# A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel

Authors: Shabir Ahmad; Aman Ullah; Ali Akgül; Manuel De la Sen
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we introduce the Yang transform homotopy perturbation method (YTHPM), which is a novel method. We provide formulae for the Yang transform of Caputo-Fabrizio fractional order derivatives. We derive an algorithm for the solution of Caputo-Fabrizio (CF) fractional order partial differential equation in series form and show its convergence to the exact solution. To demonstrate the novel approach, we include some examples with detailed solutions. We use tables and graphs to compare the exact and approximate solutions.


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## Body

## 1. Introduction and Motivation

Noninteger calculus is a popular field that is aimed at explaining real-world phenomena that are modeled with operators of fractional order. It is also a field that uses noninteger order derivatives for differentiation and integration. A fractional derivative is a sort of derivative with a noninteger order that meets specific conditions: we get the primary function when the order is zero, and we get the ordinary derivative when the order is one [1]. The memory effect and conserved illustrative physical properties are two advantages of fractional derivatives. Over time, more accurate and up-to-date studies have been revealed using these types of operators. In this sense, the theory of fractional calculus and its applications are gaining popularity around the world. Fractional order models incorporate all previous knowledge from the past due to the memory effect, making it better for them to forecast and analyze dynamical models more efficiently. Fractional order calculus has a wide range of applications in a variety of areas due to its efficient properties, including biology and physics. [2–4], economics and finance [5, 6], science and engineering [7, 8], and mathematical modeling and mechanics [9–11]. Since the kernel of Caputo and Riemann derivatives is singular, they have a problem with this kernel. Since the kernel is used to explain the physical system’s memory effect, it is clear that both derivatives cannot accurately interpret the memory’s full effect due to this limitation. Caputo and Fabrizio (CF) [12] suggested a new fractional operator with an exponential kernel in a recent attempt around the middle of the last decade. This derivative’s kernel is nonsingular, so the results are more reasonable than the classical one. We include some applications of the CF operator in [13–15].There are two types of nonlinear equations: linear and nonlinear. Partially differential equations of fractional order are notoriously difficult to solve, and finding an exact solution is even more difficult. In applied mathematics, exact and numerical solutions to this type of equation are necessary. As a result, new techniques for obtaining analytical solutions that are relatively close to the exact solutions have been developed. Integral transformations were often used to solve the differential equations. Integral transforms are useful for solving IVPs and BVPs in differential and integral equations. Several authors implemented various kinds of integral transforms and examined their implications in different types of differential equations. The Laplace transform is the integral transform that is used the most [16]. In 1998, Watugala [17] was successfully introducing Sumudu transform to solve differential equations and control engineering problems. Recently, in 2011, T. Elzaki and S. Elzaki introduced new integral transform “Elzaki Transform” and used heavily in solving partial differential equations [18]. Also in 2013, Aboodh introduces “Aboodh Transform” and applies for solving partial differential equations [19]. There are many transform which are available in literature.He formulated the homotopy perturbation method (HPM) [20] in 1999, which is a combination of the homotopy method and the classical perturbation technique and has been widely applied on both linear and nonlinear problems [21–23]. The importance of HPM is that it does not need a small parameter in the equation, so it reduces the drawbacks of traditional perturbation methods. The main purpose of this article is to apply newly introduced integral transform called “Yang Transform” discovered by Yang [24] with HPM to solve nonlinear fractional order PDEs. We solve two popular nonlinear PDEs through the proposed method. We obtain a power series solution in the context of a quickly convergent series, and just several iterations are needed to achieve very efficient results. There is no requirement for a method like discretizing the problem and no linearization for the nonlinear problem, and just a few iterations can lead to a solution that can be easily estimated using these techniques.


## 2. Preliminaries

We give the basic definitions which are needed in the rest of paper. For the sake of simplicity, we write the exponential-decay kernel asKτ,ϱ=exp−ατ−ϱ/1−α.Definition 1 [12].
Ifℙτ∈H10,T,T>0, then the Caputo-Fabrizio (CF) derivative may be expressed as follows:
(1)DCFtαℙτ=Nα1−α∫0τℙ′ϱKτ,ϱdϱ.
Nα is the normalization function with N1=N0=1. However, if ℙτ∉H10,T, then the above derivative is defined as follows:
(2)DCFtαℙτ=Nα1−α∫0τℙτ−ℙϱKτ,ϱdϱ.Definition 2 [12].
The CF fractional integral may be expressed as follows:(3)ICFtαℙτ=1−αNαℙτ+αNα∫0τℙϱdϱ,τ≥0,α∈0,1.Definition 3 [7].
ForNα=1, the following result represents the Laplace transform of CF derivative:
(4)LCFDταℙτ=vLℙτ−ℙ0v+α1−v.Definition 4. [24].
The Yang transform ofPτ is defined as follows:
(5)Yℙτ=χv=∫0∞ℙτe−τ/v,τ>0,where υ is transform variable and for some υ the integral on the right exists.Remark 5.
Yang transform of some useful functions is given below.(6)Y1=v,Yτ=v2,Yτq=Γq+1vq+1.


## 3. Main Work

First, we derive formula for Yang transform of Caputo-Fabrizio fractional derivative through the Yang-Laplace duality property. At the end of this section, we give some examples with detailed solution to check the validity and efficiency of the novel method.Lemma 6 (Laplace-Yang duality).
Let the Laplace transform ofPτ is Fυ, then χυ=F1/v.Proof.
From Equation (5), we can obtain another form of the Yang transform by substituting −τ/u=x as
(7)Yℙτ=χv=v∫0∞ℙvxe−xdx,x>0.
SinceLℙτ=Fυ, this implies that
(8)Fv=Lℙτ=∫0∞e−vτℙτdτ.
Putτ=x/v in (8), we have
(9)Fv=1v∫0∞e−xℙxvdx.
Thus, from Equation (7), we obtain
(10)Fv=χ1v.
Also from Equations (5) and (8), we obtain
(11)F1v=χv.
The relations (10) and (11) represent the duality relation between the Yang and Laplace transform.☐Lemma 7.
LetPτ be a continuous function; then, Yang transform of CF derivative of Pτ is given by
(12)YCFℙατ=Yℙτ−vℙ01+αv−1.Proof.
The Laplace transform of the CF fractional is given by(13)LCFℙατ=vLℙτ−ℙ0v+α1−v.
Also, we have that the relation between Laplace and Yang property, i.e.,χv=F1/v. To obtain the required result, we replace υ by 1/v in Equation (13), and we get
(14)YCFℙατ=1/vYℙτ−ℙ01/v+α1−1/v,YCFℙατ=Yℙτ−vℙ01+αv−1.
The proof is completed.☐


## 4. Algorithm of the Proposed Method

In this part, we discuss the algorithms of fractional order differential equations involving exponential-decay kernel. We provide some examples with detailed solution and its comparison with exact solutions.


### 4.1. Application to Caputo-Fabrizio Fractional Differential Equations

First, we develop the solution procedure of general nonlinear Caputo-Fabrizio (CF) fractional partial differential equations through YTHPM. Let us take a general nonlinear CF PDE with nonlinear termNGx,τ and linear term LGx,τ as
(15)CFDταGx,τ+LGx,τ+NGx,τ=gx,τ,Gx,0=hx,where the term gx,τ represents the source term. Implement Yang transform to Equation (15), and one can achieve
(16)YGx,τ−vGx,01+αv−1=−YLGx,τ+NGx,τ+Ygx,τ,YGx,τ=vhx−1+αv−1YLGx,τ+NGx,τ+Ygx,τ.Applying inverse of Yang transform, we achieve
(17)Gx,τ=Gx,τ−Y−11+αv−1YLGx,τ+NGx,τ,where the term Gx,τ represents the source term and the given I.C (initial condition). Now, we utilize HPM:
(18)Gx,τ=∑q=0∞ρqGqx,τ.We decompose the nonlinear termNGx,τ as
(19)NGx,τ=∑q=0∞ρqHqG,where HqG represents the He’s polynomial and is calculated through the formula:
(20)HqG1,G2,⋯,Gq=1Γq+1∂q∂ρqN∑i=0∞ρiGiρ=0q=0,1,⋯.Putting Equations (18) and (19) in Equation (17), we achieve
(21)∑q=0∞ρqGqx,τ=Gx,τ−ρY−11+αv−1YL∑q=0∞ρqGqx,τ+∑q=0∞ρqHqG=.We achieve the following terms by comparing coefficients ofρ in (21):
(22)ρ0:G0x,τ=Gx,τ,ρ1:G1x,τ=Y−11+αv−1YLG0x,τ+H0G,ρ2:G2x,τ=Y−11+αv−1YLG1x,τ+H1G,ρ3:G3x,τ=Y−11+αv−1YLG2x,τ+H2G,⋮ρq:Gqx,τ=Y−11+αv−1YLGqx,τ+HqG.Thus, we may write the acquired solution of Equation (15) as follows:
(23)Gx,τ=G0x,τ+G1x,τ+⋯.


### 4.2. Convergence and Error Analysis

The following theorems are based on the method’s mechanism and address the original problem’s (15) convergence and error analysis.Theorem 8.
LetGx,τ be the exact solution of (15) and let Gqx,τ,Gnx,τ∈H and σ∈0,1, where H denotes the Hilbert space. Then, the obtained solution ∑q=0∞Gqx,τ will converge Gx,τ if Gqx,τ≤σGq−1x,τ∀q>A, i.e., for any ω>0∃A>0, such that ∥Gq+nx,τ∥≤β, ∀m,n∈N.Proof.
We make a sequence of∑q=0∞Gqx,τ.
(24)C0x,τ=G0x,τ,C1x,τ=G0x,τ+G1x,τ,C2x,τ=G0x,τ+G1x,τ+G2x,τ,C3x,τ=G0x,τ+G1x,τ+G2x,τ+G3x,τ,⋮Cqx,τ=G0x,τ+G1x,τ+G2x,τ+⋯+Gqx,τ.
To get the desired result, we have to prove thatCqx,τ forms a “Cauchy sequence.” Further, let us take
(25)∥Cq+1x,τ−Cqx,τ∥=∥Gq+1x,τ∥≤σ∥Gqx,τ∥≤σ2∥Gq−1x,τ∥≤σ3∥Gq−2x,τ∥⋮≤σq+1∥G0x,τ.
Forq,n∈N, we acquire
(26)∥Cqx,τ−Cnx,τ∥=∥Gq+nx,τ∥=∥Cqx,τ−Cq−1x,τ+Cq−1x,τ−Cq−2x,τ+Cq−2x,τ−Cq−3x,τ+⋯+Cn+1x,τ−Cnx,τ∥≤∥Cqx,τ−Cq−1x,τ∥+∥Cq−1x,τ−Cq−2x,τ∥+⋯+∥Cn+1x,τ−Cnx,τ.≤σq∥G0x,τ∥+σq−1∥G0x,τ∥+⋯+σq+1∥G0x,τ∥=∥G0x,τ∥σq+σq−1+⋯+σq+1=∥G0x,τ∥1−σq−n1−σq+1σn+1.
Since0<σ<1, and G0x,τ is bounded, let us take β=1−σ/1−σq−nσn+1∥G0x,τ∥, and we obtain
(27)∥Gq+nx,τ∥≤β,∀q,n∈N.
Thus,Gqx,τq=0∞ forms a “Cauchy sequence” in H. It follows that the sequence Gqx,τq=0∞ is a convergent sequence with the limit lim/q⟶∞Gqx,τ=Gx,τ for ∃Gx,τ∈H. Hence, this ends the proof.☐Theorem 9.
Let∑h=0kGhx,τ is finite and Gx,τ represents the obtained series solution. Let σ>0such that ∥Gh+1x,τ∥≤σ∥Ghx,τ∥, then the following relation gives the maximum absolute error.
(28)Gx,τ−∑h=0kGhx,τ<σk+11−σG0x,τ.Proof.
Since∑h=0kGhx,τ is finite, this implies that ∑h=0kGhx,τ<∞. Consider
(29)Gx,τ−∑h=0kGhx,τ=∑h=k+1∞Ghx,τ≤∑h=k+1∞Ghx,τ≤∑h=k+1∞σhG0x,τ≤σk+11+σ+σ2+⋯G0x,τ≤σk+11−σG0x,τ.
This ends the theorem’s proof.☐


### 4.3. Test Problems

The Yang transform homotopy perturbation method is applied to well-known nonlinear fractional PDEs in this section, demonstrating its ease of use and high accuracy. The space where the solution of the following examples lies is the Hilbert spaceH.Example 1.
We take nonlinear KdV equation as follows:(30)DCFταzx,τ=−zzx−zzxxx,0<α∈0,1,subjected to I.C zx,0=x.Solution 1.
Implementing the Yang transform to Equation (30), we have
(31)Yzx,τ=υzx,0−1+αυ−αYzzx+zzxxx.
Applying Yang inverse transform, we have(32)zx,τ=zx,0−Y−11+αυ−αYzzx+zzxxx.
The solution via HPT is as follows:(33)zx,τ=∑q=0∞ρqzqx,τ.
Thus, Equation (32) can be written as
(34)∑q=0∞ρqzqx,τ=x−ρY−11+αv−αY∑q=0∞ρqHqz,where Hqz is He’s polynomial which represents the nonlinear term zzx+zzxxx. The first three terms can be written as
(35)H0z=z0∂∂xz0+z0∂3∂x3z0,H1z=z0∂∂xz1+z1∂∂xz0+z0∂3∂x3z1+z1∂3∂x3z0,H1z=z0∂∂xz2+z1∂∂xz1+z2∂∂xz0+z0∂3∂x3z2+z1∂3∂x3z1+z2∂3∂x3z0⋮.Comparing the like powers of ρ, we obtain
(36)ρ0:z0x,τ=x.Now, using He’s polynomials, we get H0z=x. The second approximation is given by
(37)ρ1:z1x,τ=−Y−11+αv−αYH0z=−Y−11+αv−αYx=−xv+α×v2−α×v=−Y−1x1−αv+α×v2ρ1:z1x,τ=−α×τ−x1−α.
The third approximation is given by(38)ρ1:z1x,τ=−Y−11+αv−αYH1z.
The second termH1z of He’s polynomial is calculated as
(39)H1z=z0∂∂xz1+z1∂∂xz0+z0∂3∂x3z1+z1∂3∂x3z0=x∂∂x−αxτ−x1−α+−αxτ−x1−α∂∂xx+x∂3∂x3−αxτ−x1−α+−αxτ−x1−α∂3∂x3x,After simple calculation, we get
(40)H1z=−2αxτ−2x1−α.
Now, substituting Equation (40) into Equation (38), we get
(41)ρ2:z2x,τ=−Y−11+αv−αY−2αxτ−2x1−α=−Y−11+αv−α−2αxτ−2x1−αv=2Y−1α3xv3+2αx1−αv2+x1−α2v=α2xτ2+4αx1−ατ+2x1−α2,Similarly, one can compute the other terms. Thus, the approximate solution is given.
(42)zx,τ=x−αxτ−x1−α+α2xτ2+4αx1−ατ+2x1−α2+··.Remark 10.
The proposed method is less computational and accurate as the obtained solution fast converges to the classical exact solution by substitutingα=1in Equation (42), i.e.,
(43)zx,τ=x1−τ+τ2−τ3+⋯=x∑i=0∞−1iτi,zx,τ=x1+τ.
Equation (43) represents the exact classical solution of (30).The absolute errors between exact solution and approximate solution forα=1 are given in Table 1. Also, the absolute errors of α=1 and α=0.9 are given in Table 2. The approximate solutions for α=0.98 and α=1 are represented by zapprox and zapprox, respectively.Example 2.
Consider the nonlinear time fractional order Burger equation as(44)DCFταzx,τ+z∂z∂x=η∂2z∂x2,α∈0,1,subjected to the initial condition zx,0=nx.Solution 2.
Implementing Yang transform to Equation (44), we get
(45)Yzx,τ=vzx,0−1+αv−αY−η∂2z∂x2+z∂z∂x..
Applying inverse Yang transform, we obtain(46)zx,τ=xx−Y−11+αv−αY−η∂2z∂x2+z∂z∂x.
Using the HPT method, the approximate solution is(47)∑q=0∞ρqzqx,τ=nx−ρY−11+αv−αY∑q=0∞ρqHqz−η∂2z∂x2∑q=0∞ρqzq.,where He’s polynomial Hqz represents the nonlinear term z∂z/∂x. The first three terms of Hqz are given by
(48)H0z=z0∂z0∂x,H1z=z0∂z1∂x+z1∂z0∂x,H1z=z0∂z2∂x+z1∂z1∂x+z2∂z0∂x,⋮.
Similarly, other terms can be calculated. Comparing the like powers ofρ in (47), we achieve
(49)ρ0:z0x,τ=nx,ρ1:z1x,τ=−Y−11+αv−αYH0z−η∂2∂x2z0.Now, we compute H0z as follows:
(50)H0z=z0∂z0∂x=nx∂∂xnx=n2x.Thus, Equation (49) can be written as
(51)ρ1:z1x,τ=−Y−11+αv−αYn2x−0=−n2xY−1v+αv2−αv,ρ1:z1x,τ=−n2x1−α+ατ.Similarly, the third approximation is given by
(52)ρ2:z2x,τ=2n3xα−2α+α2τ22−2αα−1τ+1.One can calculate more terms. The acquired series solution is represented as follows:
(53)zx,τ=nx−n2x1−α+ατ+2n3xα−2α+α2τ22−2αα−1τ+1+⋯.Remark 11.
The proposed method is less computational and accurate as the obtained solution fastly converges to the classical exact solution by substitutingα=1 in Equation (53), i.e.,
(54)zx,τ=nx1−nτ+n2τ2−n3τ3+⋯=x∑i=0∞−1ini+1τi,zx,τ=nx1+nτ.
Equation (54) is the classical solution of the considered problem.Table 1
Absolute error between exact and approximate solutions of Example1 at τ=0.5 and α=1.
xzapproxx,τzexactx,τzexact−zapprox0.10.07500.06670.00830.20.15000.13330.01670.30.22500.20000.02500.40.30000.26670.03330.50.37500.33330.04170.60.45000.40000.05000.70.52500.46670.05830.80.60000.53330.06670.90.67500.60000.075010.75000.66670.08333Table 2
Absolute error between approximate solution of Example1 at α=0.98 and α=1.
xzapproxx,τzapprox′x,τzexact−zapprox′0.10.07700.075000.00200.20.15400.15000.00400.30.23100.22500.00600.40.30800.30000.00800.50.38510.37500.01010.60.46210.45000.01210.70.53910.52500.01410.80.61610.60000.01610.90.69310.67500.01811.00.77010.75000.0201The absolute errors between exact solution and approximate solution forα=1 and τ=0.5 are given in Table 3. Also, the absolute errors of for x=0.5, α=1, and α=0.9 are given in Table 4. The approximate solutions for α=0.98 and α=1 are represented by zapprox and zapprox′, respectively.Remark 12.
The proposed YTHPM method is a powerful new method which needs less computation time and is much easier and more convenient than the other methods like HAM (homotopy analysis method). Computational time means HPM takes less time, to calculate the iterative terms for the series solution, because HPM only depends on single parameter. However, HAM depends on two parameters, i.e.,h (auxiliary parameter) and p (embedding parameter). So, HAM takes a little more time for calculating the successive terms of the series solution. Also, the convergence of the HAM depends on h which is different for various approximations. When h=−1 in HAM, then the solution directly converges to the HPM solution; otherwise, more terms will be calculated than HPM case. So, computational labor is much in HAM instead of HPM. Actually, the Yang transform is closely related to Laplace transform which is convergent. Therefore, Yang transform combined with HPM is an efficient and valid computational method for solving DEs because the Yang transform allows one in many situations to overcome the deficiency mainly caused by unsatisfied boundary or initial conditions that appear in other semianalytical methods such as HAM. The comparison between HPM and HAM is already given in [25, 26].Table 3
Absolute error between exact and approximate solution of the Example2 at τ=0.1 and α=1.
xzapproxx,τzexactx,τzexact−zapprox0.10.16800.16670.00130.20.33600.33330.00270.30.50400.50000.00400.40.67200.66670.00530.50.84000.83330.00670.61.00801.00000.00800.71.17601.16670.00930.81.34401.33330.01070.91.51201.50000.01201.01.68001.66670.0133Table 4
Absolute error between approximate solution of Example2 at α=0.98 and α=1.
τzapproxx,τzapprox′x,τzapprox−zapprox′0.10.19880.16800.03080.20.39750.33600.06150.30.59630.50400.09230.40.79500.67200.12300.50.99380.84000.15380.61.19251.00800.18450.71.39131.17600.21530.81.59001.34400.24600.91.78881.51200.27681.01.98761.68000.3076


## 4.1. Application to Caputo-Fabrizio Fractional Differential Equations

First, we develop the solution procedure of general nonlinear Caputo-Fabrizio (CF) fractional partial differential equations through YTHPM. Let us take a general nonlinear CF PDE with nonlinear termNGx,τ and linear term LGx,τ as
(15)CFDταGx,τ+LGx,τ+NGx,τ=gx,τ,Gx,0=hx,where the term gx,τ represents the source term. Implement Yang transform to Equation (15), and one can achieve
(16)YGx,τ−vGx,01+αv−1=−YLGx,τ+NGx,τ+Ygx,τ,YGx,τ=vhx−1+αv−1YLGx,τ+NGx,τ+Ygx,τ.Applying inverse of Yang transform, we achieve
(17)Gx,τ=Gx,τ−Y−11+αv−1YLGx,τ+NGx,τ,where the term Gx,τ represents the source term and the given I.C (initial condition). Now, we utilize HPM:
(18)Gx,τ=∑q=0∞ρqGqx,τ.We decompose the nonlinear termNGx,τ as
(19)NGx,τ=∑q=0∞ρqHqG,where HqG represents the He’s polynomial and is calculated through the formula:
(20)HqG1,G2,⋯,Gq=1Γq+1∂q∂ρqN∑i=0∞ρiGiρ=0q=0,1,⋯.Putting Equations (18) and (19) in Equation (17), we achieve
(21)∑q=0∞ρqGqx,τ=Gx,τ−ρY−11+αv−1YL∑q=0∞ρqGqx,τ+∑q=0∞ρqHqG=.We achieve the following terms by comparing coefficients ofρ in (21):
(22)ρ0:G0x,τ=Gx,τ,ρ1:G1x,τ=Y−11+αv−1YLG0x,τ+H0G,ρ2:G2x,τ=Y−11+αv−1YLG1x,τ+H1G,ρ3:G3x,τ=Y−11+αv−1YLG2x,τ+H2G,⋮ρq:Gqx,τ=Y−11+αv−1YLGqx,τ+HqG.Thus, we may write the acquired solution of Equation (15) as follows:
(23)Gx,τ=G0x,τ+G1x,τ+⋯.


## 4.2. Convergence and Error Analysis

The following theorems are based on the method’s mechanism and address the original problem’s (15) convergence and error analysis.Theorem 8.
LetGx,τ be the exact solution of (15) and let Gqx,τ,Gnx,τ∈H and σ∈0,1, where H denotes the Hilbert space. Then, the obtained solution ∑q=0∞Gqx,τ will converge Gx,τ if Gqx,τ≤σGq−1x,τ∀q>A, i.e., for any ω>0∃A>0, such that ∥Gq+nx,τ∥≤β, ∀m,n∈N.Proof.
We make a sequence of∑q=0∞Gqx,τ.
(24)C0x,τ=G0x,τ,C1x,τ=G0x,τ+G1x,τ,C2x,τ=G0x,τ+G1x,τ+G2x,τ,C3x,τ=G0x,τ+G1x,τ+G2x,τ+G3x,τ,⋮Cqx,τ=G0x,τ+G1x,τ+G2x,τ+⋯+Gqx,τ.
To get the desired result, we have to prove thatCqx,τ forms a “Cauchy sequence.” Further, let us take
(25)∥Cq+1x,τ−Cqx,τ∥=∥Gq+1x,τ∥≤σ∥Gqx,τ∥≤σ2∥Gq−1x,τ∥≤σ3∥Gq−2x,τ∥⋮≤σq+1∥G0x,τ.
Forq,n∈N, we acquire
(26)∥Cqx,τ−Cnx,τ∥=∥Gq+nx,τ∥=∥Cqx,τ−Cq−1x,τ+Cq−1x,τ−Cq−2x,τ+Cq−2x,τ−Cq−3x,τ+⋯+Cn+1x,τ−Cnx,τ∥≤∥Cqx,τ−Cq−1x,τ∥+∥Cq−1x,τ−Cq−2x,τ∥+⋯+∥Cn+1x,τ−Cnx,τ.≤σq∥G0x,τ∥+σq−1∥G0x,τ∥+⋯+σq+1∥G0x,τ∥=∥G0x,τ∥σq+σq−1+⋯+σq+1=∥G0x,τ∥1−σq−n1−σq+1σn+1.
Since0<σ<1, and G0x,τ is bounded, let us take β=1−σ/1−σq−nσn+1∥G0x,τ∥, and we obtain
(27)∥Gq+nx,τ∥≤β,∀q,n∈N.
Thus,Gqx,τq=0∞ forms a “Cauchy sequence” in H. It follows that the sequence Gqx,τq=0∞ is a convergent sequence with the limit lim/q⟶∞Gqx,τ=Gx,τ for ∃Gx,τ∈H. Hence, this ends the proof.☐Theorem 9.
Let∑h=0kGhx,τ is finite and Gx,τ represents the obtained series solution. Let σ>0such that ∥Gh+1x,τ∥≤σ∥Ghx,τ∥, then the following relation gives the maximum absolute error.
(28)Gx,τ−∑h=0kGhx,τ<σk+11−σG0x,τ.Proof.
Since∑h=0kGhx,τ is finite, this implies that ∑h=0kGhx,τ<∞. Consider
(29)Gx,τ−∑h=0kGhx,τ=∑h=k+1∞Ghx,τ≤∑h=k+1∞Ghx,τ≤∑h=k+1∞σhG0x,τ≤σk+11+σ+σ2+⋯G0x,τ≤σk+11−σG0x,τ.
This ends the theorem’s proof.☐


## 4.3. Test Problems

The Yang transform homotopy perturbation method is applied to well-known nonlinear fractional PDEs in this section, demonstrating its ease of use and high accuracy. The space where the solution of the following examples lies is the Hilbert spaceH.Example 1.
We take nonlinear KdV equation as follows:(30)DCFταzx,τ=−zzx−zzxxx,0<α∈0,1,subjected to I.C zx,0=x.Solution 1.
Implementing the Yang transform to Equation (30), we have
(31)Yzx,τ=υzx,0−1+αυ−αYzzx+zzxxx.
Applying Yang inverse transform, we have(32)zx,τ=zx,0−Y−11+αυ−αYzzx+zzxxx.
The solution via HPT is as follows:(33)zx,τ=∑q=0∞ρqzqx,τ.
Thus, Equation (32) can be written as
(34)∑q=0∞ρqzqx,τ=x−ρY−11+αv−αY∑q=0∞ρqHqz,where Hqz is He’s polynomial which represents the nonlinear term zzx+zzxxx. The first three terms can be written as
(35)H0z=z0∂∂xz0+z0∂3∂x3z0,H1z=z0∂∂xz1+z1∂∂xz0+z0∂3∂x3z1+z1∂3∂x3z0,H1z=z0∂∂xz2+z1∂∂xz1+z2∂∂xz0+z0∂3∂x3z2+z1∂3∂x3z1+z2∂3∂x3z0⋮.Comparing the like powers of ρ, we obtain
(36)ρ0:z0x,τ=x.Now, using He’s polynomials, we get H0z=x. The second approximation is given by
(37)ρ1:z1x,τ=−Y−11+αv−αYH0z=−Y−11+αv−αYx=−xv+α×v2−α×v=−Y−1x1−αv+α×v2ρ1:z1x,τ=−α×τ−x1−α.
The third approximation is given by(38)ρ1:z1x,τ=−Y−11+αv−αYH1z.
The second termH1z of He’s polynomial is calculated as
(39)H1z=z0∂∂xz1+z1∂∂xz0+z0∂3∂x3z1+z1∂3∂x3z0=x∂∂x−αxτ−x1−α+−αxτ−x1−α∂∂xx+x∂3∂x3−αxτ−x1−α+−αxτ−x1−α∂3∂x3x,After simple calculation, we get
(40)H1z=−2αxτ−2x1−α.
Now, substituting Equation (40) into Equation (38), we get
(41)ρ2:z2x,τ=−Y−11+αv−αY−2αxτ−2x1−α=−Y−11+αv−α−2αxτ−2x1−αv=2Y−1α3xv3+2αx1−αv2+x1−α2v=α2xτ2+4αx1−ατ+2x1−α2,Similarly, one can compute the other terms. Thus, the approximate solution is given.
(42)zx,τ=x−αxτ−x1−α+α2xτ2+4αx1−ατ+2x1−α2+··.Remark 10.
The proposed method is less computational and accurate as the obtained solution fast converges to the classical exact solution by substitutingα=1in Equation (42), i.e.,
(43)zx,τ=x1−τ+τ2−τ3+⋯=x∑i=0∞−1iτi,zx,τ=x1+τ.
Equation (43) represents the exact classical solution of (30).The absolute errors between exact solution and approximate solution forα=1 are given in Table 1. Also, the absolute errors of α=1 and α=0.9 are given in Table 2. The approximate solutions for α=0.98 and α=1 are represented by zapprox and zapprox, respectively.Example 2.
Consider the nonlinear time fractional order Burger equation as(44)DCFταzx,τ+z∂z∂x=η∂2z∂x2,α∈0,1,subjected to the initial condition zx,0=nx.Solution 2.
Implementing Yang transform to Equation (44), we get
(45)Yzx,τ=vzx,0−1+αv−αY−η∂2z∂x2+z∂z∂x..
Applying inverse Yang transform, we obtain(46)zx,τ=xx−Y−11+αv−αY−η∂2z∂x2+z∂z∂x.
Using the HPT method, the approximate solution is(47)∑q=0∞ρqzqx,τ=nx−ρY−11+αv−αY∑q=0∞ρqHqz−η∂2z∂x2∑q=0∞ρqzq.,where He’s polynomial Hqz represents the nonlinear term z∂z/∂x. The first three terms of Hqz are given by
(48)H0z=z0∂z0∂x,H1z=z0∂z1∂x+z1∂z0∂x,H1z=z0∂z2∂x+z1∂z1∂x+z2∂z0∂x,⋮.
Similarly, other terms can be calculated. Comparing the like powers ofρ in (47), we achieve
(49)ρ0:z0x,τ=nx,ρ1:z1x,τ=−Y−11+αv−αYH0z−η∂2∂x2z0.Now, we compute H0z as follows:
(50)H0z=z0∂z0∂x=nx∂∂xnx=n2x.Thus, Equation (49) can be written as
(51)ρ1:z1x,τ=−Y−11+αv−αYn2x−0=−n2xY−1v+αv2−αv,ρ1:z1x,τ=−n2x1−α+ατ.Similarly, the third approximation is given by
(52)ρ2:z2x,τ=2n3xα−2α+α2τ22−2αα−1τ+1.One can calculate more terms. The acquired series solution is represented as follows:
(53)zx,τ=nx−n2x1−α+ατ+2n3xα−2α+α2τ22−2αα−1τ+1+⋯.Remark 11.
The proposed method is less computational and accurate as the obtained solution fastly converges to the classical exact solution by substitutingα=1 in Equation (53), i.e.,
(54)zx,τ=nx1−nτ+n2τ2−n3τ3+⋯=x∑i=0∞−1ini+1τi,zx,τ=nx1+nτ.
Equation (54) is the classical solution of the considered problem.Table 1
Absolute error between exact and approximate solutions of Example1 at τ=0.5 and α=1.
xzapproxx,τzexactx,τzexact−zapprox0.10.07500.06670.00830.20.15000.13330.01670.30.22500.20000.02500.40.30000.26670.03330.50.37500.33330.04170.60.45000.40000.05000.70.52500.46670.05830.80.60000.53330.06670.90.67500.60000.075010.75000.66670.08333Table 2
Absolute error between approximate solution of Example1 at α=0.98 and α=1.
xzapproxx,τzapprox′x,τzexact−zapprox′0.10.07700.075000.00200.20.15400.15000.00400.30.23100.22500.00600.40.30800.30000.00800.50.38510.37500.01010.60.46210.45000.01210.70.53910.52500.01410.80.61610.60000.01610.90.69310.67500.01811.00.77010.75000.0201The absolute errors between exact solution and approximate solution forα=1 and τ=0.5 are given in Table 3. Also, the absolute errors of for x=0.5, α=1, and α=0.9 are given in Table 4. The approximate solutions for α=0.98 and α=1 are represented by zapprox and zapprox′, respectively.Remark 12.
The proposed YTHPM method is a powerful new method which needs less computation time and is much easier and more convenient than the other methods like HAM (homotopy analysis method). Computational time means HPM takes less time, to calculate the iterative terms for the series solution, because HPM only depends on single parameter. However, HAM depends on two parameters, i.e.,h (auxiliary parameter) and p (embedding parameter). So, HAM takes a little more time for calculating the successive terms of the series solution. Also, the convergence of the HAM depends on h which is different for various approximations. When h=−1 in HAM, then the solution directly converges to the HPM solution; otherwise, more terms will be calculated than HPM case. So, computational labor is much in HAM instead of HPM. Actually, the Yang transform is closely related to Laplace transform which is convergent. Therefore, Yang transform combined with HPM is an efficient and valid computational method for solving DEs because the Yang transform allows one in many situations to overcome the deficiency mainly caused by unsatisfied boundary or initial conditions that appear in other semianalytical methods such as HAM. The comparison between HPM and HAM is already given in [25, 26].Table 3
Absolute error between exact and approximate solution of the Example2 at τ=0.1 and α=1.
xzapproxx,τzexactx,τzexact−zapprox0.10.16800.16670.00130.20.33600.33330.00270.30.50400.50000.00400.40.67200.66670.00530.50.84000.83330.00670.61.00801.00000.00800.71.17601.16670.00930.81.34401.33330.01070.91.51201.50000.01201.01.68001.66670.0133Table 4
Absolute error between approximate solution of Example2 at α=0.98 and α=1.
τzapproxx,τzapprox′x,τzapprox−zapprox′0.10.19880.16800.03080.20.39750.33600.06150.30.59630.50400.09230.40.79500.67200.12300.50.99380.84000.15380.61.19251.00800.18450.71.39131.17600.21530.81.59001.34400.24600.91.78881.51200.27681.01.98761.68000.3076


## 5. Conclusion

We used a novel method to solve approximately nonlinear PDEs of fractional order in the Caputo-Fabrizio context in this paper. The Yang homotopy perturbation transform method (YHPTM) is a new method that combines the Yang transform and the HPM. The nonlinear term was decomposed into He’s polynomial using the HPM. A general procedure for solving nonlinear PDEs described by the CF derivative has been developed. We have demonstrated the estimated solution’s convergence and provided a result for the absolute error calculation. We solved well-known nonlinear PDEs such as the KdV equation and Burger’s equation to test the accuracy and validity of the proposed technique. By substitutingα=1, we were able to achieve the required solution in series form, which quickly converges to the exact solution of the problems, as seen in the remarks following each problem’s solution (also see Figures 1–4). We determined the numerical values of the absolute errors between the estimated and exact solutions, indicating that the exact and approximate solutions are in agreement. We have shown 3D graphs that demonstrate the suggested technique’s high precision and speed of convergence (see Figures 5 and 6). The approach also has the advantage of not requiring linearization, discretization, or additional memory. As a result, we have concluded that the proposed novel method is effective, reliable, and computationally effective. This approach will be used to solve the Atangana-Baleanu fractional-order PDEs. Further, we will use the Yang transform with HAM to solve nonlinear PDEs of fractional order in the future.Figure 1
Approximate solutionzx,τ for fractional orders α=0.7 and 0.8.Figure 2
Approximate solutionzx,τ for fractional orders α=0.9 and 1.Figure 3
Approximate solutionzx,τ for fractional orders α=0.7 and 0.8.Figure 4
Approximate solutionzx,τ for fractional orders α=0.9 and 1.Figure 5
Comparison between exact and approximate solutions of Example1.Figure 6
Comparison between exact and approximate solutions of the Example2.



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