# Solving Nonlinear Fractional Partial Differential Equations Using the Elzaki Transform Method and the Homotopy Perturbation Method

Authors: Mohamed. Z. Mohamed; Mohammed Yousif; Amjad E. Hamza
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

In this paper, we combine the Elzaki transform method (ETM) with the new homotopy perturbation method (NHPM) for the first time. This hybrid approach can solve initial value problems numerically and analytically, such as nonlinear fractional differential equations of various normal orders. The Elzaki transform method (ETM) is used to solve nonlinear fractional differential equations, and then the homotopy is applied to the transformed equation, which includes the beginning conditions. To obtain the solution to an equation, we use the inverse transforms of the Elzaki transform method (ETM). The initial conditions have a big impact on the equation’s result. We give three beginning value issues that were solved as precise or approximation solutions with high rigor to demonstrate the method’s power and correctness. It is clear that solving nonlinear partial differential equations with the crossbred approach is the best alternative.


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## Body

## 1. Introduction

Many scholars have been investigating and debating the linear and nonlinear fractional differential equations (FDEs) in recent years. In view of the many applications of fractional differential equations in assorted fields, which engender in the physical sciences as well as in engineering, these kinds of equations play a worthy turn and also help to evolve mathematical tools to realize fractional modeling.The Elzaki transform is a transform similar to integral by other metamorphoses defined by integrals, which are known as Laplace transforms and Sumudu transform. In solving linear and non-linear differential equations. Using these method help in whereas the conversion was known by Tarig M. Elzaki [1]. Admit for its performance in solving linear order, nonlinear partial differential equations, and integral equations, the interesting convert it is evidence in [2–4]. The homotopy analysis transform method (HARM) is one of the more technicalities utilize in the solutions for the nonlinear factor [5]. To solve it, the homotopy perturbation technique combines the Laplace transform method and the well-known base flow equation [6]. By placing the solution in a rapid approximation series, HPM paired with the Sumudu transform tool improves the answer in a closed shape [7]. The theoretical formulation of initial value issues for fractional differential equations may be done in two methods [8]. The homotopy perturbation method (HPM) was introduced by Ji-Huan He in 1999 [9] for solving differential and integral equations. The HPM is applied to algebraic equations [10], nonlinear reaction-diffusion-convection problem [11], singular boundary and initial value problems [12, 13], nonlinear wave equations [14], and other modifications which can be seen in [15–25].The major goal of this research is to combine the Elzaki transform (ET) with the new homotopy perturbation method (NHPM) to approximate the solution of specific initial value problems represented by highly nonlinear fractional partial differential equations with starting conditions. Our suggested technique, which is a combination of the Elzaki transform (ET) and the new homotopy perturbation method (NHPM), discovers precise or approximate solutions to initial value problems with a high rate of convergence of the solution series.The methodology of this paper includes the following: item 1 contains definitions of fractional derivatives and discussion of an advanced method on the solution. Item 2 contains an explanation of the method to a solution. Item 3 uses the method to resolve some of the FPDEs to illustrate the approximate accuracy.Definition 1.
Letuδ,forδ>0,be in the spaceCθ,θ∈ℝ,if there exist a real numberν>θ,per seuδ=δνu1δ,whereu1δ∈C0,∞, visiblyCθ⊂Cv,ifv≤θ,[9 , 26 , 27].Definition 2.
Let taking the operator of orderυ>0, Riemann-Liouville fractional integral to uδ∈Cη,η≥−1, is known as [27]
(1)Jυhτ=1Γν∫0ττ−ξυ−1hξdξυ,τ,ξ>0,Jυuδ=1Γν∫0δδ−ξυ−1uξdξυ,δ,ξ>0,Definition 3.
Caputo fractional derivative, the left party ofj∈C−1w,w∈N∪0, is known as [26]
(2)Dυhτ=∂υhτ∂τυ=Jϑ−v∂ϑhτ∂τϑ,ϑ−1<υ≤ϑ,ϑ∈N.Definition 4.
If the set [1]
(3)A=uδ<Neδ/κiifδ∈−1i×0,∞.
Then Elzaki integral transform (EIT) of a functionuδ is defined as
(4)Euδ=β∫0∞uδe−δ/βdδ=Uβκ1≤β≤κ2.Definition 5.
Ifϑ−1<υ≤ϑ,ϑ∈N, then (EIT) of the fractional derivative D∗vuμ,δ is
(5)ED∗vuμ,δ=Uμ,ββν−∑κ=0r−1β2−ν+κuκμ,0,ϑ−1<υ≤ϑ,where Uμ,β be the (EIT) uμ,δ [28].


## 2. Elzaki Transform Homotopy Perturbation Method (ETHPM)

This section intends to discuss the utilization of (ETHPM) algorithm to solve linear and nonlinear fraction partial differential equations.(6)Dδνuμ,δ+Ruμ,δ+Nuμ,δ=Φμ,δ,μ,δ≥0,r−1<ν≤r,where Dδν=∂ν/∂δν represents the order ν fractional derivative, R represents a linear operator, N represents a nonlinear function, and Φ represents the source function. The beginning and boundary conditions are determined using Equation (6).
(7)uμ,0=φμ,0<ν≤1,uμ,0=ϕμ,∂uμ,0∂δ=φμ,(8)1<ν≤2.Using the linearity of (ET) and applying it to both aspect equations (6), the conclusion is
(9)EDδνuμ,δ+ERuμ,δ+ENuμ,δ=EΦμ,δ,ν>0.Using the property of (ET), to get(10)uμ,δβν−C+ERuμ,δ+ENuμ,δ=EΦμ,δ,ν>0.where C=∑κ=0n−1β2−ν+κuκμ,0,(11)uμ,β=βνEΦμ,δ+βνC−βνERuμ,δ−βνENuμ,δ.Taking the inverse (ET) to both aspect equations (11), we get
(12)uμ,δ=Gμ,δ−E−1βνERuμ,δ+Nuμ,δ.This is now applied to the HPM, whereGμ,δ is the term from the start conditions.
(13)uμ,δ=∑κ=0∞Pκuκμ,δ.The nonlinear operator is decomposition as, accordingly:(14)Nuμ,δ=∑κ=0∞PκHnu,where Hnu are given by
(15)Hnu0,u1,⋯un=1n!∂∂pnN∑i=0∞piuip=0,n=1,2,⋯.When we make up the equation (14) and equation (13) into equation (12), we get
(16)∑n=0∞Pnunμ,δ=Gμ,δ−PE−1βνER∑n=0∞unμ,δ+N∑n=0∞PnHun.Using He’s polynomials, this is the conjugation of the ET and the HPM. We find after matching the coefficient(17)P0:u0μ,δ=Gμ,δ,P1:u1μ,δ=−E−1βνERu0μ,δ+H0u,P2:u2μ,δ=−E−1βνERu1μ,δ+H1u,P3:u3μ,δ=−E−1βνERu2μ,δ+H2u,⋮⋮⋮Then the solution isuμ,δ=limp⟶1unμ,δ=u0μ,δ+u1μ,δ+u2μ,δ+⋯.


## 3. Illustrative Application

Here, we append examples to explain the solve method described.Example 1.
Let the homogeneous nonlinear fractional partial differential equations [29].
(18)Dδνu+uuμ=0,0<ν≤1.with initial condition:
(19)uμ,0=−μ.
Taking the (ET) to both aspect equations (18) and subject to equation (19), we get
(20)uμ,β=−μβ2−βνEuμ,δuδμ,δ.
Taking the inverse of (ET), we get:(21)uμ,δ=−μ−E−1βαEuμ,δuμμ,δ.
When, we apply the (HPM), we get:(22)∑κ=0∞Pκuκμ,δ=−μ−E−1βνE∑κ=0∞PκHκu.
After matching the coefficient we find:(23)P0:u0μ,δ=−μ,P1:u1μ,δ=−E−1βνEH0u=−μδνν!,P2:u2μ,δ=−E−1βvEH1u=−2μδ2ν2ν!,p3:u3μ,δ=−E−1βαEH2u=−μ42ν!+1ν2!2ν!δ3α3ν!,⋮⋮⋮
The convergent solution of equation (18), is presented by
(24)uμ,δ=−μ1+δνv!+2δ2ν2ν!+42ν!+1ν2!2ν!δ3ν3ν!+⋯.
The convergent solution of equation (18) at ν⟶1(25)uμ,δ=−μ1+δ+δ2+δ3+⋯=μδ−1,we have the exact solution [29], see Table 1 and Figure 1.Example 2.
Let a homogeneous nonlinear diffusion problem [30].
(26)Dδνu=uμμ−uμ+uuμμ−u2+u,0<ν≤1have initial condition:
(27)uμ,0=eμ.
Taking the (ET) to both aspect equations (26) and subject to equation (27), we get
(28)uμ,β=β2eμ+βνEuμμ−uμ+u+uuμμ−u2.
Taking the inverse of (ET), we get(29)uμ,δ=eμ+E−1βνEuμμ−uμ+u−uuμμ−u2.
When, we apply the (HPM), we get(30)∑κ=0∞Pnunμ,δ=eμ+E−1βνE∑κ=0∞Pnunμ,δ+∑κ=0∞PnHnu.
The elementary few components of He’s polynomials [5, 6] are offered by
(31)H0u=u0u0μμ−u02,H1u=u0u1μμ+u1u0μμ−2u0u1,H2u=u0u2μμ+u1u1μμ+u2u0μμ−u12−2u2u0,⋮⋮
After matching the coefficient we find(32)P0:u0μ,δ=eμ,P1:u1μ,δ=E−1βνEu0μμ−u0μ+u0+H0u=δνν!eμ,P2:u2μ,δ=E−1βνEu1μμ−u1μ+u1+H1u=δ2ν2ν!eμ,P3:u3μ,δ=E−1βνEu2μμ−u2μ+u2+H2u=δ3ν3ν!eμ,⋮⋮⋮
The convergent solution of equation (26), is presented by
(33)uμ,δ=eμ1+δνν!+δ2ν2ν!+δ3ν3ν!+⋯,=eμ∑n=1∞δnνnν!.
If we takeν⟶1, we get exact solution of uμ,δ=eμ+δ [30]; see Table 2 and Figure 2.Example 3.
Let a homogeneous nonlinear fractional partial differential equations [31].
(34)Dδνu−2μ2δuuμ=0,δ≥0,1<ν≤2.with initial condition
(35)uμ,0=0,uμμ,0=μ.
Taking the (ET) to both aspect equations (34) and subject to equation (35), we get
(36)uμ,0=0,uμμ,0=μ.
Taking the inverse of (ET), we get(37)uμ,δ=μδ+E−1βνE2μ2δuuμ.
After matching the coefficient, we find(38)P0:u0μ,δ=μδ,P1:u1μ,δ=E−1βνE2μ2δH0u=2μ3δν+1ν+1!,P2:u2μ,δ=E−1βνE2μ2δH1u=16μ5δ2ν+12ν+1!,P3:u3μ,δ=E−1βνE2μ2δH2u=32×62ν+1!+24ν+12!μ72ν+1!δ3ν+13ν+1!,⋮⋮⋮
The convergent solution of equation (34), is presented by
(39)uμ,δ=μδ+2μ3ν+1!δν+1+16μ52ν+1!δ2ν+1+32×62ν+1!+24α+12!2ν+1!μ73ν+1!δ3ν+1+⋯.
Whenν⟶2, equation (39) becomes
(40)uμ,δ=μδ+13μδ3+215μδ5+17315μδ7+⋯.
Therefore, we conclude thatuμ,δ=tanμδ [31]; see Table 3 and Figure 3.Table 1
Solution for the first three approximations with exact solution, with mesh pointsδ=0.2, for equation (18).
δμν=0.75ν=0.85ν=0.95ν=1Exact0.20000000.1-0.1480375-0.136454922-0.128120827-0.1248-0.1250.2-0.296074999-0.272909844-0.256241654-0.2496-0.250.3-0.444112499-0.409364766-0.384362481-0.3744-0.3750.4-0.592149999-0.545819688-0.512483308-0.4992-0.50.5-0.740187498-0.68227461-0.640604134-0.624-0.6250.6-0.888224998-0.818729532-0.768724961-0.7488-0.750.7-1.036262498-0.955184454-0.896845788-0.8736-0.8750.8-1.184299998-1.091639376-1.024966615-0.9984-10.9-1.332337497-1.228094299-1.153087442-1.1232-1.1251-1.480374997-1.364549221-1.281208269-1.248-1.25Figure 1
Create 2D and 3D comparison charts of precise data and predicted computational outputs for Example1, using δ=0.2.
(a)
3D approximate solution Example1(b)
2D approximate solution Example1Table 2
Solution for the first three approximations with exact solution, with mesh pointsδ=0.2, for equation (26).
δμν=0.75ν=0.85ν=0.95ν=1Exact0.201.4031840451.3159197811.2489648851.2213333331.2214027580.11.5507581991.4543162731.3803196691.3497820811.3498588080.21.7138528631.6072680511.5254891551.4917399021.4918246980.31.8941003421.7763059071.685926251.6486275571.6487212710.42.0933046141.963121631.8632366621.8220152311.82211880.52.3134593822.1695849342.0591949722.0136382452.0137527070.62.5567680292.3977621742.2757623982.2254144282.2255409280.72.825665672.6499370232.5151064192.4594633072.4596031110.83.1228435232.9286333322.779622472.7181273212.7182818280.93.4512758433.2366403893.0719579173.0039952663.00416602413.8142496923.577040833.3950385513.3199282063.320116923Figure 2
Create 2D and 3D comparison charts of precise data and predicted computational outputs for Example2, using δ=0.2.
(a)
3D approximate solution Example2(b)
2D approximate solution Example2Table 3
Solution for the first three approximations with exact solution, with mesh pointsδ=0.2, for equation (34).
δμν=1.75ν=1.85ν=1.95ν=2Exact0.20000000.10.0200054120.0200040870.0200030770.0200026670.0200026670.20.0400433450.040032720.040024630.0400213470.0400213470.30.0601465860.0601105850.0600832070.0600721040.0600721040.40.0803484510.0802626420.0801974970.0801711040.0801711050.50.1006830620.1005142710.1003864090.100334670.1003346720.60.1211856310.1208914150.1206691370.1205793310.1205793370.70.1418927610.1414207290.1410652450.1409218780.1409218950.80.1628427610.1621297380.1615947370.1613794170.1613794610.90.1840759890.1830470010.1822781420.1819694270.18196952910.205635210.2042022830.2031365960.2027098210.202710036Figure 3
Create 2D and 3D comparison charts of precise data and predicted computational outputs for Example3, using δ=0.2.
(a)
3D approximate solution Example3(b)
2D approximate solution Example3In all figures, the exact and (NHPM) solutions atnu=1,2 are plotted in 2D plots of all examples, and it is observed that the exact and derived results are in good contact, confirming the proposed method’s high accuracy.


## 4. Conclusion

The Elzaki transform method has been used with the new homotopy perturbation approach to solve nonlinear problems quickly, easily, and accurately, resulting in approximations that swiftly converge to the true answer. The strategy presented is well-suited to such problems and is quite effective. As indicated by the approximation solution’s faster convergence, the relevance of ETHPM has been established.



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