# Numerical Solution of Nonlinear Fredholm Integro-Differential Equations Using Spectral Homotopy Analysis Method

Authors: Z. Pashazadeh Atabakan; A. Kazemi Nasab; A. Kılıçman; Zainidin K. Eshkuvatov
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Spectral homotopy analysis method (SHAM) as a modification of homotopy analysis method (HAM) is applied to obtain solution of  high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and  exact solutions.


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## Body

## 1. Introduction

The integro-differential equations stem from the mathematical modeling of many complex real-life problems. Many scientific phenomena have been formulated using integro-differential equations [1, 2]. Solving nonlinear integro-differential equation is much more difficult than linear one analytically. So different types of numerical methods have been used to obtain an efficient approximation solution [3, 4]. In 1992 Liao [5] proposed the homotopy analysis method (HAM) concept in topology for solving nonlinear differential equations. Liao [6, 7] found that the convergence of series solutions of nonlinear equations cannot be guaranteed by the early HAM. Further, Liao [6] introduced a nonzero auxiliary parameter to solve this limitation. Unlike the special cases of HAM such as Lyapunove’s artificial small parameter method [8], Adomian decomposition method [9–12], and the δ-expansion method [13], this method need not a small perturbation parameter. In the HAM the perturbation techniques [14] need not be converted a nonlinear problem to infinite number of linear problems. The homotopy analysis method is applicable for solving problems having strong nonlinearity [15], even if they do not have any small or large parameters, so it is more powerful than traditional perturbation methods.The convergence region and the rate of approximation in series can been adjusted by this method. Also it can give us freedom to use different base function to approximate a non linear problem. The convergence of HAM for solving Volterra-Fredholm integro-differential equations is presented in [16].In 2010, Motsa et al. [17] suggested the so-called spectral homotopy analysis method (SHAM) using the Chebyshev pseudospectral method to solve the linear high-order deformation equations. Since the SHAM combines the HAM with the numerical techniques, it provides us larger freedom to choose auxiliary linear operators. Thus, one can choose more complicated auxiliary linear operators in the frame of the SHAM. In theory, any continuous function in a bounded interval can be best approximated using Chebyshev polynomial. So, the SHAM provides larger freedom to choose the auxiliary linear operator L and initial guess. Further, it is easy to employ the optimal convergence-control parameter in the frame of the SHAM. Thus, the SHAM has great potential to solve more complicated nonlinear problems in science and engineering, although further modifications in theory and more applications are needed. Chebyshev polynomial is considered a kind of special function. There are many other special functions such as Hermite polynomial, Legendre polynomial, Airy function, Bessel function, Riemann zeta function, and hypergeometric functions. Since the HAM provides us extremely large freedom to choose the auxiliary linear operator L and the initial guess, it should be possible to develop a generalized spectral HAM which can use a proper special function for a given nonlinear problem. The spectral homotopy analysis method has been used for solving partial and ordinary differential equations [18–20]. Spectral homotopy analysis method and its convergency for solving a class of optimal control problems are presented in [21]. Motsa et al. [17–19] found that the spectral homotopy analysis method is more efficient than the homotopy analysis method as it does not depend on the rule of solution expression and the rule of ergodicity. This method is more flexible than homotopy analysis method, since it allows for a wider range of linear and nonlinear operators, and one is not restricted to use the method of higher-order differential mapping for solving boundary value problems in bounded domains, unlike the homotopy analysis method. The range of admissible h values is much wider in spectral homotopy analysis method than in homotopy analysis method. The main restriction of HAM in solving integral equations is to choose the best initial guess, as the series solution is convergent. In SHAM the initial approximation is taken to be the solution of the nonhomogeneous linear part of the given equation. In 2012, Pashazadeh Atabakan et al. solved linear Volterra and Fredholm integro-differential equations using spectral homotopy analysis method; see [22].In this paper, we apply spectral homotopy analysis method (SHAM) to solve higher-order nonlinear Fredholm type of integro-differential equations. Fredholm integro-differential equation is given by(1)∑j=02‍aj(x)y(j)(x)=f(x)+μ∫-11‍k(x,t)[y(t)]rdt,y(-1)=y(1)=0,
where μ is constant value, f(x), k(x,t), [y(t)]r, and aj(x)r≥1 are functions that have suitable derivatives on interval -1≤t≤x≤1 and a2(x)≠0.The paper is organized in the following way. Section2 includes a brief introduction in homotopy analysis method. Spectral homotopy analysis method for solving nonlinear Fredholm integral equations is presented in Section 3. The existence and uniqueness of the solution and convergence of the proposed method are proved in Section 4. In Section 5, numerical examples are presented. In Section 6, concluding remarks are given.


## 2. Homotopy Analysis Solution

In this section, we give a brief introduction to HAM. We consider the following differential equation in a general form as follows:(2)N[y(η)]=0,
where N is nonlinear operator, η denotes independent variables, and y(η) is an unknown function, respectively. For simplicity we disregard all initial and all boundary conditions which can be dealt in similar way. The so-called zero-order deformation equation was constructed by Liao as follows:
(3)(1-p)L[ψ(η;p)-y0(η)]=phH(η)(N[ψ(η;p)]),
where p∈[0,1] is the embedding parameter, h is a nonzero convergence-parameter, H(η) is an auxiliary function, y0(η) is called an initial guess of y(η), and ψ(η;p) is an unknown function. In addition, L is an auxiliary linear operator, and N is nonlinear operator as follows:
(4)L(ψ(x;p))=ak(x)∂2ψ(x;p)∂x2
with the property L(∑j=02cjtj)=0, where cj, are constants and
(5)N[ψ(x;p)]=∑j=02‍aj(x)∂jψ(x;p)∂xj-f(x)-μ∫-11‍k(x,t)ψr(t)dt
is a nonlinear operator. Obviously, when p=0 and p=1, it holds ψ(η;0)=y0(η) and ψ(η;1)=y(η). In this way, as p increase from 0 to 1, ψ(η;p) alter from initial guess y0(η) to the solution y(η), and ψ(η;p) is expanded in Taylor series with respect to p as follows:
(6)ψ(η;p)=y0(η)+∑m=1+∞‍ym(η)pm,
where
(7)ym(η)=Dm[ψ(η;p)],Dmψ=1m!∂mψ∂pm|p=0.
The series (6) converges at p=1 if the auxiliary linear operator, the initial guess, the convergence-parameter, and the auxiliary function are properly selected as follows:
(8)ψ(η)=y0(η)+∑m=1+∞‍ym(η).
The admissible and valid values of the convergence-parameter h are found from the horizontal portion of the h-curves. Liao proved that y(η) is one of the solutions of original nonlinear equation. As H(η)=1, so (3) becomes
(9)(1-p)L[ψ(η;p)-y0(η)]=ph(N[ψ(η;p)]).
Define the vector ym={y0(η),y1(η),…,ym(η)}. Operating on both side of (9) with Dm, we have the so called mth-order deformation equation as follows:
(10)L[ym(η)-χmym-1(η)]=hH(η)Rm(ym-1(η)),
where
(11)Rm(ym-1)=1(m-1)!∂m-1N[ψ(η;p)]∂pm-1|p=0,χm={0,m≤11,otherwise,ym(η) for m≥0 that is governed by the linear equation (10) can be solved by symbolic computation software such as MAPLE, MATLAB, and similar CAS.


## 3. Spectral-Homotopy Analysis Solution

Consider the non linear Fredholm integro-differential equation:(12)∑j=02‍aj(x)y(j)(x)=f(x)+μ∫-11‍k(x,t)[y(t)]rdt,y(-1)=y(1)=0.
We begin by defining the following linear operator:
(13)L(ψ(x;p))=∑j=02‍aj(x)∂jψ(x;p)∂xj,
where p∈[0,1] is the embedding parameter and ψ(x;p) is an unknown function. The zeroth-order deformation equation is given by
(14)(1-p)L[ψ(η;p)-y0(η)]=ph(N[ψ(η;p)]-f(η)),
where h is the nonzero convergence controlling auxiliary parameter and N is a nonlinear operator given by
(15)N[ψ(η;p)]=∑j=02‍aj(η)∂jψ(η;p)∂ηj-f(η)-μ∫-11‍k(η,t)ψr(t)dt.
Differentiating (14) m times with respect to the embedding parameter p, setting p=0, and finally dividing them by m!, we have the so called mth-order deformation equation
(16)L[ym(η)-χmym-1(η)]=hRm,
subject to boundary conditions
(17)ym(-1)=ym(1)=0,
where
(18)Rm(η)=∑j=02‍aj(η)∂jψ(η;p)∂ηj-f(η)(1-χm)-μ∫-11‍k(η,t)ψr(t)dt.
The initial approximation that is used in the higher-order equation (18) is obtained on solving the following equation:
(19)∑j=02‍aj(x)y0(j)(x)=f(x)
subject to boundary conditions
(20)y0(-1)=y0(1)=0,
where we use the Chebyshev pseudospectral method to solve (19)-(20).We first approximatey0(η) by a truncated series of Chebyshev polynomial of the following form:
(21)y0(η)≈y0N(ηj)=∑k=0N‍y^kTk(ηj),j=0,…,N,
where Tk is the kth Chebyshev polynomials, y^k are coefficients and Gauss-Lobatto collocation points η0,η1,…,ηN which are the extrema of the Nth-order Chebyshev polynomial defined by
(22)ηj=cos(πjN).Derivatives of the functionsy0(η) at the collocation points are represented as
(23)dsy0(ηk)dηs=∑j=0N‍Dkjsy0(ηj),k=0,…,N,
where s is the order of differentiation and D is the Chebyshev spectral differentiation matrix. Following [23], we express the entries of the differentiation matrix D as
(24)Dkj=(-12)ckcj×(-1)k+jsin(π(j+k)/2N)sin(π(j-k)/2N),j≠k,Dkj=(-12)xksin2(πk/N),k≠0,N,k=j,D00=-DNN=2N2+16.Substituting (21)–(23) into (19) will result in
(25)AY0=F
subject to the boundary conditions
(26)y0(η0)=y0(ηN)=0,
where
(27)A=∑j=02‍ajDj,Y0=[y0(η0),y0(η1),…,y0(ηN)]T,F=[f(η0),f(η1),…,f(ηN)]T,ar=diag(ar(η0),ar(η1),…,ar(ηN)).
The values of y0(ηi), i=0,…,N are determined from the equation
(28)Y0=A-1F,
which is the initial approximation for the SHAM solution of the governing equation (12). Apply the Chebyshev pseudospectral transformation on (16)–(18) to obtain the following result:
(29)AYm=(χm+h)AYm-1-h[Sm-1-(1-χm)F],
subject to the boundary conditions
(30)ym(η0)=ym(ηN)=0,
where A and F were defined in and
(31)Ym=[ym(η0),ym(η1),…,ym(ηN)]T,sm=∫-11‍k(τ,t)[Ym]rdt.
To implement the boundary condition (30) we delete the first and the last rows of Sm-1, F and the first and the last rows and columns of A. Finally this recursive formula can be written as follows:
(32)Ym=(χm+h)Ym-1-hA-1[Sm-1-(1-χm)Fm-1],
with starting from the initial approximation we can obtain higher-order approximation Ym for m≥1 recursively. To compute the integral in (32) we use the Clenshaw-Curtis quadrature formula as follows:
(33)Sm(η)=∫-11‍k(η,t,Y~m)dt=∑j=0N‍wjk(η,ηj,Y~m),
where the nodes ηj are given by (22) and the weights wj are given by
(34)w0=wN={1N2,Nodd,1N2-1,Neven,(35)wl=2Nγl[1-∑k=1⌊N/2⌋‍2γ2k(4k2-1)cos2klπN],l=1,…,N-1,
where γ0=γN=2 and γl=1, for l=1,…,N-1. Y~ is a column vector of the elements of the vector Y that is computed as follows:
(36)Y~m=∑n1=0m‍ym-n1∑n2=0n1‍yn1-n2⋯∑nr-1=0nr-2‍ynr-2-nr-1ynr-1,
where m, r≥0 are positive integers [24].Regarding to accuracy, the stability, and the error of previous quadrature formula at the Gauss-Lobatto points we refer the reader to [25].


## 4. Convergence Analysis

Following the authors in [7, 16, 26], we present the convergence of spectral homotopy analysis method for solving Fredholm integro-differential equations.In view of (13) and (27), (12) can be written as follows:
(37)AY=F+μ∫-11‍k(x,t)G(Y)dt,
where Y, F, and G(Y) are vector functions.We obtain(38)Y=A-1F+μ∫-11‍k(x,t)A-1G(Y)dt.
By substituting F~=A-1F and G~(Y)=A-1G(Y) in (38), we obtain
(39)Y=F~+μ∫-11‍k(x,t)G~(Y)dt.
In (39), we assume that F~ is bounded for all t in C=[-1,1] and
(40)|k(x,t)|≤M.
Also, we suppose that the non linear term G~(Y) is Lipschitz continuous with
(41)∥G~(Y)-G~(Y*)∥≤L∥Y-Y*∥.
If we set α=2μLM, then the following can be proved by using the previous assumptions.Theorem 1.
The nonlinear Fredholm integro-differential equation in (32) has a unique solution whenever 0<α<1.Proof.
LetY and Y* be two different solutions of (39), then
(42)∥Y-Y*∥=∥μ∫-11‍k(x,t)[G~(Y)-G~(Y*)]dt∥≤μ∫-11‍|k(x,t)|∥G~(Y)-G~(Y*)∥dt≤2μLM∥Y-Y*∥.
So we get (1-α)∥Y-Y*∥≤0. Since 0<α<1, so ∥Y-Y*∥=0, therefore Y=Y*, and this completes the proof.Theorem 2.
If the series solutionY=∑m=0∞Ym obtained from (32) is convergent, then it converges to the exact solution of the problem (39).Proof.
We assume(43)Y=∑m=0∞‍Ym,V(Y)=∑m=0∞‍G~(Ym),
where limm→∞Ym=0. We can write
(44)∑m=1n‍[Ym-χmYm-1]=Y1+(Y2-Y1)+⋯+(Yn-Yn-1)=Yn.
Hence, from (44),
(45)∑m=1∞‍[Ym-χmYm-1]=0,
so using (45) and the definition of the linear operator L, we have
(46)∑m=1∞‍L[Ym-χmYm-1]=L[∑m=1∞‍Ym-χmYm-1]=0.
Therefore, from (16), we can obtain that
(47)∑m=1∞‍L[Ym-χmYm-1]=h∑m=1∞‍Rm(Ym-1)=0.
Since h≠0, we have
(48)∑m=1∞‍Rm(Ym-1)=0.
By applying (39) and (43),
(49)∑m=1∞‍Rm(Ym-1)=∑m=1∞‍[Ym-1-(1-χm-1)F~-μ∫-11‍k(x,t)G~(Ym-1)dt]=Y-F~-μ∫-11‍k(x,t)V(Y)dt.
Therefore, Y must be the exact solution of (39).


## 5. Numerical Examples

In this section we apply the technique described in Section3 to some illustrative examples of higher-order nonlinear Fredholm integro-differential equations.Example 1.
Consider the second-order Fredholm integro-differential equation(50)y′′(x)=6x+∫-11‍xt(y′(t))2(y(t))2dt
subject to y(-1)=y(1)=0 with the exact solution y(x)=x3-x. We employ SHAM and HAM to solve this example. From the h-curves (Figure 1), it is found that when -1.5≤h≤1.5 and -1≤h≤0, the SHAM solution and HAM solution converge to the exact solution, respectively. A numerical results of Example 1 against different order of SHAM approximate solutions is shown in Table 1.Table 1
The numerical results of Example1 against different order of SHAM approximate solutions with h=-0.01.
x
SHAM
Numerical
2nd order
4th order
1.00000
0
0
0
0.99965
−0.01162119
−0.01162119
−0.01162119
0.99861
−0.04513180
−0.04513187
−0.04513187
0.99687
−0.16001177
−0.16001177
−0.16001177
0.99443
−0.22774902
−0.22774902
−0.22774902
0.99130
−0.29155781
−0.29155781
−0.29155781
0.98748
−0.34334545
−0.34334545
−0.34334545
0.98297
−0.37606083
−0.37606087
−0.3760608
0.97778
−0.38445192
−0.38445192
−0.38445192
0.97191
−0.36563660
−0.36563661
−0.36563661Theh-curve y′′(0) and y′′′(0) for 10th-order (a) SHAM, (b) HAM.
(a)
(b)Example 2.
Consider the second order Fredholm integro-differential equation(51)xy′′(x)+x2y′(x)+2y(x)=(-π2x+2)sin(πx)+πx2cos(πx)+∫-11‍cos(πt)y4(t)dt
subject to y(-1)=y(1)=0 with the exact solution y(x)=sin(πx). We employ HAM and SHAM to solve this example. The numerical results of Example 2 against different order of SHAM approximate solutions with h=-0.01 is shown in Table 2. In Table 3, there is a comparison of the numerical result against the HAM and SHAM approximation solutions at different orders with h=-0.001. It is worth noting that the SHAM results become very highly accurate only with a few iterations, and fifth-order solutions are very close to the exact solution. Comparison of the numerical solution with the 4th-order SHAM solution for h=-0.01 is made in Figure 2. As it is shown in Figure 3, the rate of convergency in SHAM is faster than HAM. In Figure 4, it is found that when -2.5≤h≤0.5 and -1≤h≤1, the SHAM solution and HAM solution converge to the exact solution, respectively. In HAM we choose y0(x)=1-x2 as initial guess.Table 2
The numerical results of Example2 against different order of SHAM approximate solutions with h=-0.01.
x
2nd  order
3rd  order
4th  order
Numerical
1.00000
0
0
0
0
0.99965
0.00437807
0.00437807
0.00437807
0.00437807
0.99861
0.00109471
0.00109471
0.00109471
0.00109471
0.99687
0.00984768
0.00984768
0.00984768
0.00984768
0.99443
0.01749926
0.01749926
0.01749926
0.01749926
0.99130
0.02732631
0.02732631
0.02732631
0.02732631
0.98748
0.03931949
0.03931949
0.03931950
0.03931950
0.98297
0.05346606
0.05346607
0.05346607
0.06974900
0.97778
0.06974898
0.06974899
0.06974899
0.06974900
0.97191
0.0881459
0.08814599
0.08814599
0.08814600Table 3
Numerical result of Example2 against the HAM and the SHAM solutions with h=-0.001.
x
SHAM
HAM
Numerical
5th  order
6th  order
7th  order
3rd  order
4th  order
−0.97191
−0.0881460
−0.0881460
−0.0881460
−0.05395836
−0.05794467
−0.0881460
−0.97778
−0.06974902
−0.06974902
−0.06974902
−0.04280765
−0.04597139
−0.06974902
−0.98297
−0.05346609
−0.05346609
−0.05346609
−0.03289259
−0.03532441
−0.05346607
−0.98748
−0.03931951
−0.03931951
−0.03931951
−0.02424140
−0.02603420
−0.03931950
−0.99130
−0.02732631
−0.02732631
−0.02732631
−0.01687877
−0.01812740
−0.02732630
−0.99443
0.01749926
0.01749926
0.01749926
−0.00609972
−0.01162680
−0.01749926
−0.99687
−0.00984768
−0.00984768
−0.00984768
−0.00609972
−0.01162680
−0.00984768
−0.99861
−0.00437807
−0.00437807
−0.00437807
−0.00271424
−0.00655115
−0.00437807
−0.99965
−0.00109471
0.00109471
−0.00109471
−0.00067905
−0.00072931
−0.00109471
−1.00000
0
0
0
0
0
0Figure 2
Comparison of the numerical solution of Example2 with the 4th-order SHAM solution for h=-0.01.Comparison of the absolute error of third-order (a) SHAM, (b) HAM.
(a)
(b)Theh-curve y′′(-1) and y′′′(1) for 6th-order (a) SHAM, (b) HAM.
(a)
(b)Example 3.
Consider the first-order Fredholm integro-differential equation [27, 28]
(52)y′(x)=-12ex+2+32ex+∫01‍ex-ty3(t)dt
subject to the boundary condition y(0)=1. In order to apply the SHAM for solving the given problem, we should transform using an appropriate change of variables as
(53)x=ζ+12,ζ∈[-1,1].
Then, we use the following transformation:
(54)y(x)=Y(ζ)+e(x+1)/2.
We make the governing boundary condition homogeneous. Substituting (54) into the governing equation and boundary condition results in
(55)Y′(ζ)=14∫-11‍e(ζ-t)/2(Y3(t)+3et+1Y(t)+3e(t+1)/2Y2(t))dt
subject to the boundary condition Y(-1)=0. A comparison between absolute errors in solutions by SHAM, Lagrange interpolation, and Rationalized Haar functions is tabulated in Table 4. It is also worth noting that the SHAM results are very close to exact solutions only with two iterations.Table 4
A comparison of absolute errors between SHAM, LIM, and RHFS.
x
SHAM
LIM
RHFS
2nd order(h=-1)
6th  order
k
=
32
0.0
0
0
8.0
×
1
0
-
5
0.1
0
1.0
×
1
0
-
7
2.0
×
1
0
-
5
0.2
2.0
×
1
0
-
19
7.0
×
1
0
-
7
5.0
×
1
0
-
5
0.3
1.2
×
1
0
-
19
1.0
×
1
0
-
6
1.0
×
1
0
-
5
0.4
0
3.0
×
1
0
-
6
2.0
×
1
0
-
5
0.5
1.0
×
1
0
-
19
4.0
×
1
0
-
6
7.0
×
1
0
-
5


## 6. Conclusion

In this paper, we presented the application of spectral homotopy analysis method (SHAM) for solving nonlinear Fredholm integro-differential equations. A comparison was made between exact analytical solutions and numerical results obtained by the spectral homotopy analysis method, Rationalized Haar functions, and Lagrange interpolation solutions. In Example1, the numerical results indicate that the rate of convergency in SHAM is faster than HAM. In this example, we found that the forth-order SHAM approximation sufficiently gives a match with the numerical results up to eight decimal places. In contrast, HAM solutions have a good agreement with the numerical results in 20th order with six decimal places. As we can see in Table 4, the spectral homotopy analysis results are more accurate and efficient than Lagrange interpolation solutions and rationalized Haar functions solutions [27, 28]. As it is shown in Figures 1 and 4 the rang of admissible values of h is much wider in SHAM than HAM.In this paper, we employed the spectral homotopy analysis method to solve nonlinear Fredholm integro-difflerential equations; however, it remains to be generalized and verified for more complicated integral equations that we consider it as future works.



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