# A Modified Techniques of Fractional-Order Cauchy-Reaction Diffusion Equation via Shehu Transform

Authors: Mounirah Areshi; A. M. Zidan; Rasool Shah; Kamsing Nonlaopon
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.


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## Body

## 1. Introduction

In recent centuries, fractional partial differential equation (FPDE) fundamental signification is well-known in different engineering fields. Numerous physical phenomena can be modelled using FPDEs in various engineering and science fields such as physics, finance, aerospace, mechanics, biology, signal processing, biochemistry, and polymers [1, 2]. It is challenging to create computational models for any natural phenomenon using a traditional differential operator since it can describe those phenomena, primarily hereditary properties. They have a memory effect and the integrated capacity to describe and explain physical phenomena that cannot be adequately explained using an integer-order differential equation. The nonlocality of fractional operators in partial differential equations is one purpose for their importance in modeling different areas of chemical, biological, psychological, thermoplasticity, physical, and mechanical systems [3–5].Reaction-diffusion (RD) equations define various nonlinear schemes in chemistry, physics, biology, ecology, and other field sciences. RD equations are commonly used in ecology as a model for geographic impacts. They agree on three significant forms of environmental concepts: a minimum patch size required to maintain a population, the establishment of spatial trends in population distributions in homogeneous ecosystems, and the propagation of wavefronts associated with biological invasions. RD equations can be studied using techniques from the concept of PDEs and dynamical schemes [6–9].The equation of reaction-diffusion write of the following form:(1)∂μ∂η=Δμ+gμ,∇μ:φ,η.The term ofΔμ is diffusion term and gμ,∇μ:φ,η is the reaction function. Moreover, the diffusion general term is Aμ, where A is a 2nd order nonlinear elliptic operators. In this article, we consider the fractional-order one-dimensional, reaction–diffusion equation(2)∂δμφ,η∂ηδ=D∂2μφ,η∂φ2+rφ,ημφ,η,φ,η∈Ω⊂R2,Ifδ=1, then, it becomes classical reaction-diffusion equation, where μ is the concentration, r is the reaction parameter, and D>0 is the diffusion coefficient, with the initial and boundaries conditions(3)μφ,0=fφ,φ∈R,(4)μ0,η=g0η.∂μ∂φ0,η=g1t,η∈R.The model defined by equations (2) and (3) is called the characteristic Cauchy model in the domain Ω=R×R+, and the model given by equations (2) and (4) is called the noncharacteristic Cauchy equation in the domain Ω=R+×R, using different analytical and numerical methods to solve reaction diffusion equation, such as Picard technique [10], homotopy perturbation technique [3], differential transformation technique and variation iteration technique [11], Adomian decomposition method [12], homotopy analysis method [13], fractional iteration algorithm I [14], new Sumudu transform iterative method [15], and finite-difference discretization scheme [16].Daftardar-Gejji and Jafari in 2006 [17] suggested DJM solves linear and nonlinear differential equations. The DJM is simple and easy to comprehend and use, and it provides faster numerical solutions than the variational iterative technique [18] and the Adomian decomposition technique [19, 20]. The new iterative transformation method (NITM) is a combination of Shehu transform and the new iterative approach that provides the solution in the form of convergent series in an easy way. Another technique is the homotopy perturbation transform method (HPTM) to some applicable fractional models arising in real-life problems. The HPTM is applied directly to fractional models without any linearization, discretization, or variable transforms. The HPTM is an iterative technique that converges to solutions in closed form or approximate solutions. The nonlinear terms are decomposed successfully via He’s polynomials, and the fractional derivatives are computed in the Caputo sense. Applications of three fractional models are demonstrated, and the analytical and numerical simulations of the three fractional models are provided to bolster the efficiency, simplicity, and high accuracy of the HPTM. Many researchers use this method to solve different fractional-order partial differential equations, such as fractional order gas dynamic equation [21], convection-diffusion problems [22], and Klein-Gordon equations [23].This paper uses the Iterative transform method and homotopy perturbation transform method to solve fractional Cauchy-reaction diffusion equation equations using the fractional operator of Caputo type. The fractional calculus fundamental definitions are defined in Section2, writing the general methodologies in Sections 3 and 4, many test models show the effectiveness of suggested techniques in Sections 5 and 6. Finally, the conclusion is given in Section 7.


## 2. Basic Definitions

### 2.1. Definition

The Riemann fractional integral is given as [24, 25](5)I0δhη=1Γδ∫0ηη−sδ−1hsds.


### 2.2. Definition

The Caputo fractional derivative offη is defined as [24, 25](6)Dηδfη=Ij−δfj,j−1<δ<j,j∈ℕ,(7)djdηjhη,δ=j,j∈ℕ.


### 2.3. Definition

The integral transform of Shehu transform in Set A, the function is defined by [26–28](8)A=μη:∃,ρ1,ρ2>0,μη<Meηρi,ifη∈0,∞.The transformation of Shehu is defined asS. for a function μη is given as(9)Sμη=Vs,v=∫0∞e−sηvμηdη,η>0,s>0.The Shehu transform of a functionμη is Vs,v: then, μη is called the inverse of Vs,v which is given as(10)S−1Vs,v=μη,forη≥0,S−1istheShehuinversetransformation.


### 2.4. Definition

Thenth derivative of Shehu transformation is defined as [27–29](11)Sμjη=sjvjVs,v−∑k=0j−1svj−k−1μk0.


### 2.5. Definition

The Shehu transform of fractional derivative is given as [26–28].(12)Sμδη=sδvδμs,v−∑k=0j−1svδ−k−1μk0,0<δ≤n.


## 2.1. Definition

The Riemann fractional integral is given as [24, 25](5)I0δhη=1Γδ∫0ηη−sδ−1hsds.


## 2.2. Definition

The Caputo fractional derivative offη is defined as [24, 25](6)Dηδfη=Ij−δfj,j−1<δ<j,j∈ℕ,(7)djdηjhη,δ=j,j∈ℕ.


## 2.3. Definition

The integral transform of Shehu transform in Set A, the function is defined by [26–28](8)A=μη:∃,ρ1,ρ2>0,μη<Meηρi,ifη∈0,∞.The transformation of Shehu is defined asS. for a function μη is given as(9)Sμη=Vs,v=∫0∞e−sηvμηdη,η>0,s>0.The Shehu transform of a functionμη is Vs,v: then, μη is called the inverse of Vs,v which is given as(10)S−1Vs,v=μη,forη≥0,S−1istheShehuinversetransformation.


## 2.4. Definition

Thenth derivative of Shehu transformation is defined as [27–29](11)Sμjη=sjvjVs,v−∑k=0j−1svj−k−1μk0.


## 2.5. Definition

The Shehu transform of fractional derivative is given as [26–28].(12)Sμδη=sδvδμs,v−∑k=0j−1svδ−k−1μk0,0<δ≤n.


## 3. The HPTM General Implementation

Consider the general fractional partial differential equation solve to HPTM:(13)Dηδμφ,η+Mμφ,η+Nμφ,η=hφ,η,η>0,0<δ≤1,μφ,0=gφ,μ∈R.Applying Shehu transformation of equation (13), we get(14)SDηδμφ,η+Mμφ,η+Nμφ,η=Shφ,η,η>0,0<δ≤1,μφ,η=vsgφ+vδsδShφ,η−vδsδSMμφ,η+Nμφ,η.Now, using inverse Shehu transformation, we achieve as(15)μφ,η=Fφ,η−S−1vδsδSMμφ,η+Nμφ,η,where(16)Fφ,η=S−1vsgφ+vδsδShφ,η=gμ+S−1vδsδShφ,η.The parameterp show the producer of perturbation is given as(17)μφ,η=∑k=0∞pkμkφ,η,where the perturbation p is parameter and p∈0,1.The nonlinear components can be defined as(18)Nμφ,η=∑k=0∞pkHkμk,where He’s polynomials Hn in terms of μ0,μ1,μ2,⋯,μn, and can be calculated as(19)Hnμ0,μ1,⋯,μn=1δn+1DpkN∑k=0∞pkμkp=0,where Dpk=∂k/∂k.Substituting equations (18) and (19) in equation (15), we obtain as(20)∑k=0∞pkμkφ,η=Fφ,η−p×S−1vδsδSM∑k=0∞pkμkφ,η+∑k=0∞pkHkμk.Thep coefficients comparison on both sides, we have(21)p0:μ0φ,η=Fφ,η,p1:μ1φ,η=S−1vδsδSMμ0φ,η+H0μ,p2:μ2φ,η=S−1vδsδSMμ1φ,η+H1μ,⋮pk:μkφ,η=S−1vδsδSMμk−1φ,η+Hk−1μ,k>0,k∈N.Theμkφ,η components simply calculated to the convergence series form. We can achieve p⟶1,(22)μφ,η=limM⟶∞∑k=1Mμkφ,η.


## 4. The NITM General Implementation

Consider the general fractional partial differential equation solve to NITM:(23)Dηδμφ,η+Mμφ,η+Nμφ,η=hφ,η,n∈N,n−1<δ≤n,where linear and nonlinear functions represent by M and N.With initial condition(24)μkφ,0=gkφ,k=0,1,2,⋯,n−1,Apply the Shehu transformation of equation (23), we obtain as(25)SDηδμφ,η+SNμφ,η+Mμφ,η=Ehφ,η.Using the differentiation property of Shehu transform is given as(26)Sμφ,η=vsμφ,0+vδsδShφ,η−vδsδSNμφ,η+Mμφ,η,The Shehu inverse transform applies in equation (26), we have(27)μφ,η=S−1vsμφ,0+vδsδShφ,η−S−1vδsδSNμφ,η+Mμφ,η.In the iterative method, we get(28)μφ,η=∑m=0∞μmφ,η.(29)N∑m=0∞μmφ,η=∑m=0∞Nμmφ,η,the nonlinear term N is expressed as(30)N∑m=0∞μmφ,η=μ0φ,η+N∑k=0mμkφ,η−M∑k=0mμkφ,η.Substituting equations (28), (29), and (30) in equation (27), we can obtain the solution(31)∑m=0∞μmφ,η=S−1vδsδ∑k=0ms2−φ+kukφ,0+Shφ,η−S−1vδsδEN∑k=0mμkφ,η−M∑k=0mμkφ,η.The iterative method can be defined as(32)μ0φ,η=S−1vδsδ∑k=0ms2−φ+kukφ,0+vδsδEgφ,η,(33)μ1φ,η=−S−1vδsδSNμ0φ,η+Mμ0φ,η,(34)μm+1φ,η=−S−1vδsδS−N∑k=0mμkφ,η−M∑k=0mμkφ,η,m≥1.Finally, equations (23) and (24) provide the m-terms solution in series form is achieved as(35)μφ,η≅μ0φ,η+μ1φ,η+μ2φ,η+..⋯,+μmφ,η,m=1,2,⋯.Similarly, the convergence proof of the NITM, see [17].


### 4.1. Example

Consider the fractional Cauchy-reaction diffusion equation is given as:(36)Dηδμφ,η=Dη2μφ,η−μφ,η,0<δ≤1,with the initial and boundaries conditions(37)μφ,0=e−φ+φ=gφ,μ0,η=1=f0η,∂μ0,η∂η=e−η−1=f1η,φ,η∈R.Applying Shehu transform in equation (36), we get(38)Sμφ,η=vse−φ+φ+sδSDη2μ−μ.Using inverse Shehu transform in the above equation, we get(39)μφ,η=e−φ+φ+S−1sδSDη2μ−μ.First, we implement the NITM, we get the following(40)μ0φ,η=e−φ+φ,(41)μ1φ,η=S−1sδSDη2μ0−μ0=φ−ηδΓδ+1,(42)μ2φ,η=S−1sδSDη2μ1−μ1=φ−ηδ2Γ2δ+1,(43)μ3φ,η=S−1sδSDη2μ2−μ2=φ−ηδ3Γ3δ+1,⋮(44)μnφ,η=S−1sδSDη2μn−μn=φ−ηδnΓnδ+1,n≥0.The series form solution is given as(45)μφ,η=μ0φ,η+μ1φ,η+μ2φ,η+μ3φ,η+⋯μnφ,η.The approximate solutions is achieved as(46)μφ,η=e−φ+φ1−ηδΓδ+1+η2δΓ2δ+1−η3δΓ3δ+1+⋯+−ηδnΓnδ+1.Now we implement the HPTM, we obtain as(47)∑n=0∞pnμnφ,η=e−φ+φ+pS−1sδS∑n=0∞pnHnμ.Where the nonlinear function of the polynomial is shown byHnμ. For instance, the He’s polynomial component is obtained through the recursive relation Hnμ=Dη2μn−μn, ∀n∈N. Now, on both sides, the coefficient of corresponding power p is calculated, and the following solution is obtained as follows:(48)p0:μ0φ,η=e−φ+φ,p1:μ1φ,η=S−1sδSH0μ=φ−ηδΓδ+1,p2:μ2φ,η=S−1sδSH1μ=φ−ηδ2Γ2δ+1,p3:μ3φ,η=S−1sδSH2μ=φ−ηδ3Γ3δ+1,⋮pn:μnφ,η=S−1sδSHn−1μ=φ−ηδnΓnδ+1.Then the series form result is defined as(49)μφ,η=∑n=0∞pnμnφ,η.The approximate solution, we can achieve as(50)μφ,η=e−φ+φ1−ηδΓδ+1+η2δΓ2δ+1−η3δΓ3δ+1+.⋯+−ηδnΓnδ+1=φ∑k=0∞ηδkΓkδ+1.μφ,η=e−φ+φEδηδ.The actual solution is(51)μφ,η=e−φ+φe−η.Figure1 shows the exact and the analytical solution figures at δ=1 of Example 1. Figure 1 shows that both the analytical and actual solutions are in close contact with each other. Figure 2 shows that the first graph analytical solution of fractional-order at δ=0.8, and the second graph fractional order δ=0.6. In Figure 3, the first graph shows the approximate solution of fractional-order at δ=0.4, and the second graph shows the different fractional-order δ of Example 1. In Figure 4, the 2D graphs show that the first actual and approximate solution and second graph show the different fractional-order δ of Example 1. The solution is very rapidly convergent by using the homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful to find the analytical result as well as approximate results of various physical fractional problems.Figure 1
The first graph shows the exact and second analytical solution figures atδ=1 of Example 1.Figure 2
The first graph shows the analytical solution of fractional-order atδ=0.8 and second graph fractional order 0.6 of Example 1.Figure 3
The first graph shows the analytical solution of fractional-order atδ=0.4, and the second graph shows the different fractional-order δ of Example 1.Figure 4
The first graph shows the exact and analytical solution, and the second graph shows the different fractional-orderδ of Example 1.


### 4.2. Example

Consider the fractional Cauchy-reaction diffusion equation is given as:(52)Dηδμφ,η=Dη2μφ,η−1+4φ2μφ,η,0<δ≤1,with initial condition(53)μφ,0=eφ2,and the exact solution is given as(54)μφ,η=eφ2+1.First, we implement the NITM, we obtain as(55)μ0φ,η=eφ2,(56)μ1φ,η=S−1sδSDη2μ0φ,η−1+4φ2μ0φ,η=eφ2ηδΓδ+1,(57)μ2φ,η=S−1sδSDη2μ1φ,η−1+4φ2μ1φ,η=eφ2η2δΓ2δ+1,(58)μ3φ,η=S−1sδSDη2μ2φ,η−1+4φ2μ2φ,η=eφ2η3δΓ3δ+1,⋮(59)μnφ,η=S−1sδSDη2μnφ,η−1+4φ2μnφ,η=eφ2ηδnΓnδ+1,n≥0.The series form of iterative transform method is defined as(60)μφ,η=μ0φ,η+μ1φ,η+μ2φ,η+μ3φ,η+⋯μnφ,η.The approximate solution is given as(61)μφ,η=eφ21+ηδΓδ+1+η2δΓ2δ+1+η3δΓ3δ+1+⋯+ηnδΓnδ+1.μφ,η=eφ2Eδηδ.Now, we implement HPTM, we get(62)∑n=0∞pnμnφ,η=eφ2+pS−1sδS∑n=0∞pnHnμ.On both sidesp Comparisons coefficient, we have(63)p0:μ0φ,η=eφ2,p1:μ1φ,η=S−1sδSH0μ=eφ2ηδΓδ+1,p2:μ2φ,η=S−1sδSH1μ=eφ2η2δΓ2δ+1,p3:μ3φ,η=S−1sδSH2μ=eφ2η3δΓ3δ+1,⋮pn:μnφ,η=S−1sδSHn−1μ=eφ2ηnδΓnδ+1.The series form solution is express as(64)μφ,η=∑n=0∞pnμnφ,η.μφ,η=eφ21+ηδΓδ+1+η2δΓ2δ+1+η3δΓ3δ+1+⋯+ηnδΓnδ+1,μφ,η=eφ2Eδηδ.Now,δ=1, then the exact result of equation (52) is μφ,η=eφ2+η.Figure5 shows the exact and the analytical solution figures at δ=1 of Example 2. Figure 5 shows that both the analytical and actual solutions are in close contact with each other. Figure 6 shows the first graph analytical solution of fractional-order at δ=0.8 and the second graph fractional order δ=0.6. In Figure 7, the first graph shows the approximate solution of fractional-order at δ=0.4, and the second graph shows the different fractional-order δ of Example 2. In Figure 8, the 2D graphs show that the first actual and approximate solution and second graph show the different fractional-order δ of Example 2. The solution is very rapidly convergent by using the homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful to find the analytical solution as well as approximate results of various physical fractional problems.Figure 5
The first graph shows the exact and second analytical solution figures atδ=1 of Example 2.Figure 6
The first graph shows the analytical solution of fractional-order atδ=0.8 and second graph fractional order 0.6 of Example 2.Figure 7
The first graph shows the analytical solution of fractional-order atδ=0.4, and the second graph shows the different fractional-order δ of Example 2.Figure 8
The first graph shows the exact and analytical solution, and the second graph shows the different fractional-orderδ of Example 2.


### 4.3. Example

Consider fractional-order Cauchy-reaction diffusion equation:(65)Dηδμφ,η=Dη2μφ,η+2ημφ,η,0<δ≤1,with initial condition(66)μφ,0=eφ,the actual solution is(67)μφ,η=eφ+η+η2.Apply the Shehu transform, we obtain as(68)μφ,η=eφ+S−1sδSDη2μφ,η+2ημφ,η.First, we implement the NITM(69)μ0φ,η=eφ,(70)μ1φ,η=S−1sδSDη2μ0φ,η+2ημ0φ,η=eφηδΓδ+1+2ηδ+1Γδ+2,(71)μ2φ,η=S−1sδSDη2μ1φ,η+2ημ1φ,η=eφη2δΓ2δ+1+2δ+2η2δ+1Γ2δ+2+4δ+2η2δ+2Γ2δ+3,(72)μ3φ,η=S−1sδSDη2μ2φ,η+2ημ2φ,η,=eφη3δΓ3δ+1+6δ+1η3δ+1Γ3δ+2+4δ+2δ+3η3δ+2Γ3δ+3+8δ+22δ+3η3δ+3Γ3δ+4.⋮The series form of proposed technique can be defined as(73)μφ,η=μ0φ,η+μ1φ,η+μ2φ,η+μ3φ,η+⋯μnφ,η.The approximate solution is given as(74)μφ,η=eφ+eφηδΓδ+1+2ηδ+1Γδ+2+eφη2δΓ2δ+1+2δ+2η2δ+1Γ2δ+2+4δ+2η2δ+2Γ2δ+3+eφη3δΓ3δ+1+6δ+1η3δ+1Γ3δ+2+4δ+2δ+3η3δ+2Γ3δ+3+8δ+22δ+3η3δ+3Γ3δ+4+⋯.Now, we apply HPTM, we get(75)∑n=0∞pnμnφ,η=eφ+pS−1sδS∑n=0∞pnHnμ.0The coefficients ofp Comparison on both sides, we achieve as(76)p0:μ0φ,η=eφ,p1:μ1φ,η=S−1sδSH0μ=eφηδΓδ+1+2ηδ+1Γδ+2,p2:μ2φ,η=S−1sδSH1μ=eφη2δΓ2δ+1+2δ+2η2δ+1Γ2δ+2+4δ+2η2δ+2Γ2δ+3,p3:μ3φ,η=S−1sδSH2μ=eφη3δΓ3δ+1+6δ+1η3δ+1Γ3δ+2+4δ+2δ+3η3δ+2Γ3δ+3+8δ+22δ+3η3δ+3Γ3δ+4.Finally, we obtain the analytical resultμφ,η to the truncate series.(77)μφ,η=limN⟶∞∑n=1Nμnφ,η.μφ,η=eφ+eφηδΓδ+1+2ηδ+1Γδ+2+eφη2δΓ2δ+1+2δ+2η2δ+1Γ2δ+2+4δ+2η2δ+2Γ2δ+3+eφη3δΓ3δ+1+6δ+1η3δ+1Γ3δ+2+4δ+2δ+3η3δ+2Γ3δ+3+8δ+22δ+3η3δ+3Γ3δ+4+⋯.Now, forδ=1, the actual result of the above series is(78)μφ,η=eφ+η+η2.Figure9 shows the exact and the analytical solution figures at δ=1 of Example 3. Figure 9 shows that both the analytical and actual solutions are in close contact with each other. Figure 10 shows that the first graph analytical solution of fractional-order at δ=0.8 and second graph fractional order δ=0.6. In Figure 11, the first graph shows the approximate solution of fractional-order at δ=0.4, and the second graph shows the different fractional-order δ of Example 3. The solution is very rapidly convergent by using the Homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful to find the analytical result as well as approximate results of various physical fractional problems.Figure 9
The first graph shows the exact and second analytical solution figures atδ=1 of Example 3.Figure 10
The first graph shows the analytical solution of fractional-order atδ=0.8 and second graph fractional order 0.6 of Example 3.Figure 11
The first graph shows the analytical solution of fractional-order atδ=0.4, and the second graph shows the different fractional-order δ of Example 3.


### 4.4. Example

Consider the fractional Cauchy-reaction diffusion equation is given as:(79)Dηδμφ,η=Dη2μφ,η−4φ2−2η+2μφ,η,0<δ≤1,with initial condition(80)μφ,0=eφ2,the exact result is(81)μφ,η=eφ2+η2.First, we apply the NITM(82)μ0φ,η=eφ2,(83)μ1φ,η=S−1sδSDη2μ0φ,η−4φ2−2η+2μ0φ,η=2eφ2ηδ+1Γδ+2,(84)μ2φ,η=S−1sδSDη2μ1φ,η−4φ2−2η+2μ1φ,η=22eφ2δ+2η2δ+1Γ2δ+3,(85)μ3φ,η=S−1sδSDη2μ2φ,η−4φ2−2η+2μ2φ,η=23eφ2δ+22δ+3η3δ+1Γ3δ+4.⋮The series form result is defined as(86)μφ,η=μ0φ,η+μ1φ,η+μ2φ,η+μ3φ,η+⋯μnφ,η.The approximate solution is(87)μφ,η=eφ2+2eφ2ηδ+1Γδ+2+22eφ2δ+2η2δ+1Γ2δ+3+23eφ2δ+22δ+3η3δ+1Γ3δ+4+24eφ2δ+22δ+33δ+4η4δ+1Γ4δ+5+⋯.Now, implement the HPTM, we obtain as(88)∑n=0∞pnμnφ,η=eφ2+pS−1sδS∑n=0∞pnHnμ.The comparison of coefficient ofp on both sides, we get(89)p0:μ0φ,η=eφ2,p1:μ1φ,η=S−1sδSH0μ=2eφ2ηδ+1Γδ+2,p2:μ2φ,η=S−1sδSH1μ=22eφ2δ+2η2δ+1Γ2δ+3,p3:μ3φ,η=S−1sδSH2μ=23eφ2δ+22δ+3η3δ+1Γ3δ+4,p4:μ4φ,η=S−1sδSH3μ=24eφ2δ+22δ+33δ+4η4δ+1Γ4δ+5.Finally, we obtain the approximate resultμφ,η by truncate series(90)μφ,η=limN⟶∞∑n=1Nμnφ,η.μφ,η=eφ2+2eφ2ηδ+1Γδ+2+22eφ2δ+2η2δ+1Γ2δ+3+23eφ2δ+22δ+3η3δ+1Γ3δ+4+24eφ2δ+22δ+33δ+4η4δ+1Γ4δ+5+⋯.The special case forδ=1, the close form above equation is expressed as(91)μφ,η=eφ2+η2.Figure12 shows the exact and the analytical solution figures at δ=1 of Example 4. Figure 12 shows that both the analytical and actual solutions are in close contact with each other. Figure 13 shows the first graph analytical solution of fractional-order at δ=0.8 and second graph fractional order δ=0.6. In Figure 14, the first graph shows the approximate solution of fractional-order at δ=0.4, and the second graph shows the different fractional-order δ of Example 4. The solution is very rapidly convergent by apply the homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful in finding the analytical result as well as approximate results of various physical fractional problems.Figure 12
The first graph shows the exact and second analytical solution figures atδ=1 of Example 4.Figure 13
The first graph shows the analytical solution of fractional-order atδ=0.8 and second graph fractional order 0.6 of Example 4.Figure 14
The first graph shows the analytical solution of fractional-order atδ=0.4, and the second graph shows the different fractional-order δ of Example 4.


## 4.1. Example

Consider the fractional Cauchy-reaction diffusion equation is given as:(36)Dηδμφ,η=Dη2μφ,η−μφ,η,0<δ≤1,with the initial and boundaries conditions(37)μφ,0=e−φ+φ=gφ,μ0,η=1=f0η,∂μ0,η∂η=e−η−1=f1η,φ,η∈R.Applying Shehu transform in equation (36), we get(38)Sμφ,η=vse−φ+φ+sδSDη2μ−μ.Using inverse Shehu transform in the above equation, we get(39)μφ,η=e−φ+φ+S−1sδSDη2μ−μ.First, we implement the NITM, we get the following(40)μ0φ,η=e−φ+φ,(41)μ1φ,η=S−1sδSDη2μ0−μ0=φ−ηδΓδ+1,(42)μ2φ,η=S−1sδSDη2μ1−μ1=φ−ηδ2Γ2δ+1,(43)μ3φ,η=S−1sδSDη2μ2−μ2=φ−ηδ3Γ3δ+1,⋮(44)μnφ,η=S−1sδSDη2μn−μn=φ−ηδnΓnδ+1,n≥0.The series form solution is given as(45)μφ,η=μ0φ,η+μ1φ,η+μ2φ,η+μ3φ,η+⋯μnφ,η.The approximate solutions is achieved as(46)μφ,η=e−φ+φ1−ηδΓδ+1+η2δΓ2δ+1−η3δΓ3δ+1+⋯+−ηδnΓnδ+1.Now we implement the HPTM, we obtain as(47)∑n=0∞pnμnφ,η=e−φ+φ+pS−1sδS∑n=0∞pnHnμ.Where the nonlinear function of the polynomial is shown byHnμ. For instance, the He’s polynomial component is obtained through the recursive relation Hnμ=Dη2μn−μn, ∀n∈N. Now, on both sides, the coefficient of corresponding power p is calculated, and the following solution is obtained as follows:(48)p0:μ0φ,η=e−φ+φ,p1:μ1φ,η=S−1sδSH0μ=φ−ηδΓδ+1,p2:μ2φ,η=S−1sδSH1μ=φ−ηδ2Γ2δ+1,p3:μ3φ,η=S−1sδSH2μ=φ−ηδ3Γ3δ+1,⋮pn:μnφ,η=S−1sδSHn−1μ=φ−ηδnΓnδ+1.Then the series form result is defined as(49)μφ,η=∑n=0∞pnμnφ,η.The approximate solution, we can achieve as(50)μφ,η=e−φ+φ1−ηδΓδ+1+η2δΓ2δ+1−η3δΓ3δ+1+.⋯+−ηδnΓnδ+1=φ∑k=0∞ηδkΓkδ+1.μφ,η=e−φ+φEδηδ.The actual solution is(51)μφ,η=e−φ+φe−η.Figure1 shows the exact and the analytical solution figures at δ=1 of Example 1. Figure 1 shows that both the analytical and actual solutions are in close contact with each other. Figure 2 shows that the first graph analytical solution of fractional-order at δ=0.8, and the second graph fractional order δ=0.6. In Figure 3, the first graph shows the approximate solution of fractional-order at δ=0.4, and the second graph shows the different fractional-order δ of Example 1. In Figure 4, the 2D graphs show that the first actual and approximate solution and second graph show the different fractional-order δ of Example 1. The solution is very rapidly convergent by using the homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful to find the analytical result as well as approximate results of various physical fractional problems.Figure 1
The first graph shows the exact and second analytical solution figures atδ=1 of Example 1.Figure 2
The first graph shows the analytical solution of fractional-order atδ=0.8 and second graph fractional order 0.6 of Example 1.Figure 3
The first graph shows the analytical solution of fractional-order atδ=0.4, and the second graph shows the different fractional-order δ of Example 1.Figure 4
The first graph shows the exact and analytical solution, and the second graph shows the different fractional-orderδ of Example 1.


## 4.2. Example

Consider the fractional Cauchy-reaction diffusion equation is given as:(52)Dηδμφ,η=Dη2μφ,η−1+4φ2μφ,η,0<δ≤1,with initial condition(53)μφ,0=eφ2,and the exact solution is given as(54)μφ,η=eφ2+1.First, we implement the NITM, we obtain as(55)μ0φ,η=eφ2,(56)μ1φ,η=S−1sδSDη2μ0φ,η−1+4φ2μ0φ,η=eφ2ηδΓδ+1,(57)μ2φ,η=S−1sδSDη2μ1φ,η−1+4φ2μ1φ,η=eφ2η2δΓ2δ+1,(58)μ3φ,η=S−1sδSDη2μ2φ,η−1+4φ2μ2φ,η=eφ2η3δΓ3δ+1,⋮(59)μnφ,η=S−1sδSDη2μnφ,η−1+4φ2μnφ,η=eφ2ηδnΓnδ+1,n≥0.The series form of iterative transform method is defined as(60)μφ,η=μ0φ,η+μ1φ,η+μ2φ,η+μ3φ,η+⋯μnφ,η.The approximate solution is given as(61)μφ,η=eφ21+ηδΓδ+1+η2δΓ2δ+1+η3δΓ3δ+1+⋯+ηnδΓnδ+1.μφ,η=eφ2Eδηδ.Now, we implement HPTM, we get(62)∑n=0∞pnμnφ,η=eφ2+pS−1sδS∑n=0∞pnHnμ.On both sidesp Comparisons coefficient, we have(63)p0:μ0φ,η=eφ2,p1:μ1φ,η=S−1sδSH0μ=eφ2ηδΓδ+1,p2:μ2φ,η=S−1sδSH1μ=eφ2η2δΓ2δ+1,p3:μ3φ,η=S−1sδSH2μ=eφ2η3δΓ3δ+1,⋮pn:μnφ,η=S−1sδSHn−1μ=eφ2ηnδΓnδ+1.The series form solution is express as(64)μφ,η=∑n=0∞pnμnφ,η.μφ,η=eφ21+ηδΓδ+1+η2δΓ2δ+1+η3δΓ3δ+1+⋯+ηnδΓnδ+1,μφ,η=eφ2Eδηδ.Now,δ=1, then the exact result of equation (52) is μφ,η=eφ2+η.Figure5 shows the exact and the analytical solution figures at δ=1 of Example 2. Figure 5 shows that both the analytical and actual solutions are in close contact with each other. Figure 6 shows the first graph analytical solution of fractional-order at δ=0.8 and the second graph fractional order δ=0.6. In Figure 7, the first graph shows the approximate solution of fractional-order at δ=0.4, and the second graph shows the different fractional-order δ of Example 2. In Figure 8, the 2D graphs show that the first actual and approximate solution and second graph show the different fractional-order δ of Example 2. The solution is very rapidly convergent by using the homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful to find the analytical solution as well as approximate results of various physical fractional problems.Figure 5
The first graph shows the exact and second analytical solution figures atδ=1 of Example 2.Figure 6
The first graph shows the analytical solution of fractional-order atδ=0.8 and second graph fractional order 0.6 of Example 2.Figure 7
The first graph shows the analytical solution of fractional-order atδ=0.4, and the second graph shows the different fractional-order δ of Example 2.Figure 8
The first graph shows the exact and analytical solution, and the second graph shows the different fractional-orderδ of Example 2.


## 4.3. Example

Consider fractional-order Cauchy-reaction diffusion equation:(65)Dηδμφ,η=Dη2μφ,η+2ημφ,η,0<δ≤1,with initial condition(66)μφ,0=eφ,the actual solution is(67)μφ,η=eφ+η+η2.Apply the Shehu transform, we obtain as(68)μφ,η=eφ+S−1sδSDη2μφ,η+2ημφ,η.First, we implement the NITM(69)μ0φ,η=eφ,(70)μ1φ,η=S−1sδSDη2μ0φ,η+2ημ0φ,η=eφηδΓδ+1+2ηδ+1Γδ+2,(71)μ2φ,η=S−1sδSDη2μ1φ,η+2ημ1φ,η=eφη2δΓ2δ+1+2δ+2η2δ+1Γ2δ+2+4δ+2η2δ+2Γ2δ+3,(72)μ3φ,η=S−1sδSDη2μ2φ,η+2ημ2φ,η,=eφη3δΓ3δ+1+6δ+1η3δ+1Γ3δ+2+4δ+2δ+3η3δ+2Γ3δ+3+8δ+22δ+3η3δ+3Γ3δ+4.⋮The series form of proposed technique can be defined as(73)μφ,η=μ0φ,η+μ1φ,η+μ2φ,η+μ3φ,η+⋯μnφ,η.The approximate solution is given as(74)μφ,η=eφ+eφηδΓδ+1+2ηδ+1Γδ+2+eφη2δΓ2δ+1+2δ+2η2δ+1Γ2δ+2+4δ+2η2δ+2Γ2δ+3+eφη3δΓ3δ+1+6δ+1η3δ+1Γ3δ+2+4δ+2δ+3η3δ+2Γ3δ+3+8δ+22δ+3η3δ+3Γ3δ+4+⋯.Now, we apply HPTM, we get(75)∑n=0∞pnμnφ,η=eφ+pS−1sδS∑n=0∞pnHnμ.0The coefficients ofp Comparison on both sides, we achieve as(76)p0:μ0φ,η=eφ,p1:μ1φ,η=S−1sδSH0μ=eφηδΓδ+1+2ηδ+1Γδ+2,p2:μ2φ,η=S−1sδSH1μ=eφη2δΓ2δ+1+2δ+2η2δ+1Γ2δ+2+4δ+2η2δ+2Γ2δ+3,p3:μ3φ,η=S−1sδSH2μ=eφη3δΓ3δ+1+6δ+1η3δ+1Γ3δ+2+4δ+2δ+3η3δ+2Γ3δ+3+8δ+22δ+3η3δ+3Γ3δ+4.Finally, we obtain the analytical resultμφ,η to the truncate series.(77)μφ,η=limN⟶∞∑n=1Nμnφ,η.μφ,η=eφ+eφηδΓδ+1+2ηδ+1Γδ+2+eφη2δΓ2δ+1+2δ+2η2δ+1Γ2δ+2+4δ+2η2δ+2Γ2δ+3+eφη3δΓ3δ+1+6δ+1η3δ+1Γ3δ+2+4δ+2δ+3η3δ+2Γ3δ+3+8δ+22δ+3η3δ+3Γ3δ+4+⋯.Now, forδ=1, the actual result of the above series is(78)μφ,η=eφ+η+η2.Figure9 shows the exact and the analytical solution figures at δ=1 of Example 3. Figure 9 shows that both the analytical and actual solutions are in close contact with each other. Figure 10 shows that the first graph analytical solution of fractional-order at δ=0.8 and second graph fractional order δ=0.6. In Figure 11, the first graph shows the approximate solution of fractional-order at δ=0.4, and the second graph shows the different fractional-order δ of Example 3. The solution is very rapidly convergent by using the Homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful to find the analytical result as well as approximate results of various physical fractional problems.Figure 9
The first graph shows the exact and second analytical solution figures atδ=1 of Example 3.Figure 10
The first graph shows the analytical solution of fractional-order atδ=0.8 and second graph fractional order 0.6 of Example 3.Figure 11
The first graph shows the analytical solution of fractional-order atδ=0.4, and the second graph shows the different fractional-order δ of Example 3.


## 4.4. Example

Consider the fractional Cauchy-reaction diffusion equation is given as:(79)Dηδμφ,η=Dη2μφ,η−4φ2−2η+2μφ,η,0<δ≤1,with initial condition(80)μφ,0=eφ2,the exact result is(81)μφ,η=eφ2+η2.First, we apply the NITM(82)μ0φ,η=eφ2,(83)μ1φ,η=S−1sδSDη2μ0φ,η−4φ2−2η+2μ0φ,η=2eφ2ηδ+1Γδ+2,(84)μ2φ,η=S−1sδSDη2μ1φ,η−4φ2−2η+2μ1φ,η=22eφ2δ+2η2δ+1Γ2δ+3,(85)μ3φ,η=S−1sδSDη2μ2φ,η−4φ2−2η+2μ2φ,η=23eφ2δ+22δ+3η3δ+1Γ3δ+4.⋮The series form result is defined as(86)μφ,η=μ0φ,η+μ1φ,η+μ2φ,η+μ3φ,η+⋯μnφ,η.The approximate solution is(87)μφ,η=eφ2+2eφ2ηδ+1Γδ+2+22eφ2δ+2η2δ+1Γ2δ+3+23eφ2δ+22δ+3η3δ+1Γ3δ+4+24eφ2δ+22δ+33δ+4η4δ+1Γ4δ+5+⋯.Now, implement the HPTM, we obtain as(88)∑n=0∞pnμnφ,η=eφ2+pS−1sδS∑n=0∞pnHnμ.The comparison of coefficient ofp on both sides, we get(89)p0:μ0φ,η=eφ2,p1:μ1φ,η=S−1sδSH0μ=2eφ2ηδ+1Γδ+2,p2:μ2φ,η=S−1sδSH1μ=22eφ2δ+2η2δ+1Γ2δ+3,p3:μ3φ,η=S−1sδSH2μ=23eφ2δ+22δ+3η3δ+1Γ3δ+4,p4:μ4φ,η=S−1sδSH3μ=24eφ2δ+22δ+33δ+4η4δ+1Γ4δ+5.Finally, we obtain the approximate resultμφ,η by truncate series(90)μφ,η=limN⟶∞∑n=1Nμnφ,η.μφ,η=eφ2+2eφ2ηδ+1Γδ+2+22eφ2δ+2η2δ+1Γ2δ+3+23eφ2δ+22δ+3η3δ+1Γ3δ+4+24eφ2δ+22δ+33δ+4η4δ+1Γ4δ+5+⋯.The special case forδ=1, the close form above equation is expressed as(91)μφ,η=eφ2+η2.Figure12 shows the exact and the analytical solution figures at δ=1 of Example 4. Figure 12 shows that both the analytical and actual solutions are in close contact with each other. Figure 13 shows the first graph analytical solution of fractional-order at δ=0.8 and second graph fractional order δ=0.6. In Figure 14, the first graph shows the approximate solution of fractional-order at δ=0.4, and the second graph shows the different fractional-order δ of Example 4. The solution is very rapidly convergent by apply the homotopy perturbation method and the iterative method by modifying the Shehu transformation. It may be concluded that both methodologies are efficient and very powerful in finding the analytical result as well as approximate results of various physical fractional problems.Figure 12
The first graph shows the exact and second analytical solution figures atδ=1 of Example 4.Figure 13
The first graph shows the analytical solution of fractional-order atδ=0.8 and second graph fractional order 0.6 of Example 4.Figure 14
The first graph shows the analytical solution of fractional-order atδ=0.4, and the second graph shows the different fractional-order δ of Example 4.


## 5. Conclusion

This paper develops an effective and new coupling method of the homotopy perturbation method, the iterative method, and Shehu transform approach for the time-fractional Cauchy-reaction diffusion equations. With the help of the Shehu transform, the calculation of this method is very straightforward and simple. A very close contact of the achieved solutions with the exact solution of the problem is analyzed. It is also investigated that the rate of convergence of the proposed techniques is accurate the solution of fractional-order partial differential equations. Thus, we further this technique can be applied to solve other schemes of linear and nonlinear fractional-order partial differential equations.



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