# Approximate and Exact Solutions to Fractional Order Cauchy Reaction-Diffusion Equations by New Combine Techniques

Authors: Adnan Khan; Muhammad Imran Liaqat; Muhammad Younis; Ashraful Alam
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we present a simple and efficient novel semianalytic method to acquire approximate and exact solutions for the fractional order Cauchy reaction-diffusion equations (CRDEs). The fractional order derivative operator is measured in the Caputo sense. This novel method is based on the combinations of Elzaki transform method (ETM) and residual power series method (RPSM). The proposed method is called Elzaki residual power series method (ERPSM). The proposed method is based on the new form of fractional Taylor’s series, which constructs solution in the form of a convergent series. As in the RPSM, during establishing the coefficients for a series, it is required to compute the fractional derivatives every time. While ERPSM only requires the concept of the limit at zero in establishing the coefficients for the series, consequently scarce calculations give us the coefficients. The recommended method resolves nonlinear problems deprived of utilizing Adomian polynomials or He’s polynomials which is the advantage of this method over Adomain decomposition method (ADM) and homotopy-perturbation method (HTM). To study the effectiveness and reliability of ERPSM for partial differential equations (PDEs), absolute errors of three problems are inspected. In addition, numerical and graphical consequences are also recognized at diverse values of fractional order derivatives. Outcomes demonstrate that our novel method is simple, precise, applicable, and effectual.


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## Body

## 1. Introduction

Differential equations (DEs) can be resolved by a diversity of procedures, analytical and numerical. However, there are numerous analytic methods for verdict on the results of DEs; there occur quite a numeral of DEs that cannot be explained analytically. This means that the result cannot be articulated as a summation of a fixed numeral of basic functions.Numerous DEs arising in applications are so thorny that it is occasionally unreasonable to have result formulations or as a minimum if a result formula is existing, it possibly will comprise integrals that can be premeditated only by means of an algebraic quadrature formulation. In moreover instance, numerical procedures offer an influential substitute means for resolving the DEs under the prearranged preliminary condition.Earlier, numerous procedures have been offered to resolve fractional order DEs comprising the Bernstein wavelets method [1], Shehu variational iteration method [2], Chebyshev spectral collocation approach [3], Taylor wavelet technique [4], operational matrix approach [5], fractional natural decomposition method [6]. homotopy analysis approach [7], Aboodh decomposition approach [8], Sumudu decomposition method [9], Elzaki decomposition technique [10], residual power series method [11], and generalized pseudospectral method [12]. Numerical method is based on the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) and the collocation method [13].In this research, an easy and effective novel semianalytical method is initiated. The unexploited method is called ERPSM that is the merger of ETM and RPSM. The process of this efficacious method relies on transforming DE into the Elzaki space and creating a series explanation and subsequently acquiring the consequence of the actual DE by utilizing the inverse ETM.Reaction-diffusion equation is a mathematical model, characterized by the parabolic PDEs. It is exemplifying in what way chemicals might work to each other, whereas they diffuse by a medium instantaneously. Alan Tuning recognized it in 1952 [14]. Reaction-diffusion is measured immensely by experts in biology, chemistry, physics, and computer science [15].By a reaction-diffusion, we mean an equation of the following form:(1)∂Φ∂ϒ=ΔΦ+ΩΦ,ΔΦ,χ,ϒ,where Φ is the diffusion term and ΩΦ,ΔΦ,χ,ϒ is the reaction term.In this paper, we deliberate the one-dimensional time-fractional CRDEs. The time-fractional CRDEs can be utilized to explicate several categories of linear and nonlinear systems in physics, chemistry, ecology, biology, and engineering [16–18].The general form of the fractional order CRDE is as follows [19]:(2)∂ϖΦχ,ϒ∂ϒϖ=λ∂2Φχ,ϒ∂χ2+zχ,ϒΦχ,ϒ,χ≥0,ϒ≥0,0<ϖ≤1.With the initial condition,(3)Φχ,0=gχ.Fractional derivative is considered in the Caputo sense.The termλ∂2Φχ,ϒ/∂χ2 represents diffusion and zχ,ϒΦχ,ϒ represents the reaction, where zχ,ϒ is the reaction parameter, Φχ,ϒ is the concentration, and λ is the diffusion coefficient constant.Verdict on the results of fractional order CRDEs is a fascinating zone for the researchers. Chowdhury and Hashim applied homotopy-perturbation method (HPM) to acquire estimated analytical explanations for the CRDEs [20]. Ali et al. established estimated results of CRDEs by optimal homotopy asymptotic method (OHAM) [21]. Wang and Liu used a novel evaluating procedure for nonlinear time-fractional CRDE [22]. Kumar et al. applied homotopy analysis transform method (HATM) for cracking CRDEs [23]. Hosseini et al. recognized comparative explanation of CRDEs by Mittag–Leffler law [24]. Lima et al. considered problems of CRDEs by means of finite element approach [25].Elzaki transform was presented by Elzaki in 2011 [26]. It is a very useful method to resolve the entire natures of DEs.Elzaki transform was defined for functions of exponential order. We consider functions in the setξ defined as(4)ξ=Φϒ|∃M,Θ1,Θ2>0,|Φϒ<Meϒ/Θjifϒ∈−1jX0,∞.Elzaki transform is defined as(5)EΦϒ=υ∫0∞Φϒe−ϒ/υdϒ,Θ1≤υ≤Θ2,where E symbolizes Elzaki transform operator.The framework of this study is as follows. In the next section, a new form of fractional Taylor’s series is introduced that will be used in our work in the next sections and further explained and the conditions for convergence of the new form of Taylor’s formula were determined. Moreover, we presented some new results. Next, we build Elzaki residual power series solutions for CRDEs. Further, few problems are solved to illustrate the capability, the potentiality, and the simplicity of the proposed method. Eventually, our results are compiled in the conclusion.


## 2. Some New Results

In this section, we familiarize a novel formula of fractional Taylor’s series and elucidate and govern the circumstances for the convergence of the novel formula of fractional Taylor’s series and present some expedient outcomes which are pillars for the new effectual method.Lemma 1 (a new formula of fractional Taylor’s series in Elzaki transform).
Suppose thatΦϒ is a piecewise continuous and exponential order; the Elzaki transform of ΦϒEΦϒ=Ψυ has fractional Taylor’s series representation as(6)Ψυ=∑n=0∞ℵnυnϖ+2,where ℵn represents nth coefficient of the new formula of fractional Taylor’s series in Elzaki transform.Proof:.
Consider the following fractional Taylor’s series:(7)Φϒ=ℵ0+ℵ1Γϖ+1ϒϖ+ℵ2Γ2ϖ+1ϒ2ϖ+ℵ3Γ3ϖ+1ϒ3ϖ+⋯.
Applying Elzaki transform at the both sides of equation (6),(8)EΦϒ=ℵ0E1+ℵ1Γϖ+1Eϒϖ+ℵ2Γ2ϖ+1Eϒ2ϖ+ℵ3Γ3ϖ+1Eϒ3ϖ+ℵ4Γ4ϖ+1Eϒ4ϖ+⋯,Ψυ=∑n=0∞ℵnυnϖ+2.which is a new form of fractional Taylor formula in Elzaki transform form.Remark 1.
The multiple fractional Taylor’s series or generalized form of Taylor’s series representation atϒ=0 takes the following form in Elzaki transform space:(9)Ψχ,υ=∑n=0∞ℵnχυnϖ+2,whereχ=χ1,χ2,χ3…χd∈ℜd,d∈N.Lemma 2.
Assume that the functionEΦϒ=Ψυ has fractional power series (FPS) representation as follows:(10)Ψυ=∑n=0∞ℵnυnϖ+2.
Then,limυ⟶01/υ2Ψυ=ℵ0.Proof:.
From the new form of fractional Taylor’s series, we have(11)1υ2Ψυ=ℵ0+ℵ1υϖ+ℵ2υ2ϖ+ℵ3υ3ϖ+ℵ4υ4ϖ+⋯.
Taking limitυ⟶0, so the last equation becomes as(12)limυ⟶01υ2Ψυ=ℵ0=Φ0.Remark 2.
In the case of a generalized form of Taylor’s series in Elzaki transform space, we have the following:(13)limυ⟶01υ2Ψχ,υ=ℵ0χ,0,whereχ=χ1,χ2,χ3…χd∈ℜd,d∈N.Lemma 3.
Presume thatΦϒ is a piecewise continuous function on 0,∞ and exponential order, EΦϒ=Ψυ.
Then,(14)EDϒnϖΦϒ=EΦϒυnϖ−∑j=0n−1υj−nϖ+2DϒjϖΦ0,0<ϖ≤1,where DϒnϖΦ=Dϒϖ.Dϒϖ.Dϒϖ…Dϒϖn−times.Proof:.
To prove, we use the principle of mathematical induction method.
Usingn=1 in equation (14),(15)EDϒϖΦϒ=ΕΦϒυϖ−υ−ϖ+2Φ0.
Equation (14) is effective forn=1.
Usingn=2 in equation (14), we get(16)EDϒ2ϖΦϒ=EΦϒυ2ϖ−υ−2ϖ+2Φ0+υ−ϖ+2DϒϖΦ0,L.H.S=EDϒ2ϖΦϒ=EDϒϖDϒϖΦϒ.
LetDϒϖΦϒ=zϒ.
So, equation (16) becomes as(17)L.H.S=EDϒ2ϖΦϒ=EDϒϖzϒ.
By utilizing Caputo fractional derivative, the last equation becomes as(18)L.H.S=EDϒ2ϖΦϒ=EJϒ1−ϖz1ϒ.
By using Riemann–Liouville integral formula of Elzaki transform,(19)L.H.S=EDϒ2ϖΦϒ=υ1−ϖEz1ϒ.
By differential property, the above equation becomes as follows:(20)EDϒ2ϖΦϒ=υ−ϖZυ−υ2−ϖz0,whereDϒϖΦ0=z0.
From equation (20), we have(21)EDϒ2ϖΦϒ=υ−ϖEDϒϖΦϒ−υ2−ϖDϒϖΦ0,EDϒ2ϖΦϒ=EΦϒυ2ϖ−υ2−2ϖΦ0−υ2−ϖDϒϖΦ0.
So, from equation (21), we conclude that formula equation (14) is accurate when n=2. Now, suppose formula is valid for n=r. So, we have(22)EDϒrϖΦϒ=EΦϒυrϖ−∑j=0r−1υj−rϖ+2DϒjϖΦ0.
Now, we will prove forn=r+1.(23)EDϒr+1ϖΦϒ=ΕΦϒυrϖ−∑j=0r−1+1υj−r+1ϖ+2DϒjϖΦ0,(24)L.H.S=EDϒr+1ϖΦϒ,L.H.S=EDϒϖDϒrϖΦϒ.
Suppose that(25)DϒrϖΦϒ=bϒ.
So, equation (24) becomes as(26)L.H.S=EDϒϖbϒ.
By utilizing Caputo fractional derivative, so the last equation becomes as(27)L.H.S=EJϒ1−ϖb1ϒ.
By utilizing Riemann–Liouville fractional integral formula, equation (27) becomes as follows:(28)L.H.S=υ1−ϖEb1ϒ.
By differential property of E-L, so equation (28) becomes as(29)L.H.S=υϖEDϒrϖΦϒ−υ2−ϖb0,where DϒrϖΦ0=b0.
From equation (29), we get(30)L.H.S=EΦϒυr+1ϖ−∑j=0r−1υj−r+1ϖ+2DϒjϖΦ0−υ2−ϖb0,L.H.S=EΦϒυnϖ−∑j=0n−1υj−nϖ+2DϒjϖΦ0.
So, equation (14) is valid for all integers. Thus, the proof completes.Remark 3.
By making generalization, the above proved formula takes the following form:(31)EDϒnϖΦχ,ϒ=EΦχ,ϒυnϖ−∑j=0n−1υj−nϖ+2DϒjϖΦ0,whereχ=χ1,χ2,χ3,…χd∈ℜd,d∈N,and DϒnϖΦ=Dϒϖ.Dϒϖ.Dϒϖ…Dϒϖn−times.Theorem 1.
Suppose that the functionEΦϒ=Ψυ has FPS representation as follows:(32)Ψυ=∑n=0∞ℵnυnϖ+2,then we haveℵn=DϒnϖΦ0, where DϒnϖΦ=Dϒϖ.Dϒϖ.Dϒϖ…Dϒϖn−times.Proof:.
Consider new form of Taylor’s series.(33)Ψυ=ℵ0υ2+ℵ1υϖ+2+ℵ2υ2ϖ+2+ℵ3υ3ϖ+2+ℵ4υ4ϖ+2+⋯.
From the above equation, we have(34)ℵ1=1υϖ+2Ψυ−1υϖ+2υ2Ψ0−1υϖ+2ℵ2υ2ϖ+2−1υϖ+2ℵ3υ3ϖ+2−1υϖ+2ℵ4υ4ϖ+2+⋯.
Takingυ⟶0 on the above equation,(35)ℵ1=limυ⟶01υ21υϖΨυ−1υϖ−2Ψ0.
By using Lemma3,(36)ℵ1=limυ⟶01υ2EDϒϖΦϒυ.
By Lemma2, the above equation becomes as(37)ℵ1=DϒϖΦ0.
Again from equation (32),(38)ℵ2=1υ21υ2ϖΨυ−1υ2ϖ−2Ψ0−1υϖ−2ℵ1−ℵ3υϖ−ℵ4υ2ϖ+⋯.
Takingυ⟶0 on the last equation, utilizing Lemma 3, the above equation becomes as(39)ℵ2=limυ⟶01υ2EDϒ2ϖΦϒυ.
By Lemma2,(40)ℵ2=Dϒ2ϖΦ0.
Again from equation (32), we have(41)ℵ3=limυ⟶01υ21υ3ϖΨυ−Ψ01υ3ϖ−2−ℵ11υ2ϖ−ℵ21υϖ−2.
By Lemma3,(42)ℵ3=limυ⟶01υ2EDϒ3ϖΦϒυ.
By Lemma2, the last equation becomes as(43)ℵ3=Dϒ3ϖΦ0.
In the same manner, we can obtain the following form by making generalization:(44)ℵn=DϒnϖΦ0.
This completes the proof of the theorem.Remark 4.
For multiple Taylor’s series, the proved result becomes as follows:(45)ℵnχ=DϒnϖΦ0,whereχ=χ1,χ2,χ3,…χd∈ℜd,d∈N.
The following theorem describes and determines the conditions for convergence of the new form of Taylor’s formula that are introduced in Lemma1.Theorem 2.
LetΨυ=EΦϒ be represented as the new form of fractional Taylor’s formula as in Elzaki transform:(46)Ψυ=∑n=0∞ℵnυnϖ+2,if(47)1υ2EDϒn+1ϖΦϒ≤T.Then, the remainderRnυ of the new form of fractional Taylor’s formula satisfies the following inequality:(48)Rnυ≤υn+1ϖ+2T.Proof:.
Consider the following:(49)Ψnυ=ℵ0υ2+ℵ1υϖ+2+ℵ2υ2ϖ+2+ℵ3υ3ϖ+2+⋯ℵnυnϖ+2.
From equations (46) and (49), we get(50)Rnυ=Ψυ−∑k=0nℵkυkϖ+2.
By Theorem1,(51)Rnυ=Ψυ−∑k=0nυkϖ+2DϒkϖΦ0,1υn+1ϖ+2Rnυ=1υ21υn+1ϖΨυ−∑k=0n1υn+1−kϖ−2DϒkϖΦ0.
By Lemma3,(52)1υn+1ϖ+2Rnυ=1υ2EDϒn+1ϖΦϒ,1υn+1ϖ+2Rnυ=1υ2EDϒn+1ϖΦϒ.
By the given assumption, the above equation becomes as(53)−υn+1ϖ+2T≤Rnυ≤υn+1ϖ+2T,Rnυ≤υn+1ϖ+2T. Hence, the required result is proved.


## 3. Demonstrating the ERPSM for the CRDEs

We exploit our novel ERPSM to originate the results of the linear and nonlinear CRDEs. The foremost set of rules of this method for resolving the CRDEs can be accumulated by the following steps: employing the Elzaki transform to CRDE and then deploying the novel form of Taylor’s series to introduce the solution of CRDE in the novel space. The coefficients of this series are established with a new idea. At the end, employing the inverse Elzaki transform to achieve the solution of the problem in the actual space.


### 3.1. Elzaki Residual Power Series Solutions for the CRDEs

In this subsection, we systematized the stages for conquering the Elzaki residual power series solution for the linear and nonlinear CRDE by the following procedure.Step 1. Rewriting equation (2) as demonstrated:(54)∂ϖΦχ,ϒ∂ϒϖ−λ∂2Φχ,ϒ∂χ2−zχ,ϒΦχ,ϒ=0.Step 2. Manipulating Elzaki transform at both sides of equation (54), we get in this way(55)E∂ϖΦχ,ϒ∂ϒϖ−λE∂2Φχ,ϒ∂χ2−EE−1Zχ,υE−1Ψχ,υ=0,where(56)E−1Zχ,υ=zχ,ϒ,E−1Ψχ,υ=Φχ,ϒ,E∂ϖΦχ,ϒ∂ϒϖ=Ψχ,υυϖ−υ−ϖ+2gχ.So, we get the following form:(57)Ψχ,υ=υ2gχ+λυϖDχχΨχ,υ+υϖEE−1Zχ,υE−1Ψχ,υ.Step 3. Considering the solution of equation (57) as the following:(58)Ψχ,υ=∑n=0∞ℵnχυ2+nϖ.Step 4. Settingℵ0χ=limυ⟶∞1/υ2Ψχ,υ=Φχ,0.Step 5. Establishing thekth-truncated series of Ψχ,υ as(59)Ψkχ,υ=∑n=0kℵnχυ2+nϖ,ℵ0=limυ⟶01υ2Ψχ,υ,Ψkχ,υ=ℵ0υ2+∑n=1kℵnχυ2+nϖ.Step 6. Considering the Elzaki residual function (ERF) of equation (57) and the kth-truncated ERF separately such that(60)EResχ,υ=Ψχ,υ−υ2gχ−λυϖDχχΨχ,υ−υϖEE−1Zχ,υE−1Ψχ,υ,EReskχ,υ=Ψkχ,υ−υ2gχ−λυϖDχχΨkχ,υ−υϖEE−1Zχ,υE−1Ψkχ,υ.Step 7. Replacing the series form ofΨkχ,υ into equation (60).Step 8. Dividing at both sides of equation (60) with υkυ+2 as follows:(61)1υ2+kϖEReskχ,υ=1υ2+kϖΨkχ,υ−1υ2+kϖυ2gχ−λ1υ2+kϖυϖDχχΨkχ,υ−1υ2+kϖυϖEE−1Zχ,υE−1Ψkχ,υ.Step 9. Taking limit at both sides of equation (61).(62)limυ⟶01υ2+kϖEReskχ,υ=limυ⟶01υ2+kϖΨkχ,υ−limυ⟶01υ2+kϖυ2gχ−λlimυ⟶01υ2+kϖυϖDχχΨkχ,υ−limυ⟶01υ2+kϖυϖEE−1Zχ,υE−1Ψkχ,υ.Step 10. Solving the following equation forℵnχ:(63)limυ⟶01υkϖ+2EReskχ,υ=0,k=1,2,3,….Step 11. Replacing the attained values ofℵnχ into kth-truncated series of Ψχ,υ to get the kth-approximate solution of equation (57).Step 12. Manipulating the inverse Elzaki transform onΨkχ,υ to attain the kth-approximate solution of Φkχ,ϒ in the real space.


### 3.2. Applications to Linear and Nonlinear CRDEs

In this subsection, we consider three main problems of CRDEs to illustrate the execution and capability of ERPSM.


#### 3.2.1. Approximate and Closed Form Solutions of Linear CRDEs

Two applications are considered for linear CRDEs.Problem 1.
Consider the time-fractional linear CRDE [19].(64)DϒϖΦχ,ϒ=Φχχχ,ϒ−Φχ,ϒ,χ,ϒ≥0,0<ϖ≤1.
Subject to initial condition,(65)Φχ,0=e−χ+χ.
Solution. Utilizing Elzaki transform on equation (64),(66)EDϒϖΦχ,ϒ=EΦχχχ,ϒ−EΦχ,ϒ,where EΦχ,ϒ=Ψχ,υ,(67)EDϒϖΦχ,ϒ=Ψχ,υυϖ−υ−ϖ+2Φχ,0,so equation (66) becomes as(68)Ψχ,υ=υ2Φχ,0+υϖDχχΨχ,υ−υϖΨχ,υ.
Initiate a series solution to the algebraic equation (68). Hence, presume that the expansion of Ψχ,υ is the following:(69)Ψχ,υ=∑n=0∞ℵnχυ2+nϖ.
Assume thatΨχ,υ has the kth-truncated series as(70)Ψkχ,υ=∑n=0kℵnχυ2+nϖ.
By Lemma2, we have(71)limυ⟶01υ2Ψχ,υ=Φχ,0=e−χ+χ.
Thekth-truncated series becomes as follows:(72)Ψkχ,υ=υ2e−χ+χ+∑n=1kℵnχυ2+nϖ.
The ERF of the algebraic equation (68) is described as(73)EResχ,υ=Ψχ,υ−υ2e−χ+χ−υϖDχχΨχ,υ+υϖΨχ,υ.
Furthermore,kth-truncated ERF of the algebraic equation (69) is explained as follows:(74)EReskχ,υ=Ψkχ,υ−υ2e−χ+χ−υϖDχχΨkχ,υ+υϖΨkχ,υ.
By utilizing equations (72) and (74), we get undefined coefficients in the following form:(75)ℵ1χ=−χ,ℵ2χ=χ,ℵ3χ=−χ,ℵ4χ=χ,ℵ5χ=−χ.
So, we get the 5th approximate solution of Elzaki transform of equation (68).(76)Ψ5χ,υ=e−χ+χυ2−χυ2+ϖ+χυ2+2ϖ−χυ2+3ϖ+χυ2+4ϖ−χυ2+5ϖ.
Operating inverse Elzaki transform on both sides of equation (76), we get the 5th approximate solution of equation (64).(77)Φ5χ,ϒ=e−χ+χ1−ϒϖΓϖ+1+ϒ2ϖΓ2ϖ+1−ϒ3ϖΓ3ϖ+1+ϒ4ϖΓ4ϖ+1−ϒ5ϖΓ5ϖ+1.
Whenϖ=1, equation (77) becomes as(78)Φ5χ,ϒ=e−χ+χ1−ϒ1!+ϒ22!−ϒ33!+ϒ44!−ϒ55!.
Equation (78) is coinciding with the six terms of the expansion of the exact solution Φχ,ϒ=e−χ+χe−ϒ.
Table1 demonstrates the values of absolute error of the 5th order approximate and exact solutions at ϖ=1 when χ=1 which support the capability and exactness of the novel technique.
Figure1 displays the evaluations of exact solution at ϖ=1 and the 5th approximate solution of Problem 1, at χ=1, for several values of ϒandϖ. Figure 1 confirms that when values of ϖ approach to “1,” the approximate solution approaches to the exact solution, which approves the efficacy and correctness of the new method. Moreover, the approximate solution overlaps with the exact solution at ϖ=1 and this once more ratifies the usefulness and correctness of the ERPSM.Table 1
Absolute error of ERPS results.
ϒExact solutionApproximate solutionAbsolute error02.718281828462.7182818284600.062.886370989272.886370989091.7766543792 × 10−100.123.064854203293.064854191821.1469452055 × 10−80.183.254374202893.25437407111.3178722424 × 10−70.243.455613464763.455612717787.4698766506 × 10−70.303.669296667623.669293792830.000002874789631680.363.89619330183.896184641130.000008660664293720.424.137120440254.137098405160.00002203509554110.484.392945680924.392896138730.044093686996Figure 1
Evaluation of closed form and approximate consequences of Problem1.Problem 2.
Consider the time-fractional linear CRDE [20],(79)DϒϖΦχ,ϒ=Φχχχ,ϒ−1+4χ2Φχ,ϒ,χ,ϒ≥0,0<ϖ≤1.
With the initial condition,(80)Φχ,0=eχ2.
Solution. Manipulating Elzaki transform on equation (79),(81)EDϒϖΦχ,ϒ=EΦχχχ,ϒ−1+4χ2EΦχ,ϒ,Ψχ,υ=υ2eχ2+υϖDχχΨχ,υ−υϖ1+4χ2Ψχ,υ,where(82)EDϒϖΦχ,ϒ=Ψχ,υυϖ−υ−ϖ+2eχ2,EΦχ,ϒ=Ψχ,υ.
Now, establishing a series solution of equation (81), consequently assume that Ψχ,υ has the expansion as follows:(83)Ψχ,υ=∑n=0∞ℵnχυ2+nϖ.
Thekth-truncated series Ψχ,υ is as follows:(84)Ψkχ,υ=∑n=0kℵnχυ2+nϖ.
By Lemma2, we have(85)limυ⟶01υ2Ψχ,υ=Φχ,0=eχ2.
So, equation (84) becomes as(86)Ψkχ,υ=eχ2υ2+∑n=1kℵnχυ2+nϖ.
The ERF of equation (81) is defined as(87)EResχ,υ=Ψχ,υ−υ2eχ2−υϖDχχΨχ,υ+υϖ1+4χ2Ψχ,υ.
Thekth-ERF is as follows:(88)EReskχ,υ=Ψkχ,υ−υ2eχ2−υϖDχχΨkχ,υ+υϖ1+4χ2Ψkχ,υ.
To find unspecified coefficients using equations (86) and (88), so we have(89)ℵ1χ=eχ2,ℵ2χ=eχ2,ℵ3χ=eχ2,ℵ4χ=eχ2,ℵ5χ=eχ2.
The 5th approximate solution of equation (81) in Elzaki transform form is(90)Ψ5χ,υ=eχ21υ2+1υ2+ϖ+1υ2+2ϖ+1υ2+3ϖ+1υ2+4ϖ+1υ2+5ϖ.
By applying inverse Elzaki transform on equation (90), we get the 5th approximate solution of equation (79) as follows:(91)Φ5χ,ϒ=eχ21+ϒϖΓϖ+1+ϒ2ϖΓ2ϖ+1+ϒ3ϖΓ3ϖ+1+ϒ4ϖΓ4ϖ+1+ϒ5ϖΓ5ϖ+1.
Whenϖ=1, equation (91) becomes as(92)Φ5χ,ϒ=eχ21+ϒ1!+ϒ22!+ϒ33!+ϒ44!+ϒ55!.
Equation (92) represents the first six terms of the expansion of eχ2+ϒ, so closed form solution of equation (79) is eχ2+ϒ.
Table2 demonstrates the values of absolute error of the 5th order approximate and exact solutions at ϖ=1 when χ=1, which support the capability and accuracy of the new technique.
Figure2 demonstrates the exploits of exact solution at ϖ=1 and the 5th approximate solution of Problem 2, when χ=1 for numerous values of ϒandϖ. The figure endorses that when values of ϖ approach to “1,” the approximate solution approaches to exact solution, which supports the ability and precision of the new method. Moreover, the approximate solution overlaps with the exact solution at ϖ=1 and this once more ratifies the usefulness and correctness of the ERPSM.Table 2
Absolute error of ERPS results.
ϒExact solutionApproximate solutionAbsolute error0.001.367879441171.367879441170.00.061.309643974761.309643974697 × 10−110.121.254799877891.254799873814.08 × 10−90.181.203149652581.203149606534.605 × 10−80.241.154507302241.154507045652.5659 × 10−70.301.108697661851.108696691179.7068 × 10−70.361.065555767241.065552892690.000002874550.421.024926260991.024919071810.000007189180.480.986662832970.986646944530.000015888447Figure 2
The behavior of exact and approximate outcomes of Problem2.


#### 3.2.2. Approximate and Closed Form Solutions of Nonlinear CRDEs

Problem 3.
Consider the nonlinear time-fractional CRDE [21],(93)DϒϖΦχ,ϒ=Φχχχ,ϒ−Φχχ,ϒ+Φχ,ϒΦχχχ,ϒ−Φ2χ,ϒ+Φχ,ϒ,χ,ϒ≥0,0<ϖ≤1,
With the initial condition,(94)Φχ,0=eχ.
Solution. By applying Elzaki transform on equation (93), we get(95)Ψχ,υ=υ2eχ+υϖDχχΨχ,υ−υϖDχΨχ,υ+υϖEE−1Ψχ,υDχχE−1Ψχ,υ−υϖEE−1Ψχ,υ2+υϖΨχ,υ.
Here,(96)EΦχχχ,ϒ=DχχΨχ,υ,EΦχχ,ϒ=DχΨχ,υ,Φχ,ϒ=E−1Ψχ,υ,Φ2χ,ϒ=E−1Ψχ,υ2,Φχχχ,ϒ=DχχE−1Ψχ,υ,EDϒϖΦχ,ϒ=Ψχ,υυϖ−υ−ϖ+2eχ.
Define a series solution of equation (95) as follows:(97)Ψχ,υ=∑n=0∞ℵnχυ2+nϖ.
Thekth-truncated series is(98)Ψkχ,υ=∑n=0kℵnχυ2+nϖ.
By Lemma2,(99)limυ⟶01υ2Ψχ,υ=Φχ,0=eχ.
Therefore, thekth-truncated series becomes as(100)Ψkχ,υ=eχυ2+∑n=1kℵnχυ2+nϖ.
Now, define ERF in the following form:(101)EResχ,υ=Ψχ,υ−υ2eχ−υϖDχχΨχ,υ+υϖDχΨχ,υ−υϖEE−1Ψχ,υDχχE−1Ψχ,υ+υϖEE−1Ψχ,υ2−υϖΨχ,υ.
Thekth-truncated ERF is(102)EReskχ,υ=Ψkχ,υ−υ2eχ−υϖDχχΨkχ,υ+υϖDχΨkχ,υ−υϖEE−1Ψkχ,υDχχE−1Ψkχ,υ+υϖEE−1Ψkχ,υ2−υϖΨkχ,υ.
The undefined coefficients are determined in the following form by utilizing equations (100) and (102).(103)ℵ1χ=eχ,ℵ2χ=eχ,ℵ3χ=eχ,ℵ4χ=eχ,ℵ5χ=eχ.
The 5th approximate solution of equation (95) is given as(104)Ψ5χ,υ=eχ1υ2+1υ2+ϖ+1υ2+2ϖ+1υ2+3ϖ+1υ2+4ϖ+1υ2+5ϖ.
By applying inverse Elzaki transform on the above equation, we get the 5th approximate solution of equation (93).(105)Φ5χ,ϒ=eχ1+ϒϖΓϖ+1+ϒ2ϖΓ2ϖ+1+ϒ3ϖΓ3ϖ+1+ϒ4ϖΓ4ϖ+1+ϒ5ϖΓ4ϖ+1.
Forϖ=1, the last equation becomes as(106)Φ5χ,ϒ=eχ1+ϒ1!+ϒ22!+ϒ33!+ϒ44!+ϒ55!.
Equation (106) coincides with the 1st six terms of the expansion of eχ+ϒ, therefore exact solution of equation (93) is eχ+ϒ.
Table3 demonstrates the values of absolute error of the 5th order approximate and exact results at ϖ=1, and χ=1, which support the ability and accuracy of the novel technique.
Figure3 establishes the actions of exact solution at ϖ=1 and the 5th approximate solution of Problem 3, when χ=1 for certain values of ϒandϖ. The figure recommends that when values of ϖ approach to “1,” the approximate solution approaches to exact solution, which supports the capability and exactness of the new method. Moreover, the approximate solution overlaps with the exact solution at ϖ=1 and this once more ratifies the usefulness and correctness of the ERPSM.Table 3
Absolute error of ERPS results.
ϒExact solutionApproximate solutionAbsolute error0.001.367879441171.367879441170.00.061.309643974761.309643974697 × 10−110.121.254799877891.254799873814.08 × 10−90.181.203149652581.203149606534.605 × 10−80.241.154507302241.154507045652.5659 × 10−70.301.108697661851.108696691179.7068 × 10−70.361.065555767241.065552892690.000002874550.421.024926260991.024919071810.000007189180.480.986662832970.986646944530.000015888447Figure 3
2D plot of exact and approximate solution of Problem3.


## 3.1. Elzaki Residual Power Series Solutions for the CRDEs

In this subsection, we systematized the stages for conquering the Elzaki residual power series solution for the linear and nonlinear CRDE by the following procedure.Step 1. Rewriting equation (2) as demonstrated:(54)∂ϖΦχ,ϒ∂ϒϖ−λ∂2Φχ,ϒ∂χ2−zχ,ϒΦχ,ϒ=0.Step 2. Manipulating Elzaki transform at both sides of equation (54), we get in this way(55)E∂ϖΦχ,ϒ∂ϒϖ−λE∂2Φχ,ϒ∂χ2−EE−1Zχ,υE−1Ψχ,υ=0,where(56)E−1Zχ,υ=zχ,ϒ,E−1Ψχ,υ=Φχ,ϒ,E∂ϖΦχ,ϒ∂ϒϖ=Ψχ,υυϖ−υ−ϖ+2gχ.So, we get the following form:(57)Ψχ,υ=υ2gχ+λυϖDχχΨχ,υ+υϖEE−1Zχ,υE−1Ψχ,υ.Step 3. Considering the solution of equation (57) as the following:(58)Ψχ,υ=∑n=0∞ℵnχυ2+nϖ.Step 4. Settingℵ0χ=limυ⟶∞1/υ2Ψχ,υ=Φχ,0.Step 5. Establishing thekth-truncated series of Ψχ,υ as(59)Ψkχ,υ=∑n=0kℵnχυ2+nϖ,ℵ0=limυ⟶01υ2Ψχ,υ,Ψkχ,υ=ℵ0υ2+∑n=1kℵnχυ2+nϖ.Step 6. Considering the Elzaki residual function (ERF) of equation (57) and the kth-truncated ERF separately such that(60)EResχ,υ=Ψχ,υ−υ2gχ−λυϖDχχΨχ,υ−υϖEE−1Zχ,υE−1Ψχ,υ,EReskχ,υ=Ψkχ,υ−υ2gχ−λυϖDχχΨkχ,υ−υϖEE−1Zχ,υE−1Ψkχ,υ.Step 7. Replacing the series form ofΨkχ,υ into equation (60).Step 8. Dividing at both sides of equation (60) with υkυ+2 as follows:(61)1υ2+kϖEReskχ,υ=1υ2+kϖΨkχ,υ−1υ2+kϖυ2gχ−λ1υ2+kϖυϖDχχΨkχ,υ−1υ2+kϖυϖEE−1Zχ,υE−1Ψkχ,υ.Step 9. Taking limit at both sides of equation (61).(62)limυ⟶01υ2+kϖEReskχ,υ=limυ⟶01υ2+kϖΨkχ,υ−limυ⟶01υ2+kϖυ2gχ−λlimυ⟶01υ2+kϖυϖDχχΨkχ,υ−limυ⟶01υ2+kϖυϖEE−1Zχ,υE−1Ψkχ,υ.Step 10. Solving the following equation forℵnχ:(63)limυ⟶01υkϖ+2EReskχ,υ=0,k=1,2,3,….Step 11. Replacing the attained values ofℵnχ into kth-truncated series of Ψχ,υ to get the kth-approximate solution of equation (57).Step 12. Manipulating the inverse Elzaki transform onΨkχ,υ to attain the kth-approximate solution of Φkχ,ϒ in the real space.


## 3.2. Applications to Linear and Nonlinear CRDEs

In this subsection, we consider three main problems of CRDEs to illustrate the execution and capability of ERPSM.


### 3.2.1. Approximate and Closed Form Solutions of Linear CRDEs

Two applications are considered for linear CRDEs.Problem 1.
Consider the time-fractional linear CRDE [19].(64)DϒϖΦχ,ϒ=Φχχχ,ϒ−Φχ,ϒ,χ,ϒ≥0,0<ϖ≤1.
Subject to initial condition,(65)Φχ,0=e−χ+χ.
Solution. Utilizing Elzaki transform on equation (64),(66)EDϒϖΦχ,ϒ=EΦχχχ,ϒ−EΦχ,ϒ,where EΦχ,ϒ=Ψχ,υ,(67)EDϒϖΦχ,ϒ=Ψχ,υυϖ−υ−ϖ+2Φχ,0,so equation (66) becomes as(68)Ψχ,υ=υ2Φχ,0+υϖDχχΨχ,υ−υϖΨχ,υ.
Initiate a series solution to the algebraic equation (68). Hence, presume that the expansion of Ψχ,υ is the following:(69)Ψχ,υ=∑n=0∞ℵnχυ2+nϖ.
Assume thatΨχ,υ has the kth-truncated series as(70)Ψkχ,υ=∑n=0kℵnχυ2+nϖ.
By Lemma2, we have(71)limυ⟶01υ2Ψχ,υ=Φχ,0=e−χ+χ.
Thekth-truncated series becomes as follows:(72)Ψkχ,υ=υ2e−χ+χ+∑n=1kℵnχυ2+nϖ.
The ERF of the algebraic equation (68) is described as(73)EResχ,υ=Ψχ,υ−υ2e−χ+χ−υϖDχχΨχ,υ+υϖΨχ,υ.
Furthermore,kth-truncated ERF of the algebraic equation (69) is explained as follows:(74)EReskχ,υ=Ψkχ,υ−υ2e−χ+χ−υϖDχχΨkχ,υ+υϖΨkχ,υ.
By utilizing equations (72) and (74), we get undefined coefficients in the following form:(75)ℵ1χ=−χ,ℵ2χ=χ,ℵ3χ=−χ,ℵ4χ=χ,ℵ5χ=−χ.
So, we get the 5th approximate solution of Elzaki transform of equation (68).(76)Ψ5χ,υ=e−χ+χυ2−χυ2+ϖ+χυ2+2ϖ−χυ2+3ϖ+χυ2+4ϖ−χυ2+5ϖ.
Operating inverse Elzaki transform on both sides of equation (76), we get the 5th approximate solution of equation (64).(77)Φ5χ,ϒ=e−χ+χ1−ϒϖΓϖ+1+ϒ2ϖΓ2ϖ+1−ϒ3ϖΓ3ϖ+1+ϒ4ϖΓ4ϖ+1−ϒ5ϖΓ5ϖ+1.
Whenϖ=1, equation (77) becomes as(78)Φ5χ,ϒ=e−χ+χ1−ϒ1!+ϒ22!−ϒ33!+ϒ44!−ϒ55!.
Equation (78) is coinciding with the six terms of the expansion of the exact solution Φχ,ϒ=e−χ+χe−ϒ.
Table1 demonstrates the values of absolute error of the 5th order approximate and exact solutions at ϖ=1 when χ=1 which support the capability and exactness of the novel technique.
Figure1 displays the evaluations of exact solution at ϖ=1 and the 5th approximate solution of Problem 1, at χ=1, for several values of ϒandϖ. Figure 1 confirms that when values of ϖ approach to “1,” the approximate solution approaches to the exact solution, which approves the efficacy and correctness of the new method. Moreover, the approximate solution overlaps with the exact solution at ϖ=1 and this once more ratifies the usefulness and correctness of the ERPSM.Table 1
Absolute error of ERPS results.
ϒExact solutionApproximate solutionAbsolute error02.718281828462.7182818284600.062.886370989272.886370989091.7766543792 × 10−100.123.064854203293.064854191821.1469452055 × 10−80.183.254374202893.25437407111.3178722424 × 10−70.243.455613464763.455612717787.4698766506 × 10−70.303.669296667623.669293792830.000002874789631680.363.89619330183.896184641130.000008660664293720.424.137120440254.137098405160.00002203509554110.484.392945680924.392896138730.044093686996Figure 1
Evaluation of closed form and approximate consequences of Problem1.Problem 2.
Consider the time-fractional linear CRDE [20],(79)DϒϖΦχ,ϒ=Φχχχ,ϒ−1+4χ2Φχ,ϒ,χ,ϒ≥0,0<ϖ≤1.
With the initial condition,(80)Φχ,0=eχ2.
Solution. Manipulating Elzaki transform on equation (79),(81)EDϒϖΦχ,ϒ=EΦχχχ,ϒ−1+4χ2EΦχ,ϒ,Ψχ,υ=υ2eχ2+υϖDχχΨχ,υ−υϖ1+4χ2Ψχ,υ,where(82)EDϒϖΦχ,ϒ=Ψχ,υυϖ−υ−ϖ+2eχ2,EΦχ,ϒ=Ψχ,υ.
Now, establishing a series solution of equation (81), consequently assume that Ψχ,υ has the expansion as follows:(83)Ψχ,υ=∑n=0∞ℵnχυ2+nϖ.
Thekth-truncated series Ψχ,υ is as follows:(84)Ψkχ,υ=∑n=0kℵnχυ2+nϖ.
By Lemma2, we have(85)limυ⟶01υ2Ψχ,υ=Φχ,0=eχ2.
So, equation (84) becomes as(86)Ψkχ,υ=eχ2υ2+∑n=1kℵnχυ2+nϖ.
The ERF of equation (81) is defined as(87)EResχ,υ=Ψχ,υ−υ2eχ2−υϖDχχΨχ,υ+υϖ1+4χ2Ψχ,υ.
Thekth-ERF is as follows:(88)EReskχ,υ=Ψkχ,υ−υ2eχ2−υϖDχχΨkχ,υ+υϖ1+4χ2Ψkχ,υ.
To find unspecified coefficients using equations (86) and (88), so we have(89)ℵ1χ=eχ2,ℵ2χ=eχ2,ℵ3χ=eχ2,ℵ4χ=eχ2,ℵ5χ=eχ2.
The 5th approximate solution of equation (81) in Elzaki transform form is(90)Ψ5χ,υ=eχ21υ2+1υ2+ϖ+1υ2+2ϖ+1υ2+3ϖ+1υ2+4ϖ+1υ2+5ϖ.
By applying inverse Elzaki transform on equation (90), we get the 5th approximate solution of equation (79) as follows:(91)Φ5χ,ϒ=eχ21+ϒϖΓϖ+1+ϒ2ϖΓ2ϖ+1+ϒ3ϖΓ3ϖ+1+ϒ4ϖΓ4ϖ+1+ϒ5ϖΓ5ϖ+1.
Whenϖ=1, equation (91) becomes as(92)Φ5χ,ϒ=eχ21+ϒ1!+ϒ22!+ϒ33!+ϒ44!+ϒ55!.
Equation (92) represents the first six terms of the expansion of eχ2+ϒ, so closed form solution of equation (79) is eχ2+ϒ.
Table2 demonstrates the values of absolute error of the 5th order approximate and exact solutions at ϖ=1 when χ=1, which support the capability and accuracy of the new technique.
Figure2 demonstrates the exploits of exact solution at ϖ=1 and the 5th approximate solution of Problem 2, when χ=1 for numerous values of ϒandϖ. The figure endorses that when values of ϖ approach to “1,” the approximate solution approaches to exact solution, which supports the ability and precision of the new method. Moreover, the approximate solution overlaps with the exact solution at ϖ=1 and this once more ratifies the usefulness and correctness of the ERPSM.Table 2
Absolute error of ERPS results.
ϒExact solutionApproximate solutionAbsolute error0.001.367879441171.367879441170.00.061.309643974761.309643974697 × 10−110.121.254799877891.254799873814.08 × 10−90.181.203149652581.203149606534.605 × 10−80.241.154507302241.154507045652.5659 × 10−70.301.108697661851.108696691179.7068 × 10−70.361.065555767241.065552892690.000002874550.421.024926260991.024919071810.000007189180.480.986662832970.986646944530.000015888447Figure 2
The behavior of exact and approximate outcomes of Problem2.


### 3.2.2. Approximate and Closed Form Solutions of Nonlinear CRDEs

Problem 3.
Consider the nonlinear time-fractional CRDE [21],(93)DϒϖΦχ,ϒ=Φχχχ,ϒ−Φχχ,ϒ+Φχ,ϒΦχχχ,ϒ−Φ2χ,ϒ+Φχ,ϒ,χ,ϒ≥0,0<ϖ≤1,
With the initial condition,(94)Φχ,0=eχ.
Solution. By applying Elzaki transform on equation (93), we get(95)Ψχ,υ=υ2eχ+υϖDχχΨχ,υ−υϖDχΨχ,υ+υϖEE−1Ψχ,υDχχE−1Ψχ,υ−υϖEE−1Ψχ,υ2+υϖΨχ,υ.
Here,(96)EΦχχχ,ϒ=DχχΨχ,υ,EΦχχ,ϒ=DχΨχ,υ,Φχ,ϒ=E−1Ψχ,υ,Φ2χ,ϒ=E−1Ψχ,υ2,Φχχχ,ϒ=DχχE−1Ψχ,υ,EDϒϖΦχ,ϒ=Ψχ,υυϖ−υ−ϖ+2eχ.
Define a series solution of equation (95) as follows:(97)Ψχ,υ=∑n=0∞ℵnχυ2+nϖ.
Thekth-truncated series is(98)Ψkχ,υ=∑n=0kℵnχυ2+nϖ.
By Lemma2,(99)limυ⟶01υ2Ψχ,υ=Φχ,0=eχ.
Therefore, thekth-truncated series becomes as(100)Ψkχ,υ=eχυ2+∑n=1kℵnχυ2+nϖ.
Now, define ERF in the following form:(101)EResχ,υ=Ψχ,υ−υ2eχ−υϖDχχΨχ,υ+υϖDχΨχ,υ−υϖEE−1Ψχ,υDχχE−1Ψχ,υ+υϖEE−1Ψχ,υ2−υϖΨχ,υ.
Thekth-truncated ERF is(102)EReskχ,υ=Ψkχ,υ−υ2eχ−υϖDχχΨkχ,υ+υϖDχΨkχ,υ−υϖEE−1Ψkχ,υDχχE−1Ψkχ,υ+υϖEE−1Ψkχ,υ2−υϖΨkχ,υ.
The undefined coefficients are determined in the following form by utilizing equations (100) and (102).(103)ℵ1χ=eχ,ℵ2χ=eχ,ℵ3χ=eχ,ℵ4χ=eχ,ℵ5χ=eχ.
The 5th approximate solution of equation (95) is given as(104)Ψ5χ,υ=eχ1υ2+1υ2+ϖ+1υ2+2ϖ+1υ2+3ϖ+1υ2+4ϖ+1υ2+5ϖ.
By applying inverse Elzaki transform on the above equation, we get the 5th approximate solution of equation (93).(105)Φ5χ,ϒ=eχ1+ϒϖΓϖ+1+ϒ2ϖΓ2ϖ+1+ϒ3ϖΓ3ϖ+1+ϒ4ϖΓ4ϖ+1+ϒ5ϖΓ4ϖ+1.
Forϖ=1, the last equation becomes as(106)Φ5χ,ϒ=eχ1+ϒ1!+ϒ22!+ϒ33!+ϒ44!+ϒ55!.
Equation (106) coincides with the 1st six terms of the expansion of eχ+ϒ, therefore exact solution of equation (93) is eχ+ϒ.
Table3 demonstrates the values of absolute error of the 5th order approximate and exact results at ϖ=1, and χ=1, which support the ability and accuracy of the novel technique.
Figure3 establishes the actions of exact solution at ϖ=1 and the 5th approximate solution of Problem 3, when χ=1 for certain values of ϒandϖ. The figure recommends that when values of ϖ approach to “1,” the approximate solution approaches to exact solution, which supports the capability and exactness of the new method. Moreover, the approximate solution overlaps with the exact solution at ϖ=1 and this once more ratifies the usefulness and correctness of the ERPSM.Table 3
Absolute error of ERPS results.
ϒExact solutionApproximate solutionAbsolute error0.001.367879441171.367879441170.00.061.309643974761.309643974697 × 10−110.121.254799877891.254799873814.08 × 10−90.181.203149652581.203149606534.605 × 10−80.241.154507302241.154507045652.5659 × 10−70.301.108697661851.108696691179.7068 × 10−70.361.065555767241.065552892690.000002874550.421.024926260991.024919071810.000007189180.480.986662832970.986646944530.000015888447Figure 3
2D plot of exact and approximate solution of Problem3.


## 3.2.1. Approximate and Closed Form Solutions of Linear CRDEs

Two applications are considered for linear CRDEs.Problem 1.
Consider the time-fractional linear CRDE [19].(64)DϒϖΦχ,ϒ=Φχχχ,ϒ−Φχ,ϒ,χ,ϒ≥0,0<ϖ≤1.
Subject to initial condition,(65)Φχ,0=e−χ+χ.
Solution. Utilizing Elzaki transform on equation (64),(66)EDϒϖΦχ,ϒ=EΦχχχ,ϒ−EΦχ,ϒ,where EΦχ,ϒ=Ψχ,υ,(67)EDϒϖΦχ,ϒ=Ψχ,υυϖ−υ−ϖ+2Φχ,0,so equation (66) becomes as(68)Ψχ,υ=υ2Φχ,0+υϖDχχΨχ,υ−υϖΨχ,υ.
Initiate a series solution to the algebraic equation (68). Hence, presume that the expansion of Ψχ,υ is the following:(69)Ψχ,υ=∑n=0∞ℵnχυ2+nϖ.
Assume thatΨχ,υ has the kth-truncated series as(70)Ψkχ,υ=∑n=0kℵnχυ2+nϖ.
By Lemma2, we have(71)limυ⟶01υ2Ψχ,υ=Φχ,0=e−χ+χ.
Thekth-truncated series becomes as follows:(72)Ψkχ,υ=υ2e−χ+χ+∑n=1kℵnχυ2+nϖ.
The ERF of the algebraic equation (68) is described as(73)EResχ,υ=Ψχ,υ−υ2e−χ+χ−υϖDχχΨχ,υ+υϖΨχ,υ.
Furthermore,kth-truncated ERF of the algebraic equation (69) is explained as follows:(74)EReskχ,υ=Ψkχ,υ−υ2e−χ+χ−υϖDχχΨkχ,υ+υϖΨkχ,υ.
By utilizing equations (72) and (74), we get undefined coefficients in the following form:(75)ℵ1χ=−χ,ℵ2χ=χ,ℵ3χ=−χ,ℵ4χ=χ,ℵ5χ=−χ.
So, we get the 5th approximate solution of Elzaki transform of equation (68).(76)Ψ5χ,υ=e−χ+χυ2−χυ2+ϖ+χυ2+2ϖ−χυ2+3ϖ+χυ2+4ϖ−χυ2+5ϖ.
Operating inverse Elzaki transform on both sides of equation (76), we get the 5th approximate solution of equation (64).(77)Φ5χ,ϒ=e−χ+χ1−ϒϖΓϖ+1+ϒ2ϖΓ2ϖ+1−ϒ3ϖΓ3ϖ+1+ϒ4ϖΓ4ϖ+1−ϒ5ϖΓ5ϖ+1.
Whenϖ=1, equation (77) becomes as(78)Φ5χ,ϒ=e−χ+χ1−ϒ1!+ϒ22!−ϒ33!+ϒ44!−ϒ55!.
Equation (78) is coinciding with the six terms of the expansion of the exact solution Φχ,ϒ=e−χ+χe−ϒ.
Table1 demonstrates the values of absolute error of the 5th order approximate and exact solutions at ϖ=1 when χ=1 which support the capability and exactness of the novel technique.
Figure1 displays the evaluations of exact solution at ϖ=1 and the 5th approximate solution of Problem 1, at χ=1, for several values of ϒandϖ. Figure 1 confirms that when values of ϖ approach to “1,” the approximate solution approaches to the exact solution, which approves the efficacy and correctness of the new method. Moreover, the approximate solution overlaps with the exact solution at ϖ=1 and this once more ratifies the usefulness and correctness of the ERPSM.Table 1
Absolute error of ERPS results.
ϒExact solutionApproximate solutionAbsolute error02.718281828462.7182818284600.062.886370989272.886370989091.7766543792 × 10−100.123.064854203293.064854191821.1469452055 × 10−80.183.254374202893.25437407111.3178722424 × 10−70.243.455613464763.455612717787.4698766506 × 10−70.303.669296667623.669293792830.000002874789631680.363.89619330183.896184641130.000008660664293720.424.137120440254.137098405160.00002203509554110.484.392945680924.392896138730.044093686996Figure 1
Evaluation of closed form and approximate consequences of Problem1.Problem 2.
Consider the time-fractional linear CRDE [20],(79)DϒϖΦχ,ϒ=Φχχχ,ϒ−1+4χ2Φχ,ϒ,χ,ϒ≥0,0<ϖ≤1.
With the initial condition,(80)Φχ,0=eχ2.
Solution. Manipulating Elzaki transform on equation (79),(81)EDϒϖΦχ,ϒ=EΦχχχ,ϒ−1+4χ2EΦχ,ϒ,Ψχ,υ=υ2eχ2+υϖDχχΨχ,υ−υϖ1+4χ2Ψχ,υ,where(82)EDϒϖΦχ,ϒ=Ψχ,υυϖ−υ−ϖ+2eχ2,EΦχ,ϒ=Ψχ,υ.
Now, establishing a series solution of equation (81), consequently assume that Ψχ,υ has the expansion as follows:(83)Ψχ,υ=∑n=0∞ℵnχυ2+nϖ.
Thekth-truncated series Ψχ,υ is as follows:(84)Ψkχ,υ=∑n=0kℵnχυ2+nϖ.
By Lemma2, we have(85)limυ⟶01υ2Ψχ,υ=Φχ,0=eχ2.
So, equation (84) becomes as(86)Ψkχ,υ=eχ2υ2+∑n=1kℵnχυ2+nϖ.
The ERF of equation (81) is defined as(87)EResχ,υ=Ψχ,υ−υ2eχ2−υϖDχχΨχ,υ+υϖ1+4χ2Ψχ,υ.
Thekth-ERF is as follows:(88)EReskχ,υ=Ψkχ,υ−υ2eχ2−υϖDχχΨkχ,υ+υϖ1+4χ2Ψkχ,υ.
To find unspecified coefficients using equations (86) and (88), so we have(89)ℵ1χ=eχ2,ℵ2χ=eχ2,ℵ3χ=eχ2,ℵ4χ=eχ2,ℵ5χ=eχ2.
The 5th approximate solution of equation (81) in Elzaki transform form is(90)Ψ5χ,υ=eχ21υ2+1υ2+ϖ+1υ2+2ϖ+1υ2+3ϖ+1υ2+4ϖ+1υ2+5ϖ.
By applying inverse Elzaki transform on equation (90), we get the 5th approximate solution of equation (79) as follows:(91)Φ5χ,ϒ=eχ21+ϒϖΓϖ+1+ϒ2ϖΓ2ϖ+1+ϒ3ϖΓ3ϖ+1+ϒ4ϖΓ4ϖ+1+ϒ5ϖΓ5ϖ+1.
Whenϖ=1, equation (91) becomes as(92)Φ5χ,ϒ=eχ21+ϒ1!+ϒ22!+ϒ33!+ϒ44!+ϒ55!.
Equation (92) represents the first six terms of the expansion of eχ2+ϒ, so closed form solution of equation (79) is eχ2+ϒ.
Table2 demonstrates the values of absolute error of the 5th order approximate and exact solutions at ϖ=1 when χ=1, which support the capability and accuracy of the new technique.
Figure2 demonstrates the exploits of exact solution at ϖ=1 and the 5th approximate solution of Problem 2, when χ=1 for numerous values of ϒandϖ. The figure endorses that when values of ϖ approach to “1,” the approximate solution approaches to exact solution, which supports the ability and precision of the new method. Moreover, the approximate solution overlaps with the exact solution at ϖ=1 and this once more ratifies the usefulness and correctness of the ERPSM.Table 2
Absolute error of ERPS results.
ϒExact solutionApproximate solutionAbsolute error0.001.367879441171.367879441170.00.061.309643974761.309643974697 × 10−110.121.254799877891.254799873814.08 × 10−90.181.203149652581.203149606534.605 × 10−80.241.154507302241.154507045652.5659 × 10−70.301.108697661851.108696691179.7068 × 10−70.361.065555767241.065552892690.000002874550.421.024926260991.024919071810.000007189180.480.986662832970.986646944530.000015888447Figure 2
The behavior of exact and approximate outcomes of Problem2.


## 3.2.2. Approximate and Closed Form Solutions of Nonlinear CRDEs

Problem 3.
Consider the nonlinear time-fractional CRDE [21],(93)DϒϖΦχ,ϒ=Φχχχ,ϒ−Φχχ,ϒ+Φχ,ϒΦχχχ,ϒ−Φ2χ,ϒ+Φχ,ϒ,χ,ϒ≥0,0<ϖ≤1,
With the initial condition,(94)Φχ,0=eχ.
Solution. By applying Elzaki transform on equation (93), we get(95)Ψχ,υ=υ2eχ+υϖDχχΨχ,υ−υϖDχΨχ,υ+υϖEE−1Ψχ,υDχχE−1Ψχ,υ−υϖEE−1Ψχ,υ2+υϖΨχ,υ.
Here,(96)EΦχχχ,ϒ=DχχΨχ,υ,EΦχχ,ϒ=DχΨχ,υ,Φχ,ϒ=E−1Ψχ,υ,Φ2χ,ϒ=E−1Ψχ,υ2,Φχχχ,ϒ=DχχE−1Ψχ,υ,EDϒϖΦχ,ϒ=Ψχ,υυϖ−υ−ϖ+2eχ.
Define a series solution of equation (95) as follows:(97)Ψχ,υ=∑n=0∞ℵnχυ2+nϖ.
Thekth-truncated series is(98)Ψkχ,υ=∑n=0kℵnχυ2+nϖ.
By Lemma2,(99)limυ⟶01υ2Ψχ,υ=Φχ,0=eχ.
Therefore, thekth-truncated series becomes as(100)Ψkχ,υ=eχυ2+∑n=1kℵnχυ2+nϖ.
Now, define ERF in the following form:(101)EResχ,υ=Ψχ,υ−υ2eχ−υϖDχχΨχ,υ+υϖDχΨχ,υ−υϖEE−1Ψχ,υDχχE−1Ψχ,υ+υϖEE−1Ψχ,υ2−υϖΨχ,υ.
Thekth-truncated ERF is(102)EReskχ,υ=Ψkχ,υ−υ2eχ−υϖDχχΨkχ,υ+υϖDχΨkχ,υ−υϖEE−1Ψkχ,υDχχE−1Ψkχ,υ+υϖEE−1Ψkχ,υ2−υϖΨkχ,υ.
The undefined coefficients are determined in the following form by utilizing equations (100) and (102).(103)ℵ1χ=eχ,ℵ2χ=eχ,ℵ3χ=eχ,ℵ4χ=eχ,ℵ5χ=eχ.
The 5th approximate solution of equation (95) is given as(104)Ψ5χ,υ=eχ1υ2+1υ2+ϖ+1υ2+2ϖ+1υ2+3ϖ+1υ2+4ϖ+1υ2+5ϖ.
By applying inverse Elzaki transform on the above equation, we get the 5th approximate solution of equation (93).(105)Φ5χ,ϒ=eχ1+ϒϖΓϖ+1+ϒ2ϖΓ2ϖ+1+ϒ3ϖΓ3ϖ+1+ϒ4ϖΓ4ϖ+1+ϒ5ϖΓ4ϖ+1.
Forϖ=1, the last equation becomes as(106)Φ5χ,ϒ=eχ1+ϒ1!+ϒ22!+ϒ33!+ϒ44!+ϒ55!.
Equation (106) coincides with the 1st six terms of the expansion of eχ+ϒ, therefore exact solution of equation (93) is eχ+ϒ.
Table3 demonstrates the values of absolute error of the 5th order approximate and exact results at ϖ=1, and χ=1, which support the ability and accuracy of the novel technique.
Figure3 establishes the actions of exact solution at ϖ=1 and the 5th approximate solution of Problem 3, when χ=1 for certain values of ϒandϖ. The figure recommends that when values of ϖ approach to “1,” the approximate solution approaches to exact solution, which supports the capability and exactness of the new method. Moreover, the approximate solution overlaps with the exact solution at ϖ=1 and this once more ratifies the usefulness and correctness of the ERPSM.Table 3
Absolute error of ERPS results.
ϒExact solutionApproximate solutionAbsolute error0.001.367879441171.367879441170.00.061.309643974761.309643974697 × 10−110.121.254799877891.254799873814.08 × 10−90.181.203149652581.203149606534.605 × 10−80.241.154507302241.154507045652.5659 × 10−70.301.108697661851.108696691179.7068 × 10−70.361.065555767241.065552892690.000002874550.421.024926260991.024919071810.000007189180.480.986662832970.986646944530.000015888447Figure 3
2D plot of exact and approximate solution of Problem3.


## 4. Conclusions

There are enormous number of numerical and analytical methods for resolving the DEs; there are numerous methods that have superiority over the others. Few of them are precise and operative, but they necessitate mathematical operations that can be problematic and elongated. Our novel method, ERPSM, is considered by accurateness, rapidity, and effortlessness in finding exact and approximate solutions to DEs.To study the efficiency and reliability of ERPSM for PDEs, absolute errors of three applications are scrutinized. Consequences verify that our novel technique is simple, accurate, applicable, and efficient. The recommended techniques offered us an effortless and quick technique to perceive the coefficients of the suggested series to be a solution to the equation. Dissimilar to the traditional RPS method, while establishing the coefficients for a series, it is required to compute the fractional derivative every time, while ERPSM only requires the concept of the limit at zero in establishing the coefficients for the series.The gain of the ERPSM is that it decreases considerably the numerical calculations to construct the consequences for this category of equations related to existing methods, for instance, the differential transform method (DTM), perturbation method, and Adomian decomposition method (ADM). Consequently, we can accomplish that the ERPSM is effortless, effective, and practical for solving numerous further fractional order PDEs.



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