# Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations

Authors: Bo Xu; Sheng Zhang
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this paper, the local fractional version of homotopy perturbation method (HPM) is established for a new class of local fractional integral-differential equation (IDE). With the embedded homotopy parameter monotonously changing from 0 to 1, the special easy-to-solve fractional problem continuously deforms to the class of local fractional IDE. As a concrete example, an explicit and exact Mittag–Leffler function solution of one special case of the local fractional IDE is obtained. In the process of solving, two initial solutions are selected for the iterative operation of local fractional HPM. One of the initial solutions has a critical condition of convergence and divergence related to the fractional order, and the other converges directly to the real solution. This paper reveals that whether the sequence of approximate solutions generated by the iteration of local fractional HPM can approach the real solution depends on the selection of the initial approximate solutions and sometimes also depends on the fractional order of the selected initial approximate solutions or the considered equations.


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## Body

## 1. Introduction

Fractals, solitons, and chaos together constitute three important branches of nonlinear sciences. In fractal space, there exist some magical functions which are continuous everywhere but nondifferentiable everywhere. The local fractional calculus [1] developed in recent years provides a powerful mathematical tool to handling with such type of nondifferentiable functions. Fractional calculus, which is widely believed to have originated more than 300 years ago, has attracted much attention [2–17]. It is of theoretical and practical value to solve fractional differential equations (DEs) directly connecting with fractional dynamical processes in a great many fields. For this reason, people often construct exact solutions of fractional DEs to obtain useful clues in these fractional dynamical processes for specific applications.With the development of fractional calculus, many numerical and analytical methods for fractional DEs have been developed, such as integral transform method [1], series expansion method [3], Adomian decomposition method [4], Fan subequation method [5], variational iteration method (VIM) [6], variable separation method [7], finite difference method [8], homotopy perturbation method (HPM) [9], combined the HPM with Laplace transform [10], exp-function method [11], and Hirota bilinear method [12]. The HPM proposed by He [18] couples the homotopy method and the perturbation technique, which needs no the small parameters embedded in differential equations. More importantly, it is indicated in [19] that the HPM can truly eliminate the limitations existing in traditional perturbation methods.One of the advantages of local fractional derivative is that it has been successfully used to describe some nondifferential problems appearing in science and engineering [1]. The concept of local fractional derivative, which is based on Riemann-Liouville fractional derivative, can be retrospected to Kolwankar and Gangal’s pioneering work [13]. More specifically, if the following limit exists and is finite for a given continuous function ux:0,1⟶R,
(1)Dαux0=limx⟶x0dqux−ux0dx−x0q,0<q<1,then Dαux0 is called q-order fractional derivative of ux at the point x=x0. Later, inspired by the relation dαux=Γ1+αdux of Jumarie [20], the local fractional derivative of a local fractional continuous but nondifferentiable function ux is defined as another form ([4], see Definition 1). Recently, the theory of local fractional calculus has been significantly developed. Yang and his collaborators [1, 3, 17] have made a series of achievements in the development of local fractional calculus. Benefiting from the graceful properties of local fractional calculus, some existing methods like those [5, 11, 12], originally proposed for DEs with integer orders, have successfully been extended to fractional DEs and many methods are meeting more and more challenges for solving fractional DEs.The paper is aimed at establishing the local fractional HPM for a new class of local fractional IDEs:(2)dαuxdxα−fx−1Γ1+α∫01gx,tutdtα=0,where 0<α≤1, ux, fx, and gx,t are the local fractional continuous but nondifferentiable functions, dαux/dxα and 1/Γ1+α∫01gx,tutdtα represent the local fractional derivative and integral [1], respectively, and Γ1+α is the well-known Euler’s Gamma function:
(3)Γ1+α=∫0∞tαe−tdt.Considering a concrete application of the established local fractional HPM, we would like to solve a special case of Equation (2):
(4)dαuxdxα−3Eα3xα+Eα3Γ1+α−Eα3−13xα−1Γ1+α∫013xαtαΓ1+αutdtα=0,where the Mittag–Leffler function
(5)Eαxα=∑k=0∞xkαΓ1+kα,which is defined on a fractal set [1].The organization of the rest of this paper is as follows. Section2 recalls the local fractional derivative and integral and some basic properties. Section 3 establishes the local fractional HPM for the class of local fractional IDEs (2). Section 4 takes the local fractional IDE (4), a special case of Equation (2), to test the established local fractional HPM and discuss the influence of not only the initial approximate solutions but also the fractional order on whether the sequence of approximate solutions can approach the real solution. Section 5 employs He-Laplace method coupling the HPM with Laplace transform to solve the local fractional IDE (4) and compares the obtained results. Section 6 summarizes the whole paper.


## 2. Local Fractional Derivative and Integral and Some Basic Properties

In this section, we recall the local fractional derivative and integral and some basic properties.Definition 1 (see [1]).
Letux be a local fractional continuous but nondifferentiable function; then, α-order local fractional derivative of ux at the point x=x0 reads
(6)Dxαux0=dαuxdxαx=x0=limx⟶x0Δαux−ux0x−x0α,0<α≤1,where Δαux−ux0≅Γ1+αux−ux0.
The local fractional derivative has some basic properties [1]:
(7)Dxαλux+μvx=λDxαux+μDxαvx,Dxαuxvx=Dxαuxvx+uxDxαvx,Dxαuxvx=Dxαuxvx−uxDxαvxv2x,DxαC=0,DxαxkαΓ1+kα=xk−1αΓ1+k−1α,DxαEαqxα=qEαqxα,Dxαsinαxα=cosαxα,Dxαcosαxα=−sinαxα,where λ, μ, C, and q are constants and k is an integer, while sinαxα=∑k=0∞−1kx2k+1α/Γ1+2k+1α and cosαxα=∑k=0∞−1kx2kα/Γ1+2kα.Definition 2 (see [1]).
Let functionux∈Cαa,b; then, the definition of α-order local fractional integral of ux in the integral a,b is as follows:
(8)Iabαux=1Γ1+α∫abutdtα=1Γ1+αlimΔxk⟶0∑k=0N−1uxkΔxkα,0<α≤1,where Δxk=xk+1−xk with x0=a<x1<⋯<xN−1<xN=b.
The local fractional integral has some basic properties [1]:
(9)aIbαλfx+μgx=λaIbαfx+μaIbαgx,IabαDxαfxgx=fxgxab−IabαfxDxαgx,I0xαC=CxαΓ1+α,I0xαxkαΓ1+kα=xk+1αΓ1+k+1α,I0xαEαqxα=Eαqxα−1q,I0xαsinαxα=1−cosαxα,I0xαcosαxα=sinαxα.


## 3. Local Fractional HPM for the Class of Local Fractional IDEs

In this section, we establish the local fractional HPM for the class of IDEs (2). For convenience, we rewrite Equation (2) as follows:
(10)Aαu=Lαu+Iαu=0,Lαu=dαuxdxα−fx,Iαu=−1Γ1+α∫01gx,tutdtα,where Aα represents the local fractional operator.In view of the local fractional HPM [17], we construct the local fractional homotopy Hαu,pα, u∈R, and p∈0,1 by the following:
(11)Hαu,pα=1−pαLαu−Lαu0+pαLαu+Iαu=0,with the embedded parameter pα which monotonously changes from 0 to 1 leads to the result that the easy-to-solve equation Lαu−Lαu0=0 continuously deforms to the original equation Lαu+Iαu=0. Using the constructed homotopy Hαu,pα, we can continuously trace a curve which is implicitly defined from a starting point
(12)Hαu,0=Lαu−Lαu0=0,to a solution function
(13)Hαu,1=Lαu+Iαu=0.From the perspective of topology, the above changing process is called a deformation. In this deformation,Lαu−Lαu0 and Lαu+Iαu are homotopic.Thus, the fractional homotopyHαu,pα in Equation (11) can be written as below:
(14)Hαu,pα=1−pαdαuxdxα−fx−dαu0xdxα−fx+pαdαuxdxα−fx−pαΓ1+α∫01gx,tutdtα=dαuxdxα−dαu0xdxα+pαdαu0xdxα−fx−1Γ1+α∫01gx,tutdtα=0.Substituting the seriesu expanded by the fractional homotopy parameter pα(15)u=v0α+pαv1α+p2αv2α+p3αv3α+⋯,into Equation (14) and comparing the coefficients of the same power of pα, we obtain a set of fractional equations:
(16)p0:dαv0αxdxα−dαu0xdxα=0,pα:dαv1αxdxα+dαu0xdxα−fx−1Γ1+α∫01gx,tv0αtdtα=0,p2α:dαv2αxdxα−1Γ1+α∫01gx,tv1αtdtα=0,p3α:dαv3αxdxα−1Γ1+α∫01gx,tv2αtdtα=0,⋮.Hereu0 is assumed to be an initial approximate solution of Equation (2). Generally, the initial approximation v0α or u0 can be freely chosen. Solving above set of fractional equations, we can obtain solutions v0α, v1α, v2α, v3α, and so on.Settingpα⟶1 and using Equation (15), we finally arrive at an approximate solution of Equation (2).
(17)u=limpα⟶1∑k=0∞vkαx=v0α+v1α+v2α+v3α+⋯.As pointed by He [18], the series (17) has convergence in most cases and the convergent rate is determined by Lαu when α=1. For the case of convergence, the series (17) can reach a closed form solution.


## 4. A Concrete Example

In this section, we apply the previously established local fractional HPM to the local fractional IDE (4).Firstly, we construct such a fractional homotopy:(18)Hαu,pα=dαuxdxα−dαu0xdxα+pαdαu0xdxα−3Eα3xα+Eα3Γ1+α−Eα3−13xα−1Γ1+α∫013xαtαΓ1+αutdtα=0.Secondly, we substitute Equation (15) into Equation (18), and then, the comparison of the coefficients of pjαj=0,1,2,⋯ gives a system of fractional equations:
(19)p0:dαv0αxdxα−dαu0xdxα=0,(20)pα:dαv1αxdxα+dαu0xdxα−3Eα3xα+Eα3Γ1+α−Eα3−13xα−1Γ1+α∫013xαtαΓ1+αv0αtdtα=0,(21)p2α:dαv2αxdxα−1Γ1+α∫013xαtαΓ1+αv1αtdtα=0,(22)p3α:dαv3αxdxα−1Γ1+α∫013xαtαΓ1+αv2αtdtα=0,⋮.In view of the arbitrariness ofv0α or u0, here we set
(23)dαv0αxdxα=dαu0xdxα=3Eα3xα−Eα3Γ1+α−Eα3−13xα,and then, from Equations (20)–(23), we have
(24)v0α=Eα3xα−Eα3Γ1+α−Eα3−13Γ1+αΓ1+2αx2α;dαv1αxdxα−Eα3Γ1+α−Eα3−131−3Γ1+3αΓ1+2αΓ1+4αxα=0,namely,
(25)v1α=Eα3Γ1+α−Eα3−131−3Γ1+3αΓ1+2αΓ1+4αΓ1+αΓ1+2αx2α;dαv2αxdxα−3Γ1+3αΓ1+2αΓ1+4αEα3Γ1+α−Eα3−131−3Γ1+3αΓ1+2αΓ1+4αxα=0,namely,
(26)v2α=3Γ1+3αΓ1+2αΓ1+4αEα3Γ1+α−Eα3−131−3Γ1+3αΓ1+2αΓ1+4αΓ1+αΓ1+2αx2α;dαv3αxdxα−32Γ21+3αΓ21+2αΓ21+4αEα3Γ1+α−Eα3−131−3Γ1+3αΓ1+2αΓ1+4αxα=0,namely,
(27)v3α=32Γ21+3αΓ21+2αΓ21+4αEα3Γ1+α−Eα3−131−3Γ1+3αΓ1+2αΓ1+4αΓ1+αΓ1+2αx2α;dαv4αxdxα−33Γ31+3αΓ31+2αΓ31+4αEα3Γ1+α−Eα3−131−3Γ1+3αΓ1+2αΓ1+4αxα=0,namely,
(28)v4α=33Γ31+3αΓ31+2αΓ31+4αEα3Γ1+α−Eα3−131−3Γ1+3αΓ1+2αΓ1+4αΓ1+αΓ1+2αx2α;⋮.Finally, we obtain an approximate solution of Equation (4).
(29)u=limpα⟶1∑k=0∞vkαxpkα=Eα3xα−Eα3Γ1+α−Eα3−13Γ1+αΓ1+2αx2α+1+3Γ1+3αΓ1+2αΓ1+4α+32Γ21+3αΓ21+2αΓ21+4α+33Γ31+3αΓ31+2αΓ31+4α+⋯×Eα3Γ1+α−Eα3−131−3Γ1+3αΓ1+2αΓ1+4αΓ1+αΓ1+2αx2α,which can be simplified as follows:
(30)u=Eα3xα−limm⟶∞3m−1Γm−11+3αΓm−11+2αΓm−11+4αEα3Γ1+α−Eα3−13Γ1+αΓ1+2αx2α.Obviously, thenth-order approximate solution of Equation (4) has the following form:
(31)un=Eα3xα−3nΓn1+3αΓn1+2αΓn1+4αEα3Γ1+α−Eα3−13Γ1+αΓ1+2αx2α,n=0,1,2,⋯.When the condition(32)q=3Γ1+3αΓ1+2αΓ1+4α<1,holds, the limit of Equation (30) exists. In this case, we obtain
(33)u=Eα3xα.In Figure1, we show the curve of the condition q with fractional order α, where the dashed line represents q=1. We can see from Figure 1 that there exists a unique value α0∈0.7,0.75 such that qα0=1. At the same time, with the help of Mathematica, we have
(34)q0.73≈1.02522329079942,q0.74≈0.9916413061494614.Figure 1
Curve of the conditionq with fractional order α.Thus, we can more accurately determine the range ofα0 as
(35)0.73<α0<0.74.This tells thatqα<qα0=1 if and only if α0<α≤1.In Figure2, the initial solution and 5th-order and 8th-order approximate solutions u0, u5, and u8 and the exact solution uexa are shown by constraining them to a Cantor set with dimension α=ln2/ln3≈0.631 which does not satisfy the convergence condition α0<α≤1. In this case, uexa, u0, u5, and u8 form a sequence of divergent approximate solutions.Figure 2
Divergent asymptotic solutions and exact solution in the caseα=ln2/ln3.Since the initial approximationv0α or u0 possesses arbitrariness as mentioned earlier, if we set again
(36)dαv0αxdxα=dαu0xdxα=3Eα3xα,then the similar process yields
(37)v0α=u0=Eα3xα,(38)v1α=v2α=v3α=⋯=0,from which we finally reach solution (33). So, for any 0<α≤1, the solution (33) is always the exact solution of Equation (4). That is to say, if we chose an appropriate initial approximation v0α or u0, then the operation can be considerably simplified. More importantly, the convergence condition that the sequence of approximate solutions depends on fractional order can be removed.It should be noted that whenα=1 and ux, fx, and gx,t are all the continuous and differentiable functions, solution (33) reduces to u=e3x which is the known exact solution of the following IDE [19]:
(39)duxdx−3e3x+132e3+1x−∫013xtutdt=0.


## 5. He-Laplace Method and Comparison

As Deng and Ge [21] pointed out, He-Laplace method has a simple and reliable algorithm and it can be coupled with the HPM or the VIM for solving various nonlinear models, shedding a bright light on fractal calculus. A newest typical example of He-Laplace method to illustrate its simplicity, directness, strength, and great prospects can be found in [22]. In what follows, we employ the local fractional version of He-Laplace method [23] to solve the local fractional IDE (4).Taking the local fractional Laplace transform on Equation (4), we can gain
(40)Ldαuxdxα=L3Eα3xα+Eα3Γ1+α−Eα3−13xα+L1Γ1+α∫013xαtαΓ1+αutdtα,(41)sαLux−u0=L3Eα3xα+Eα3Γ1+α−Eα3−13xα+L1Γ1+α∫013xαtαΓ1+αutdtα,(42)Lux=u0sα+1sαL3Eα3xα+Eα3Γ1+α−Eα3−13xα+1sαL1Γ1+α∫013xαtαΓ1+αutdtα.Then, the inverse local fractional Laplace transform of Equation (42) gives
(43)ux=Eα3xα−Eα3Γ1+α−Eα3−13Γ1+αΓ1+2αxα+L−11sαL1Γ1+α∫013xαtαΓ1+αutdtα,where u0=1 has been assumed and L and L−1 are the local fractional Laplace transform operator and inverse operator [1], respectively.Dealing Equation (43) with the local fractional HPM, we introduce
(44)∑n=0∞pnαunx=Eα3xα−Eα3Γ1+α−Eα3−13Γ1+αΓ1+2αxα+pαL−11sαL1Γ1+α∫013xαtαΓ1+α∑n=0∞pnαunxdtα,and compare the coefficients of the same powers of pα; then, He’s polynomials can be obtained:
(45)p0:u0x=Eα3xα−Eα3Γ1+α−Eα3−13Γ1+αΓ1+2αx2α,pα:u1x=L−11sαL1Γ1+α∫013xαtαΓ1+αu0xdtα=1−3Γ1+3αΓ1+2αΓ1+4αEα3Γ1+α−Eα3−13Γ1+αΓ1+2αx2α,p2α:u2x=L−11sαL1Γ1+α∫013xαtαΓ1+αu1xdtα=3Γ1+3αΓ1+2αΓ1+4α1−3Γ1+3αΓ1+2αΓ1+4αEα3Γ1+α−Eα3−13Γ1+αΓ1+2αx2α,p3α:u3x=L−11sαL1Γ1+α∫013xαtαΓ1+αu2xdtα=32Γ21+3αΓ21+2αΓ21+4α1−3Γ1+3αΓ1+2αΓ1+4αEα3Γ1+α−Eα3−13Γ1+αΓ1+2αx2α,p4α:u4x=L−11sαL1Γ1+α∫013xαtαΓ1+αu3xdtα=33Γ31+3αΓ31+2αΓ31+4α1−3Γ1+3αΓ1+2αΓ1+4αEα3Γ1+α−Eα3−13Γ1+αΓ1+2αx2α,⋮.We therefore obtain an approximate solution of Equation (4):
(46)u=limpα⟶1∑n=0∞pnαunαx=Eα3xα−Eα3Γ1+α−Eα3−13Γ1+αΓ1+2αx2α+1+3Γ1+3αΓ1+2αΓ1+4α+32Γ21+3αΓ21+2αΓ21+4α+33Γ31+3αΓ31+2αΓ31+4α+⋯×Eα3Γ1+α−Eα3−131−3Γ1+3αΓ1+2αΓ1+4αΓ1+αΓ1+2αx2α,which is the same as solution (29). It is not difficult to see that solution (46) has the same nth-order approximate solution (31), and when the condition (32) holds, the limit of solution (46) gives the same exact solution (33).Through comparison, we find that the computational difficulty of solving the local fractional IDE (4) by the above two methods is about the same. When using the local fractional HPM, we need to introduce an appropriate initial approximate solution v0x, while the local fractional He-Laplace method uses a known initial value u0. Generally speaking, it is more difficult to choose an initial approximate solution v0x than to find an initial value u0. For the latest valuable work on the modified HPM, we suggest to refer to Refs. [24, 25].


## 6. Conclusion

In summary, we have established the local fractional HPM for the class of local fractional IDEs (1). Based on the established local fractional HPM, an explicit and exact Mittag–Leffler function solution (33) of the local fractional IDE (4) is obtained by selecting two different initial solutions (24) and (37). The comparison shows that the local fractional He-Laplace method [23] can also obtain the same solution (33), but when the initial approximate solution is not easy to choose, the local fractional He-Laplace method [23] has certain advantages over the method used in this paper. The obtained results show that if we choose the approximate solutions appropriately, for example solution (37), then the calculation can be considerably simplified and that the sequence of approximate solutions generated by the local fractional HPM can directly approach the real solution without the influence of fractional order. However, for the selected initial approximate solution (24), we obtained a sequence of approximate solutions converging the real solution (33) in a certain range of the fractional order α, i.e., α0<α≤1. At the same time, there is a divergence interval 0,α0 which depends on the fractional order α. Here, qα0=1, and an approximate range of α0 is 0.73<α0<0.74. That is to say, α0 is the critical value of convergence and divergence related to fractional order α of the obtained sequence of approximate solutions. This is different from the HPM for integer-order DEs, all the sequences of approximate solutions of which either converge or diverge, and there is no such a critical value of convergence and divergence. When the nth-order approximate solution (31) is constrained to a Cantor set with dimension α=ln2/ln3≈0.631, this paper shows in Figure 2 a sequence of divergent approximate solutions. This paper fails to describe a sequence of convergent approximate solutions of (31), which is due to the complexity of the numerical simulation of fractal set. How to constrain solution (31) to other fractal sets and show some sequences of convergent approximate solutions? This is a question worth exploring. Besides, the research on qualitative behaviors of Equation (4) and other fractional IDEs is worth discussing. Some recent meaningful results of this research can be found in [26–28]. In 2007, Wang and He [29] took three concrete IDEs as examples to illustrate the effectiveness of the VIM for various IDEs. Based on this fact, we conclude that the local fractional version of VIM can also solve the local fractional IDE (4). In fact, the main steps of the local fractional VIM for Equation (4) can be summarized as follows: (i) identifying Lagrange multiplier λα=−1, (ii) determining the iterative formula of solution:
(47)un+1x=unx−1Γ1+α∫0xdαunξdξα+Funξdξα,n=0,1,2⋯,with
(48)Funξ=−3Eα3ξα+Eα3Γ1+α−Eα3−13ξα−1Γ1+α∫013ξαtαΓ1+αuntdtα,and (iii) selecting the initial approximate solution u0x=Eα3xα to obtain the exact solution ux=limn⟶∞unx=Eα3xα by using the determined iterative formula (47). Nevertheless, it is still worth trying to find an appropriate initial approximate solution and get the approximate solution (29) or (46) by Equation (47).



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