# An Accelerated Homotopy Perturbation Method for Solving Nonlinear Two-Dimensional Volterra-Fredholm Integrodifferential Equations

Authors: F. A. Hendi; M. M. Al-Qarni
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

We propose and apply coupling of the variational iteration method (VIM) and homotopy perturbation method (HPM) to solve nonlinear mixed Volterra-Fredholm integrodifferential equations (VFIDE). In this approach, we use a new formula called variational homotopy perturbation method (VHPM) and variational accelerated homotopy perturbation method (VAHPM). This approach is based on the form of He’s polynomials and on a new form of He’s polynomials. We discuss the convergence of the technique. Some numerical examples are introduced to verify the efficiency of this technique.


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## Body

## 1. Introduction

In recent years, there has been a clear interest in integrodifferential equations which are a combination of differential and Volterra-Fredholm integral equations. Integrodifferential equations play an important role in many branches of linear and nonlinear functional analyses and their applications. The mentioned integrodifferential equations are usually difficult to solve analytically, so approximation strategies are required to obtain the solution of the linear and nonlinear integrodifferential equations [1].Many researchers studied and discussed the linear VFIDE [2]. Al-Jubory [3] introduced some approximation methods to solve Volterra-Fredholm integral and integrodifferential equations. Dadkhah et al. in [4] used a numerical solution of nonlinear VFIDE by using Legendre wavelets. Rabbani and Kiasoltani [5] studied the solving of a nonlinear system of VFIDE by using the discrete collocation method. Gherjalar and Mohammadikia [6] solved integral and integrodifferential equations by using the B-splines function. In this work, we used the HPM and VIM to solve the two-dimensional nonlinear VFIDE as follows:(1)∑j=0kPjx1,ȷ1ujx1,ȷ1=f˙x1,ȷ1+∫ax1∫ΩFx1,ȷ1,y,τγuly,τdydτ,x1,ȷ1∈J´=a,x1×Ω,with initial conditions(2)ura,ȷ1=gr,r=0,1,…,k-1,Ω=a,b,where uj(x1,ȷ1)=dju/dx1j. The functions f˙(x1,ȷ1), F(x1,ȷ1,y,τ) and γuly,τ,l>0, are analytic functions on J′, and functions Pjx1,ȷ1,j=0,1,…,k,Pk(x1,ȷ1)≠0 are given.For VIM and HPM, which were proposed by He in [7, 8], the solution is considered as the summation of an infinite series which is assumed to be convergent to the exact solution. In recent years, HPM has been applied with great success, so relations and algorithms have been deduced and continuously improved to obtain an accurate solution for a large variety of linear and nonlinear problems. For instance, He in [7] used a strategy to solve some integrodifferential equations where he chose an initial approximate solution in the form of an exact solution with unknown constants.In this paper, a new approach based on VIM with HPM is introduced to solve the two-dimensional nonlinear VFIDE.


## 2. The HPM

In this section, we will present the HPM. We consider a general integral equation(3)Lu=0,where L is an integral operator. Define a convex homotopy H˘(ϑ,℘) by(4)H˘ϑ,℘=1-℘Ϝϑ+℘Lϑ=0,℘∈0,1,where Ϝ(ϑ) is a functional operator with solution ϑ0. Then,(5)H˘ϑ,0=Ϝϑ=0,H˘ϑ,1=Lϑ=0,and the process of changing ℘ from 0 to 1 is just that of changing ϑ from ϑ0 to u. In topology, this is called deformation, and Ϝ(ϑ) and L(ϑ) are called homotopies.According to the HPM, we can use the embedding parameter℘ as a “small parameter” and assume that the solution of (4) can be written as a power series in ℘:(6)ϑ=u0+℘u1+℘2u2+⋯=∑i=0∞℘iui=u.When ℘→1, the approximate solution of (3) is obtained with(7)u=lim℘→1⁡ϑ=u0+u1+u2+⋯=∑i=0∞ui.Series (7) is convergent for most cases; however, the rate of convergence depends upon the nonlinear operator L [9].


## 3. The HPM for Solving Nonlinear Mixed VFIDE

In what follows, we display an outline for utilizing the HPM for solving the nonlinear VFIDE. Equation (1) can be written as follows:(8)ux1,ȷ1=L-1f˙x1,ȷ1Pkx1,ȷ1+∑r=0k-11r!x1-ȷ1rgr+L-1∫ax1∫ΩFx1,ȷ1,y,τγuly,τPkx1,ȷ1dydτ-L-1∑j=0k-1Pjx1,ȷ1Pkx1,ȷ1ujx1,ȷ1,where L-1 is the multiple integration operator given as follows: (9)L-1·=∫ax1∫ax1⋯∫ax1·dx1dx1⋯dx1,ntimes.So, (8) takes the form(10)ux1,ȷ1=L-1f˙x1,ȷ1Pkx1,ȷ1+∑r=0k-11r!x1-ȷ1rgr+∫ax1∫Ωx1-ȷ1kk!Fx1,ȷ1,y,τγuly,τPkx1,ȷ1dydτ-∑j=0k-1∫ax1x1-ȷ1k-1k-1!Pjx1,ȷ1Pkx1,ȷ1ujx1,ȷ1dȷ1since (11)∑j=0k-1L-1Pjx1,ȷ1Pkx1,ȷ1ujx1,ȷ1=∑j=0k-1∫ax1x1-ȷ1k-1k-1!Pjx1,ȷ1Pkx1,ȷ1ujx1,ȷ1dȷ1.To illustrate the HPM, for nonlinear mixed VFIDE, let us consider (8):(12)H˘ϑ,℘=ϑx1,ȷ1-L-1f˙x1,ȷ1Pkx1,ȷ1-∑r=0k-11r!x1-ȷ1rgr-℘L-1∫ax1∫ΩFx1,ȷ1,y,τγϑly,τPkx1,ȷ1dydτ-L-1∑j=0k-1Pjx1,ȷ1Pkx1,ȷ1ϑjx1,ȷ1=0.By the HPM, we can expand ϑ(x1,ȷ1) into the form(13)ϑx1,ȷ1=u0x1,ȷ1+℘u1x1,ȷ1+℘2u2x1,ȷ1+⋯=∑i=0∞℘iuix1,ȷ1=ux1,ȷ1.When ℘→1, the approximate solution is obtained with (14)ux1,ȷ1=lim℘→1⁡ϑx1,ȷ1=u0x1,ȷ1+u1x1,ȷ1+u2x1,ȷ1+⋯ and in sum, according to [10], He’s HPM considers the nonlinear term γ(u) as(15)γu=∑i=0∞℘iH˘i=H˘0+℘H˘1+℘2H˘2+⋯,where Hn’s are the so-called He’s polynomials [10], which can be calculated by using the formula(16)H˘n=1n∂n∂℘nγ∑i=0∞℘iui℘=0,n=0,1,2,….Substituting (13) and (15) into (12) and equating the terms with identical powers of ℘, we have(17)℘0:u0x1,ȷ1=L-1f˙x1,ȷ1Pkx1,ȷ1+∑r=0k-11r!x1-ȷ1rgr,℘i+1:ui+1x1,ȷ1=L-1∫ax1∫ΩFx1,ȷ1,y,τPkx1,ȷ1H˘idydτ-L-1∑j=0k-1Pjx1,ȷ1Pkx1,ȷ1Lij,i≥0.The nonlinear termsγ(ul(x1,ȷ1)) and Dj(u(x1,ȷ1))(Dj=∂ju(x1,ȷ1)/∂x1j is a derivative operator) are usually represented by an infinite series of the so-called He’s polynomials as follows:(18)γulx1,ȷ1=∑i=0∞H˘,Djux1,ȷ1=∑i=0∞Lij.The componentsui(x1,ȷ1),i≥0, can be computed by using the recursive relations (17).


## 4. A New Formula to He’s Polynomials

He’s polynomials are not unique; another formula of He’s polynomials(H˘n), called accelerated He’s polynomials, is represented by (H¯n); in [11], the author proved that(19)γu=∑n=0∞H˘n=∑n=0∞H¯nin which H˘n can be written in the new mathematical form(20)H¯n=γδn-∑i=0n-1H¯i,where the partial sum δn=∑i=0nui(x1,ȷ1) and H¯0=γ(u0). Substituting (13) and γ(u)=∑n=0∞℘nH¯n into (12) and equating the terms with identical powers of ℘, we obtain the following accelerated recursive formula:(21)℘0:u0x1,ȷ1=L-1f˙x1,ȷ1Pkx1,ȷ1+∑r=0k-11r!x1-ȷ1rgr,℘i+1:ui+1x1,ȷ1=L-1∫ax1∫ΩFx1,ȷ1,y,τPkx1,ȷ1H¯idydτ-L-1∑j=0k-1Pjx1,ȷ1Pkx1,ȷ1Lij,i≥0.For example, ifγ(u)=u3, the first four polynomials using formulas (16) and (20) are computed to be as follows.Using formula (16),(22)H˘0=u03.H˘1=3u02u1.H˘2=3u0u12+3u02u2.H˘3=u13+6u0u1u2+3u02u3.H˘4=3u12u2+3u0u22+6u0u1u3+3u02u4.Using formula (20),(23)H¯0=u03.H¯1=3u02u1+3u0u12+u13.H¯2=3u02u2+3u0u22+3u12u2+3u1u22+6u0u1u2+u23.H¯3=3u02u3+3u0u32+3u12u3+3u1u32+3u22u3+3u2u32+6u0u1u3+6u0u2u3+6u1u2u3+u33.H¯4=3u02u4+3u0u42+3u12u4+3u1u42+3u22u4+3u2u42+3u32u4+3u3u42+6u0u1u4+6u0u2u4+6u0u3u4+6u1u2u4+6u1u3u4+6u2u3u4+u43.Clearly, the first four polynomials computed using the suggested formula (20) include the first four polynomials computed using formula (16) in addition to other terms that should appear in H˘5,H˘6,H˘7,… using formula (16). Thus, the solution that was obtained using formula (20) enforces many terms to the calculation processes earlier, yielding faster convergence.


## 5. The VIM

Consider the differential equation(24)ıu+ℵu=f˙x1,where ı is a linear operator, ℵ is a nonlinear operator, and f˙(x1) is a given continuous function. The VIM presents a correction functional for (24) in the form (25)ui+1x1=uix1+∫0x1λςıuiς+ℵu~iς-f˙ςdς,where λ is a Lagrange multiplier [8, 9] which can be identified optimally via variational theory, ui is the nth approximate solution, and u~i denotes a restricted variation (i.e., ϱui=0).


## 6. Adapting VIM with HPM for Solving (1) and (2)

This modified version of VHPM is obtained by the coupling of VIM with HPM. First, by using formula (16), we obtain(26)∑i=0∞℘iui+1x1,ȷ1=uix1,ȷ1+℘∫0ȷ1λς∑i=0∞℘iuikx1,ς-f˙x1,ςPkx1,ς-∫ax1∫ΩFx1,ς,y,τPkx1,ς∑i=0∞℘iH˘idydτ+∑j=0k-1Pjx1,ςPkx1,ς∑i=0∞℘iuijx1,ςdςwhich is called VHPM.Second, by using formula (20),(27)∑i=0∞℘iui+1x1,ȷ1=uix1,ȷ1+℘∫0ȷ1λς∑i=0∞℘iuikx1,ς-f˙x1,ςPkx1,ς-∫ax1∫ΩFx1,ς,y,τPkx1,ς∑i=0∞℘iH¯idydτ+∑j=0k-1Pjx1,ςPkx1,ς∑i=0∞℘iuijx1,ςdςwhich is called VAHPM.The following is the algorithm for calculatingu0,u1,u2,…,un-1,un:Step1: input nonlinear term γ(ul) and n that is the order of He’s polynomials, endpoint a,b, initial conditions g1,g2,…,gr, free term, and F(x1,ȷ1,y,τ).Step2: set u=u0+℘u1+℘2u2+⋯+℘nun.Step3: let ∑k=0n℘kH˘k=γ(u0+℘u1+℘2u2+⋯+℘nun).Step4: For i=0,1,…,n, do(a)ith-order derivative of both sides of the equality with respect to ℘:(28)∂i∑k=0n℘kH˘k∂℘i=∂γu0+℘u1+℘2u2+⋯+℘nun∂℘i;(b) let℘=0 of the above equality and determine H˘i by solving the equation with respect to H˘i.End do.Step5: put u0= initial conditions.Step6: for i=1,…,n, doStep7: calculate ui by applying (26),end do.Step8: set u=limi→∞ui as the approximate of the exact solution.


## 7. Convergence Analysis

In this section, the sufficient condition that guarantees the existence of a unique solution is introduced in Theorem1, convergence of the methods is proved in Theorems 2 and 3, and finally the maximum absolute error of the truncated series (ui(x1,ȷ1)=∑i=0∞ui(x1,ȷ1)) is estimated in Theorem 4.Considering (10), we set(29)L-1f˙x1,ȷ1Pkx1,ȷ1+∑r=0k-11r!x1-ȷ1rgr=f1x1,ȷ1,k1x1,ȷ1=∫Ωx1-ȷ1kk!Fx1,ȷ1,y,τPkx1,ȷ1dyk2x1,ȷ1=x1-ȷ1k-1k-1!Pjx1,ȷ1Pkx1,ȷ1.We can write (10) as(30)ux1,ȷ1=f1x1,ȷ1+∫ax1k1x1,ȷ1γuly,τdτ-∑j=0k-1∫ax1k2x1,ȷ1ujx1,ȷ1dȷ1.We assumef1(x1,ȷ1) is bounded for all x1,ȷ1 in J´ and(31)k1x1,ȷ1≤N1k2x1,ȷ1≤N1j,j=0,1,…,k-1,∀x1,ȷ1∈J´.Also, we suppose the nonlinear termsγ(ul(x1,ȷ1)) and Dj(u(x1,ȷ1)) are Lipschitz continuous with(32)γulx1,ȷ1-γul∗x1,ȷ1≤dux1,ȷ1-u∗x1,ȷ1Djux1,ȷ1-Dju∗x1,ȷ1≤Zjux1,ȷ1-u∗x1,ȷ1,j=0,1,…,k-1.Hence, we set(33)Γ=b-adN1+kZN,Z=max⁡Zj,N=max⁡N1j,j=0,1,…,k-1.Theorem 1.
Two-dimensional nonlinear VFIDE has a unique solution whenever0<Γ<1.Proof.
Letu and u∗ be two different solutions of (30). Then,(34)ux1,ȷ1-u∗x1,ȷ1=∫ax1k1x1,ȷ1γulx1,ȷ1-γul∗x1,ȷ1dτ-∑j=0k-1∫ax1k2x1,ȷ1Djux1,ȷ1-Dju∗x1,ȷ1dȷ1≤∫ax1k1x1,ȷ1γulx1,ȷ1-γul∗x1,ȷ1dτ+∑j=0k-1∫ax1k2x1,ȷ1Djux1,ȷ1-Dju∗x1,ȷ1dȷ1≤b-adN1+kZNux1,ȷ1-u∗x1,ȷ1=Γux1,ȷ1-u∗x1,ȷ1,from which we get 1-Γu-u∗≤0. Since 0<Γ<1, therefore u-u∗=0. Therefore, u=u∗ and this completes the proof.Theorem 2.
The series solutionu(x1,ȷ1)=∑i=0∞ui(x1,ȷ1) of (1) using HPM convergence when 0<Γ<1 and u1(x1,ȷ1)<∞.Proof.
Denote with(CJ´,·) the Banach space of all continuous functions on J´ with the norm f˙(x1,ȷ1)=max⁡f˙x1,ȷ1 for all x1,ȷ1 in J´. Define the sequence of partial sums δn, and let δn and δm be arbitrary partial sums with n≥m. We are going to prove that δn=∑i=0nui(x1,ȷ1) is a Cauchy sequence in this Banach space:(35)δn-δm=max∀x1,ȷ1∈J´δn-δm=max∀x1,ȷ1∈J´∑i=m+1nuix1,ȷ1=max∀x1,ȷ1∈J´∑i=m+1n∫ax1k1x1,ȷ1H¯idτ-∑j=0k-1∫ax1k2x1,ȷ1Lijdȷ1=max∀x1,ȷ1∈J´∫ax1k1x1,ȷ1∑i=mn-1H¯idτ-∑j=0k-1∫ax1k2x,ȷ1∑i=mn-1Lijdȷ1.
From (20), we have (36)∑i=mn-1H¯i=γδn-1-γδm-1∑i=mn-1Lij=Djδn-1-Djδm-1. So, (37)δn-δm=max∀x1,ȷ1∈J´∫ax1k1x1,ȷ1γδn-1-γδm-1dτ-∑j=0k-1∫ax1k2x1,ȷ1Djδn-1-Djδm-1dȷ1≤max∀x1,ȷ1∈J´∫ax1k1x1,ȷ1γδn-1-γδm-1dτ+∑j=0k-1∫ax1k2x1,ȷ1Djδn-1-Djδm-1dȷ1≤Γδn-1-δm-1. Let n=m+1; then,(38)δm+1-δm≤Γδm-δm-1≤Γ2δm-1-δm-2≤⋯≤Γmδ1-δ0.So, (39)δn-δm≤δm+1-δm+δm+2-δm+1+⋯+δn-δn-1≤Γm+Γm+1+⋯+Γn-1δ1-δ0≤Γm1+Γ+Γ2+⋯+Γn-m-1δ1-δ0≤Γm1-Γn-m1-Γu1x1,ȷ1. Since 0<Γ<1, we have 1-Γn-m<1; then,(40)δn-δm≤Γm1-Γmax∀x1,ȷ1∈J´u1x1,ȷ1.
Butu1x1,ȷ1<∞ (since f˙1(x1,ȷ1) is bounded), so, as m→∞, then δn-δm→0. We conclude that δn is a Cauchy sequence in C[J´], and therefore the series is convergent and the proof is complete.Theorem 3.
When using VIM for solving two-dimensional nonlinear VFIDE where0<Γ<1 and Pk(x1,ȷ1)=1, then u(x1,ȷ1)=limn→∞un(x1,ȷ1) converges.Proof.
One has(41)un+1x1,ȷ1=unx1,ȷ1-∫0ȷ1unx1,ς-f1x1,ς-∫ax1k1x1,ςγunlx1,ςdτ+∑j=0k-1∫ax1k2x1,ςunjx1,ςdȷ1dς(42)ux1,ȷ1=ux1,ȷ1-∫0ȷ1ux1,ς-f1x1,ς-∫ax1k1x1,ςγulx1,ςdτ+∑j=0k-1∫ax1k2x1,ςujx1,ςdȷ1dς.
By subtracting relation (41) from (42),(43)un+1x1,ȷ1-ux1,ȷ1=unx1,ȷ1-ux1,ȷ1-∫0ȷ1unx1,ς-ux1,ς-∫ax1k1x1,ςγunlx1,ς-γulx1,ςdτ+∑j=0k-1∫ax1k2x1,ςDjunx1,ς-Djux1,ςdȷ1dς.
Hence, we set(44)en+1x1,ȷ1=un+1x1,ȷ1-ux1,ȷ1,enx1,ȷ1=unx1,ȷ1-ux1,ȷ1.Then,(45)en+1x1,ȷ1=enx1,ȷ1-∫0ȷ1unx1,ς-ux1,ς-∫ax1k1x1,ςγunlx1,ς-γulx1,ςdτ+∑j=0k-1∫ax1k2x1,ς·Djunx1,ς-Djux1,ςdȷ1dς≤enx1,ȷ11-b-adN1+kZN=1-Γenx1,ȷ1.Therefore, (46)en+1=max∀x1,ȷ1∈J´en+1≤1-Γmax∀x1,ȷ1∈J´en=en. Since 0<Γ<1, then en→0. So, the series converges and the proof is complete.Theorem 4.
The maximum absolute truncation error of the seriesu(x1,ȷ1)=∑i=0∞ui(x1,ȷ1) to (1) is estimated to be (47)max∀x1,ȷ1∈J´ux1,ȷ1-∑i=0muix1,ȷ1≤Γm+11-Γmax∀x1,ȷ1∈J´γu0.Proof.
From Theorem2 and inequality (40), we have(48)δn-δm≤Γm1-Γmax∀x1,ȷ1∈J´u1x1,ȷ1. As n→∞, then δn→u(x1,ȷ1) and we have (49)max∀x,t∈J´u1x1,ȷ1≤b-adN1+kZNmax∀x1,ȷ1∈J´γu0≤Γmax∀x1,ȷ1∈J´γu0. So, (50)ux1,ȷ1-δm≤Γm+11-Γmax∀x1,ȷ1∈J´γu0.
Finally, the maximum absolute truncation error in the intervalJ´ is(51)max∀x1,ȷ1∈J´ux1,ȷ1-∑i=0muix1,ȷ1≤Γm+11-Γmax∀x1,ȷ1∈J´γu0.


## 8. Numerical Examples

Example 1.
Consider the nonlinear integrodifferential equation(52)∂2ux1,ȷ1∂ȷ12+∂2ux1,ȷ1∂x1∂ȷ1-x1u3x1,ȷ1+∫0x1∫01u2y,τdydτ=f˙x1,ȷ1,x1∈0,1,where(53)f˙x1,ȷ1=2ȷ1+115x13ȷ15-x14ȷ15+2x1,with the initial conditions(54)I.Cs:ux1,0=0,∂ux1,0∂ȷ1=0, which has exact solution u(x1,t1)=x1ȷ12. This example is solved by using the variational iteration method with He’s polynomials VHPM (see (26)) and VAHPM (see (27)) expressing the nonlinear terms of H˘ and H¯, respectively, in Table 1.Table 1
Exact solution, approximate solution, and error by using VHPM and VAHPM.
x
1
ȷ1
Exact
A
p
p
r
.
(
V
H
P
M
)
E
r
r
.
(
V
H
P
M
)
A
p
p
r
.
(
V
A
H
P
M
)
E
r
r
.
(
V
A
H
P
M
)
0.00
E
+
00
0.00
E
+
00
0.00000
E
+
00
0.00000
E
+
00
0.00000
E
+
00
0.00000
E
+
00
0.00000
E
+
00
3.00
E
-
03
3.00
E
-
03
2.70000
E
-
08
9.90000
E
-
08
2.70000
E
-
08
3.60000
E
-
08
9.00000
E
-
09
5.00
E
-
03
5.00
E
-
03
1.25000
E
-
07
4.58330
E
-
07
3.33333
E
-
07
1.66667
E
-
07
4.16670
E
-
08
7.00
E
-
03
7.00
E
-
03
3.43000
E
-
07
1.25767
E
-
06
9.14670
E
-
07
4.57333
E
-
07
1.14333
E
-
07
9.00
E
-
03
9.00
E
-
02
7.290000
E
-
07
2.67300
E
-
06
1.94400
E
-
06
9.72000
E
-
07
2.43000
E
-
07Example 2.
Consider the nonlinear integrodifferential equation(55)ux1,ȷ1∂2ux1,ȷ1∂ȷ12-4ux1,ȷ1∂2ux1,ȷ1∂x12+4∫0x1∫01u2y,τdydτ=f˙x1,ȷ1,x1∈0,1,where(56)f˙x1,ȷ1=x1-12πsin⁡2πx1ȷ1-14πsin⁡4πȷ1,with the boundary conditions(57)B.Cs:u0,ȷ1=u1,ȷ1=0and the initial conditions(58)I.Cs:ux1,0=sin⁡πx1,0≤x1≤1∂ux1,0∂ȷ1=0,0≤x1≤1, which has exact solution u(x1,ȷ1)=sin⁡πx1cos⁡(2πȷ1). This example is solved by using VHPM (see (26)) and VAHPM (see (27)) expressing the nonlinear terms of H˘ and H¯, respectively, in Table 2.Table 2
Exact solution, approximate solution, and error by using VHPM and VAHPM.
x
1
ȷ1
Exact
A
p
p
r
.
V
H
P
M
E
r
r
.
V
H
P
M
A
p
p
r
.
V
A
H
P
M
E
r
r
.
V
A
H
P
M
0.00
E
+
00
0.00
E
+
00
0.00000
E
+
00
0.00000
E
+
00
0.00000
E
+
00
0.00000
E
+
00
0.00000
E
+
00
3
.
00
E
-
03
3.00
E
-
03
9.42295
E
-
03
9.41128
E
-
03
1.16700
E
-
05
9.42443
E
-
03
1.48000
E
-
06
5.00
E
-
03
5.00
E
-
03
1.56997
E
-
02
1.56384
E
-
02
6.13000
E
-
05
1.57054
E
-
02
5.70000
E
-
06
7.00
E
-
03
7.00
E
-
03
2.19680
E
-
02
2.17975
E
-
02
1.70500
E
-
04
2.19817
E
-
02
1.37000
E
-
05
9.00
E
-
03
9.00
E
-
02
2.82253
E
-
02
2.78610
E
-
02
3.64300
E
-
04
2.82534
E
-
02
2.81000
E
-
05


## 9. Conclusion

In this paper, we applied VHPM and VAHPM to solve nonlinear mixed VFIDE. The proposed VAHPM converges faster than the VHPM. Based on the proposed formula (27) with accelerated He’s polynomials formula (20), the convergence of the technique is proved. The presented technique is very easy to implement and it reduces the computation size.



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*Source: 9385040-2017-07-30.xml*