν,τ.


## 3.2. Convergence and Absolute Error Analysis of CSTIM

In this section, we illustrate the convergence and absolute error analysis of CSTIM for the conformable CRDEs.The following theorem explains and governs the condition for the convergence of the expansion solution.Theorem 4.
Letϒ be a Banach space; then, the expansion result of ξν,τ is convergent if ∃ζ∈0,1 s.t. ξαν,τ≤ζξα−1ν,τ, ∀α∈ℕ.Proof.
Consider(29)Skν,τ=ξ0ν,τ+ξ1ν,τ+ξ2ν,τ+⋯+ξkν,τ.
It is essential to validate that the series ofkth partial sums Sk is a Cauchy series in the Banach space. For this, we have(30)Sk+1ν,τ−Skν,τ≤ξk+1ν,τ≤ζξkν,τ≤ζ2ξk−1ν,τ≤⋯≤ζk+1ξ0ν,τ.
For everyk,ℓ∈ℕ and k≤ℓ and by using equation (30) and the triangle inequality, we obtain(31)Skν,τ−Sℓν,τ≤Sℓ+1ν,τ−Sℓν,τ+Sℓ+2ν,τ−Sℓ+1ν,τ+⋯+Skν,τ−Sk+1ν,τ≤Sℓ+1ν,τ−Sℓν,τ+Sℓ+2ν,τ−Sℓ+1ν,τ‖+⋯+‖Skν,τ−Sk+1ν,τ≤ζℓ+1ξ0ν,τ+ζℓ+2ξ0ν,τ+⋯+ζkξ0ν,τ≤ζℓ+11+ζ+ζ2+⋯+ζk−ℓ−1ξ0ν,τ≤ζℓ+11−ζk−ℓ1−ζξ0ν,τ.
Since0<ζ≤1, then 1−ζk−ℓ≤1. Therefore, from inequality (31), we have(32)Skν,τ−Sℓν,τ≤ζℓ+11−ζξ0ν,τ.
So,ξ0ν,τ is bounded. Hence, we have(33)limx⟶k,ℓ⟶∞∞Skν,τ−Sℓν,τ=0.
Therefore,Sk is a Cauchy series in Banach space, so the expansion solution of equation (12) is convergent.
We propose an absolute error study of the planned scheme in the successive theorem.Theorem 5.
Letξν,τ be the approximate solution of the truncated finite series ξν,τ=∑α=0δξαν,τ. Suppose that it is reasonable to acquire a real number Ξ∈0,1, s.t. ξα+1ν,τ≤Ξξαν,τ, ∀α∈ℕ. Moreover, the greatest absolute error is(34)ξν,τ−∑α=0δξαν,τ≤Ξδ+11−Ξξ0ν,τ.Proof.
Let the series∑α=0δξαν,τ be finite. Then,(35)ξν,τ−∑α=0δξαν,τ=∑α+1∞ξαν,τ≤∑δ+1∞ξαν,τ≤∑δ+1∞Ξδξ0ν,τ≤Ξδ+11+Ξ+Ξ2+Ξ3+⋯ξ0ν,τ≤Ξδ+11−Ξξ0ν,τ,and the proof is completed.
The applicability of the recommended method is examined in Section4.


## 4. Numerical Examples and Concluding Remarks

In this section, four problems of CRDEs are established to illustrate the performance and appropriateness of the recommended method.Example 1.
Consider the following Cauchy reaction-diffusion equation [35]:(36)Tτλξν,τ=ξννν,τ−ξν,τ,0<λ≤1,subject to the initial condition,(37)ξν,0=e−ν+ν.
Applying CST on both sides of equation (36), we obtain(38)ϱλTτλξν,τ=ϱλξννν,τ−ξν,τ.
By using the linear property of CST, the above equation becomes as(39)ϱλTτλξν,τ=ϱλξννν,τ−ϱλξν,τ.
The CD definition of ST implies that(40)ϱλξν,τ=ωϖξν,0+ωϖϱλξννν,τ−ωϖϱλξν,τ.
By applying the inverse CST on both sides of equation (40), we reach to(41)ξν,τ=ϱλ−1ωϖξν,0+ϱλ−1ωϖϱλξννν,τ−ϱλ−1ωϖϱλξν,τ.
According to the CSTIM, we have(42)ξν,τ=∑α=0∞ξαν,τ,Ψξν,τ=ϱλ−1ωϖϱλξννν,τ−ϱλ−1ωϖϱλξν,τ.
Using equations (42) in (41), it becomes(43)∑α=0∞ξαν,τ=ϱλ−1ωϖξν,0+ϱλ−1ωϖϱλDνν∑α=0∞ξαν,τ−ϱλ−1ωϖϱλ∑α=0∞ξαν,τ.
By using the initial value in the above equation, we obtain(44)∑α=0∞ξαν,τ=ϱ−1ωϖe−ν+ν+ϱ−1ωϖϱDνν∑α=0∞ξαν,τ−ϱ−1ωϖϱ∑α=0∞ξαν,τ.
By iteration, the following results are obtained from equation (44):(45)ξ0ν,τ=e−ν+ν,ξ1ν,τ=−ντλλ,ξ2ν,τ=ντ2λ2λ2,ξ3ν,τ=−ντ3λ6λ3,ξ4ν,τ=ντ4λ24λ4,ξ5ν,τ=−ντ5λ120λ5.
Therefore, we have the5th approximate solution of the problem equations (36) and (37) as follows:(46)ξ5ν,τ=e−ν+ν1−τλλ+τ2λ2!λ2−τ3λ3!λ3+τ4λ4!λ4−τ5λ5!λ5.
Whenλ=1, equation (46) becomes as(47)ξ5ν,τ=e−ν+ν1−τ+τ22!−τ33!+τ44!−τ55!.
Equation (47) matches with first six terms of e−ν+νE−τ, so the exact solution of equations (36) and (37) equals to ξν,τ=e−ν+νE−τ.
Figure1 depicts the performance of the 5th-approximate and exact solutions of equation (36) and (37) for different values of λ, when ν=1, in the interval τ∈0,1. Undoubtedly, the results in instances of fractional values of λ converge to the results in case of λ=1. As well, the approximate solutions are consistent with the exact solution at λ=1 and this confirms the efficiency and exactness of the recommended method.
Figure2 demonstrates the absolute errors in the interval τ∈0,1 over the 5th-approximation and accurate solutions of equations (36) and (37) at λ=1 when ν=1, attained by means of CSTIM. From the figure, it can be predicted that the approximate solution is very close to the exact solution and this confirms the efficacy of the CSTIM.
Error functions are accessible to perceive the accuracy and applicability of the scheme. To demonstrate the precision and ability of CSTIM, we introduced a absolute error function.
Table1 displays the absolute errors at reasonable introduced grid points in the interval 0,1 amongst the 5th-approximate and exact solutions of equations (36) and (37) at λ=1, when ν=0.5, attained by means of CSTIM. Table 1 shows that the approximate solution is quite near to the precise solution, indicating that the proposed strategy is effective.Figure 1
Approximate solutions for different values.Figure 2
The absolute errors of Example1.Table 1
The absolute errors for Example1 at λ=1 with ν=0.5.
τξ−ξ50.081.79984471770922×10−100.161.138977245851435×10−80.241.28293276668856×10−70.327.128835121106292×10−70.400.0000026896844863566470.480.000007944223070488477Example 2.
Consider the following Cauchy reaction-diffusion equation [28]:(48)Tτλξν,τ=ξννν,τ−1+4ν4ξν,τ,0<λ≤1,subject to the initial condition,(49)ξν,0=eν2.
Applying CST on both sides of equation (48), we obtain(50)ϱλTτλξν,τ=ϱλξννν,τ−1+4ν4ξν,τ.
By using the linear property of CST, the above equation becomes(51)ϱλTτλξν,τ=ϱλξννν,τ−ϱλ1+4ν4ξν,τ.
The CD definition of ST implies that(52)ϱλξν,τ=ωϖξν,0+ωϖϱλξννν,τ−ωϖϱλ1+4ν4ξν,τ.
By applying the inverse CST on both sides of equation (52), we reach to(53)ξν,τ=ϱλ−1ωϖξν,0+ϱλ−1ωϖϱλξννν,τ−ϱλ−1ωϖϱλ1+4ν4ξν,τ.
According to the CSTIM, we have(54)ξν,τ=∑α=0∞ξαν,τ,Ψξν,τ=ϱλ−1ωϖϱλξννν,τ−ϱλ−1ωϖϱλ1+4ν4ξν,τ.
Using equation (54) in (53), it becomes(55)∑α=0∞ξαν,τ=ϱλ−1ωϖξν,0+ϱλ−1ωϖϱλDνν∑α=0∞ξαν,τ−ϱλ−1ωϖϱλ1+4ν4∑α=0∞ξαν,τ.
By using the initial value in the above equation, we obtain(56)∑α=0∞ξαν,τ=ϱ−1ωϖeν2+ϱ−1ωϖϱDνν∑α=0∞ξαν,τ−ϱ−1ωϖϱ1+4ν4∑α=0∞ξαν,τ.
By iteration, the following results are obtained from equation (56):(57)ξ0ν,τ=eν2,ξ1ν,τ=eν2τλλ,ξ2ν,τ=eν2τ2λ2λ2,ξ3ν,τ=eν2τ3λ6λ3,ξ4ν,τ=eν2τ4λ24λ4,ξ5ν,τ=eν2τ5λ120λ5.
Therefore, we have the5th approximate solution of the problem equations (48) and (49) as follows:(58)ξ5ν,τ=eν21+τλλ+τ2λ2!λ2+τ3λ3!λ3+τ4λ4!λ4+τ5λ5!λ5.
Whenλ=1, equation (58) becomes as(59)ξ5ν,τ=eν21+τ+τ22!+τ33!+τ44!+τ55!.
Equation (59) matches with first six terms of eν2Eτ, so the exact solution of equations (48) and (49) equals to ξν,τ=eν2Eτ.
Figure3 depicts the performance of the 5th-approximate and exact solutions of equations (48) and (49) for different values of λ, when ν=1, in the interval τ∈0,1. Definitely, the results in instances of fractional values of λ converge to the results in case of λ=1. As well, the approximate solutions are consistent with the exact solution at λ=1, and this once more confirms the efficiency and exactness of the recommended method.
Figure4 demonstrates the absolute errors in the interval 0,1 over the 5th-approximation and accurate solutions of equations (48) and (49) at λ=1 when ν=1, attained by means of CSTIM. From the figure, it can be predicted that the approximate solution is close to the exact one, which confirms the efficacy of the CSTIM.
Table2 displays the absolute errors at reasonable introduced grid points in the interval 0,1 amongst the 5th-approximate and exact solutions of equation (48) and (49) at λ=1 when ν=0.5, attained by means of CSTIM. Table 2 shows that the approximate solution is very near to the exact solution, indicating that the proposed plan is appropriate.Figure 3
Approximate solutions for different values.Figure 4
The absolute errors of Example2.Table 2
The absolute errors for Example2 at λ=1 with ν=0.5.
τξ−ξ50.084.728961666700115×10−100.163.061777098523066×10−80.243.528519882944891×10−70.320.0000020060448380210260.400.000007743924678879990.480.000023402073148215408Example 3.
Consider the following Cauchy reaction-diffusion equation [36]:(60)Tτλξν,τ=ξννν,τ−4ν4−2τ+2ξν,τ,0<λ≤1,subject to the initial condition,(61)ξν,0=eν2.
Applying CST on both sides of equation (60), we obtain(62)ϱλTτλξν,τ=ϱλξννν,τ−4ν4−2τ+2ξν,τ.
By using the linear property of CST, the above equation becomes as(63)ϱλTτλξν,τ=ϱλξννν,τ−ϱλ4ν4−2τ+2ξν,τ.
The CD definition of ST implies that(64)ϱλξν,τ=ωϖξν,0+ωϖϱλξννν,τ−ωϖϱλ4ν4−2τ+2ξν,τ.
By applying the inverse CST on both sides of equation (64), we reach to(65)ξν,τ=ϱλ−1ωϖξν,0+ϱλ−1ωϖϱλξννν,τ−ϱλ−1ωϖϱλ4ν4−2τ+2ξν,τ.
According to the CSTIM, we have(66)ξν,τ=∑α=0∞ξαν,τ,Ψξν,τ=ϱλ−1ωϖϱλξννν,τ−ϱλ−1ωϖϱλ4ν4−2τ+2ξν,τ.
Using equation (66) in (65), it becomes(67)∑α=0∞ξαν,τ=ϱλ−1ωϖξν,0+ϱλ−1ωϖϱλDνν∑α=0∞ξαν,τ−ϱλ−1ωϖϱλ4ν4−2τ+2∑α=0∞ξαν,τ.
By using the initial value in the above equation, we obtain(68)∑α=0∞ξαν,τ=ϱλ−1ωϖeν2+ϱλ−1ωϖϱλDνν∑α=0∞ξαν,τ−ϱλ−1ωϖϱλ4ν4−2τ+2∑α=0∞ξαν,τ.
By iteration, the following results are obtained from equation (68):(69)ξ0ν,τ=eν2,ξ1ν,τ=eν2τ2λλ,ξ2ν,τ=eν2τ4λ2λ2,ξ3ν,τ=eν2τ6λ6λ3,ξ4ν,τ=eν2τ8λ24λ4,ξ5ν,τ=eν2τ10λ120λ5.
Therefore, we have the5th approximate solution of the problem equations (60) and (61) as follows:(70)ξ5ν,τ=eν21+τ2λλ+τ2λ22!λ2+τ2λ33!λ3+τ2λ44!λ4+τ2λ55!λ5.
Whenλ=1, equation (70) becomes as(71)ξ5ν,τ=eν21+τ2+τ222!+τ233!+τ244!+τ255!.
Equation (71) matches with first six terms of eν2Eτ2, so the exact solution of equations (60) and (61) equals to ξν,τ=eν2Eτ2.
Figure5 depicts the performance of the 5th-approximate and exact solutions of equations (60) and (61) for different values of λ, when ν=1, in the interval τ∈0,1. Undoubtedly, the results in instances of fractional values of λ converge to the results in case of λ=1. As well, the approximate solutions are consistent to the exact solution at λ=1, and this once more confirms the efficiency and exactness of the recommended method.
Figure6 demonstrates the absolute errors in the interval 0,1 over the 5th-approximation and accurate solutions of equations (60) and (61) at λ=1 when ν=1, attained by means of CSTIM. The approximate solution is predicted to be near to the precise solution in the figure, demonstrating that the CSTIM is efficient.
Table3 displays the absolute errors at reasonable introduced grid points in the interval 0,1 amongst the 5th-approximate and exact solutions of equations (60) and (61) at λ=1, when ν=0.5, attained by means of CSTIM. From Table 3, it can be perceived that the approximate solutions are close to the exact solutions, which confirms the efficacy of the recommended method numerically.Figure 5
Approximate solutions for different values.Figure 6
The absolute errors of Example3.Table 3
The absolute errors for Example3 at λ=1 with ν=0.5.
τξ−ξ50.082.220446049250313×10−160.165.03819208574896×10−130.246.566902577276323×10−110.322.086551154079075×10−90.403.061777054114145×10−80.482.758094046484416×10−7Example 4.
In this example, we consider the nonlinear fractional Cauchy reaction-diffusion equation [37]:(72)Tτλξν,τ=ξννν,τ−ξνν,τ+ξν,τξννν,τ−ξ2ν,τ+ξν,τ,0<λ≤1,subject to the initial condition,(73)ξν,0=eν.
Applying CST on both sides of equation (72), we obtain(74)ϱλTτλξν,τ=ϱλξννν,τ−ξνν,τ+ξν,τξννν,τ−ξ2ν,τ+ξν,τ.
By using the linear property of CST, the above equation becomes as(75)ϱλTτλξν,τ=ϱλξννν,τ−ξνν,τ+ξν,τ+ϱλξν,τξννν,τ−ξ2ν,τ.
The CD definition of ST implies that(76)ϱλξν,τ=ωϖξν,0+ωϖϱλξννν,τ−ξνν,τ+ξν,τ+ωϖϱλξν,τξννν,τ−ξ2ν,τ.
By applying the inverse CST on both sides of equation (76) and by the initial condition, we reach to(77)ξν,τ=ϱλ−1ωϖeν+ϱλ−1ωϖϱλDννξν,τ−Dνξν,τ+ξν,τ+ϱλ−1ωϖϱλξν,τξννν,τ−ξ2ν,τ.
According to the CSTIM, we have(78)ξν,τ=∑α=0∞ξαν,τ,Ψξν,τ=ϱλ−1ωϖϱλDννξν,τ−Dνξν,τ+ξν,τ,ℵξν,τ=ϱλ−1ωϖϱλξν,τξννν,τ−ξ2ν,τ.
Using equations (78) in (77), it becomes(79)∑α=0∞ξαν,τ=ϱλ−1ωϖeν+ϱλ−1ωϖϱλDνν∑α=0∞ξαν,τ−Dν∑α=0∞ξαν,τ+∑α=0∞ξαν,τ−ϱλ−1ωϖϱλ∑α=0∞ξαν,τDνν∑α=0∞ξαν,τ−∑α=0∞ξαν,τ∑α=0∞ξαν,τ.
By iteration, the following results are obtained from equation (79):(80)ξ0ν,τ=eν,ξ1ν,τ=eντλλ,ξ2ν,τ=eντ2λ2λ2,ξ3ν,τ=eντ3λ6λ3,ξ4ν,τ=eντ4λ24λ4,ξ5ν,τ=eντ5λ120λ5.
Therefore, we have the5th approximate solution of the problem equations (72) and (73) as follows:(81)ξ5ν,τ=eν1+τλλ+τλ22!λ2+τλ33!λ3+τλ44!λ4+τλ55!λ5.
Whenλ=1, equation (81) becomes as(82)ξ5ν,τ=eν1+τ+τ22!+τ33!+τ44!+τ55!.
Equation (82) matches with first six terms of eνEτ, so the exact solution of equations (72) and (73) equals to ξν,τ=eνEτ.
Figure7 depicts the performance of the 5th-approximate and exact solutions of equations (72) and (73) for different values of λ, when ν=1, in the interval τ∈0,1. Undoubtedly, the results in instances of fractional values of λ converge to the results in case of λ=1. We get the same results from Example 4 as we found in Examples 1, 2, and 3.
Figure8 demonstrates the absolute errors in the interval 0,1 over the 5th-approximation and accurate solutions of equations (72) and (73) at λ=1 when ν=1, attained by means of CSTIM. One can perceive the equivalent verdicts depicted for Examples 1, 2, and 3.
Table4 displays the absolute errors at reasonable introduced grid points in the interval 0,1 amongst the 5th-approximate and exact solutions of equations (72) and (73) at λ=1, when ν=0.5, attained by means of CSTIM. From Table 4, we yield the same outcomes as in Examples 1, 2, and 3.Figure 7
Approximate solutions for different values.Figure 8
The absolute errors of Example4.Table 4
The absolute errors for Example4 at λ=1 with ν=0.5.
τξ−ξ50.086.072105040999531×10−100.163.931399583656514×10−80.244.530709207806183×10−70.320.0000025758125588026810.400.0000099433961122663560.480.000030048856725528594


## 5. Conclusion

In this study, a novel procedure is designed by a combination of CST and an iterative method. The CSTIM has been effectively applied to obtain approximate and exact solutions for the conformable CRDEs. To assess the effectiveness and reliability of CSTIM for conformable PDEs, the absolute errors of linear and nonlinear problems are studied graphically and numerically, atν=1 and ν=0.5, respectively, when λ=1. The interpretation of the 2D plots and tables for different values of λ as well validates that the approximate solution is rapidly convergent to the exact solution. The numerical and graphical consequences confirm that the CSTIM is extremely effective and precise.The CSTIM distinguishes itself from various other numerical methods in three important aspects. The advantage of this method is that there is no need for any small or large physical parametric assumptions in the problem. Thus, it is applicable to not only weakly but also strongly nonlinear problems, going beyond some of the inherent limitations of the standard perturbation methods. Second, the CSTIM does not need He’s polynomials and Adomian polynomials when solving nonlinear problems. Thus, only a few calculations are required to solve nonlinear fractional-order DEs. As a result, it has a significant advantage over homotopy analysis and Adomian decomposition methods. Finally, unlike the other analytic approximation techniques, the CSTIM can be used to construct expansion solutions for linear and nonlinear fractional-order DEs without perturbation, linearization, or discretization. Therefore, we concluded that the recommended procedure is quick, precise, and easy to implement and yields outstanding results. Consequently, we can further apply this procedure to solve more linear and nonlinear fractional-order DEs.



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*Source: 4172218-2022-06-07.xml*