# Iterative Method for a System of Nonlinear Fredholm Integral Equations

Authors: Keyan Wang; Qisheng Wang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, the iteration method is proposed to solve a class of system of Fredholm-type nonlinear integral equations. First, the existence and uniqueness of solution are theoretically proven by the fixed-point theorem. Second, the approximation solution method is given by using the appropriate integration rule. The error analysis for the approximated solution with the exact solution is discussed for infinity-norm, and the rates of convergence are obtained. Furthermore, an iteration algorithm is constructed, and the convergence of the proposed numerical method is rigorously derived. Finally, some numerical examples are given to illustrate the theoretical results.


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## Body

## 1. Introduction

In this paper, a system of nonlinear Fredholm integral equations can be written as follows:(1)upx=fpx+∫ab∑q=1nλpqkpqx,yϕpqy,uqydy,p=1,2,…,n,where x∈a,b, λpq∈ℝ, fp, kpq, and ϕpq, p,q=1,2,…,n, are known functions, and upx are unknown functions.Integral equations appear in numerous fields of physics, chemistry, biology, engineering, etc. They can be employed to model many phenomena and processes occurring in the real world so that these equations have received a great deal of attention and research. In the past, many approximate analytical methods have been applied to solve Fredholm integral problems such as spectral collocation method [1], Taylor polynomial method [2, 3], Nyström method [4], and iterative method [5–7]. Therefore, the development of an effective computational method for investigating (1) has practical significance.The Nyström method is usually regarded as an efficient discretization technique for solving the linear and nonlinear integral equations, which was introduced by Nyström in 1930 and has been developed for the weakly singular and singular integral equations in [8–10] and references therein. From then on, the method was further investigated and applied to other problems by many authors. For instance, Han and Wang [4] have obtained the asymptotic error expansions for numerical solutions of two-dimensional nonlinear Fredholm integral equations by using the Nyström method. In [11], the Nyström method was introduced for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces. Recently, the Nyström method also has been applied to the Urysohn integral equations [12], Volterra integro-differential equations [13], Mellin convolution equations [14], and multifrequency oscillatory systems [15]. Influenced by the work mentioned above, the goal of this work is to develop the Nyström method for problem (1) and provide rigorous error analysis for the method.For Fredholm integral equations, the iterated method [5–7] is used to accelerate the approximation. Comparing with [6], under some suitable conditions for λpq and kpq, we need not change and narrow the interval of integral equations. To the best of our knowledge, there is no iterated method convergence analysis for problem (1) in the literature that is combined with the Nyström method. In this paper, we develop the method and the corresponding convergence analysis which partly fill this gap. Our purpose is two fold. First of all, we propose the Nyström method to approximate (1). Secondly, in order to accelerate the approximation, we use the iterated algorithm to solve the Nyström method equations.The organization of this paper is as follows. In Section2, the existence and uniqueness of solution for (1) is proven, the Nyström method is stated, and the error estimates are derived. In Section 3, an iterated algorithm is defined and the convergence result is deduced. In Section 4, we present two numerical examples which show the efficiency of the proposed method. In Section 5, some conclusions are presented.


## 2. The Nyström Method and Error Analysis

For simplicity, let us consider the system of nonlinear integral equation (1) with n=2:(2)u1x=f1x+∫abλ11k11x,yϕ11y,u1y+λ12k12x,yϕ12y,u2ydy,u2x=f2x+∫abλ21k21x,yϕ21y,u1y+λ22k22x,yϕ22y,u2ydy.


### 2.1. The Existence and Uniqueness of Solution

In this section, we discuss the existence and uniqueness of solution for (1). For convenience, we rewrite (2) in the form(3)u1xu2x=f1xf2x+∫abλ11k11x,yϕ11y,u1y+λ12k12x,yϕ12y,u2yλ21k21x,yϕ21y,u1y+λ22k22x,yϕ22y,u2ydy.Using the well-known Banach fixed-point theorem, one can easily prove that the solution of (2) exists and is unique in the interval a,b. The uniqueness of a solution u1;u2 to (2) is provided in the following lemma.Lemma 1.
Assume thatkpq∈Ca,b×a,b and ϕpq∈C1a,b×uq,min,uq,max, p,q=1,2. If the following condition is satisfied,(4)γ=∫abλ11k11x,y∂ϕ11y,u1y∂u1λ12k12x,y∂ϕ12y,u2y∂u2λ21k21x,y∂ϕ21y,u1y∂u1λ22k22x,y∂ϕ22y,u2y∂u2dy∞<1,then (2) exists as an unique solution.Proof.
In a suitable Banach space, we define the operator:(5)Tu1u2=f1f2+∫abλ11k11ϕ11+λ12k12ϕ12λ21k21ϕ21+λ22k22ϕ22dy,for u1,u2∈Ca,b×Ca,b. Let u1,1;u2,1 and u1,2;u2,2 be two solutions of (2); using (5), we have(6)Tu1,1u2,1−Tu1,2u2,2=∫abλ11k11ϕ11y,u1,1−ϕ11y,u1,2+λ12k12ϕ12y,u2,1−ϕ12y,u2,2λ21k21ϕ21y,u1,1−ϕ21y,u1,2+λ22k22ϕ22y,u2,1−ϕ22y,u2,2dy.
Since(7)ϕpqy,uq,1−ϕpqy,uq,2=∂ϕpq∂uq|uq=ξpqyuq,1−uq,2,where ξpq, p,q=1,2, are some values between uq,1 and uq,2. Then,(8)Tu1,1u2,1−Tu1,2u2,2=∫abλ11k11∂ϕ11∂u1|u1=ξ11λ12k12∂ϕ12∂u2|u2=ξ12λ21k21∂ϕ21∂u1|u1=ξ21λ22k22∂ϕ22∂u2|u2=ξ22u1,1−u1,2u2,1−u2,2dy,and therefore(9)Tu1,1u2,1−Tu1,2u2,2∞≤∫abλ11k11∂ϕ11∂u1λ12k12∂ϕ12∂u2λ21k21∂ϕ21∂u1λ22k22∂ϕ22∂u2dy∞u1,1−u1,2u2,1−u2,2∞≤γu1,1u2,1−u1,2u2,2∞.
Note that0≤γ<1, so T is a contraction mapping; then, there exists a unique fixed point u1;u2 such that Tu1u2=u1u2.
The existence and uniqueness of a solutionUx=u1x,u2x,…,unxT to (1) is shown in the following theorem.Theorem 1.
A solutionUx to (1) exists and is unique.Proof.
Uniqueness follows directly from the proof of the previous lemma.


### 2.2. Description of the Method

Consider an intervala,b in which we select n+1 equidistant nodes a=x0<x1<⋯<xn=b, where h=xi−xi−1, i=1,2,…,n. Let(10)∫abψx,ydy=∑j=0nωjψx,xj+Rn,a≤x≤b,be a numerical integration formula, where ωj and xj=a+jh, h=b−a/n and j=0,1,…,n, are the coefficients and nodes of quadrature and Rn is the residual error for integration. Using (10) in (2) and neglecting the error Rn, we have(11)u1x=f1x+∑j=0nωjλ11k11x,xjϕ11xj,u1xj+λ12k12x,xjϕ12xj,u2xj,u2x=f2x+∑j=0nωjλ21k21x,xjϕ21xj,u1xj+λ22k22x,xjϕ22xj,u2xj.In the following, using the collocation pointsx=xi, i=0,1,…,n, we rewrite (11) as follows:(12)u1i=f1i+∑j=0nωjλ11k11ijϕ11ju1j+λ12k12ijϕ12ju2j,u2i=f2i+∑j=0nωjλ21k21ijϕ21ju1j+λ22k22ijϕ22ju2j,where upi=upxi, fpi=fpxi, kpqij=kpqxi,xj, and ϕpqjupj=ϕpqxj,uqxj, p,q=1,2 and i,j=0,1,…,n. Let upi=upi∗, p=1,2 and i=0,1,…,n, be the solutions of (12). Now, we substitute these solutions into (11), and there is(13)u¯1xi=f1x+∑j=0nwjλ11k11ijx,xjϕ11jxj,u1j∗+λ12k12ijx,xjϕ12jxj,u2j∗,u¯2xi=f1x+∑j=0nwjλ21k21ijx,xjϕ21jxj,u1j∗+λ22k22ijx,xjϕ22jxj,u2j∗.By (13), we can obtain the Nyström approximation solution u¯1x;u¯2x of (2).Remark 1.
(I)
Fori=0,1,…,n, the solution u¯1xi;u¯2xi of (13) satisfies(14)u¯1xi=u1i∗,u¯2xi=u2i∗.(II)
It is seen that when the composite trapezoidal and Simpson rules are used to solve (13), the given method is the special case of the Nyström-type method.


### 2.3. Error Analysis

In this section, the error estimation of (13) in the infinite norm sense and the results of convergence analysis are given. Let(15)Mpq=∑j=1nωjλpqkpq∂ϕpq∂uq∞,for p,q=1,2. Now, we prove the following lemma.Lemma 2.
Letu1x;u2x and u¯1x;u¯2x be the solutions of (2) and (13), respectively. Assume that kpq∈Ca,b×a,b and ϕpq∈C1a,b×uq,min,uq,max, p,q=1,2. If the following condition is satisfied(16)β=∑p,q=12Mpq<12,then we have(17)u1−u¯1∞+u2−u¯2∞≤R1n∞+R2n∞1−2β.Proof.
Substituting (10) into (2) yields(18)u1x=f1x+∑j=0nωjλ11k11x,xjϕ11xj,u1xj+λ12k12x,xjϕ12xj,u2xj+R1n,u2x=f2x+∑j=0nωjλ21k21x,xjϕ21xj,u1xj+λ22k22x,xjϕ22xj,u2xj+R2n.
It follows from (13) and (18) that(19)u1x−u¯1x=∑j=0nωjλ11k11x,xjϕ11xj,u1xj−ϕ11xj,u¯1xj+λ12k12x,xjϕ12xj,u2xj−ϕ12xj,u¯2xj+R1n=∑j=0nωjλ11k11x,xj∂ϕ11xj,u1∂u1|u1=ξ11xju1xj−u¯1xj+λ12k12x,xj∂ϕ12xj,u2∂u2|u2=ξ12xju2xj−u¯2xj+R1n,and therefore,(20)u1−u¯1∞≤M11max0≤j≤nu1xj−u¯1xj+M12max0≤j≤nu2xj−u¯2xj+R1n∞≤M11u1−u¯1∞+M12u2−u¯2∞+R1n∞≤M11+M12u1−u¯1∞+u2−u¯2∞+R1n∞≤βu1−u¯1∞+u2−u¯2∞+R1n∞,where β=∑p,q=12Mpq and Mpq (p,q=1,2) are as described in (15). Using the same argument as estimate (20), we deduce that there holds(21)u2−u¯2∞≤βu1−u¯1∞+u2−u¯2∞+R2n∞.
By (20) and (21), we have(22)u1−u¯1∞+u2−u¯2∞≤2βu1−u¯1∞+u2−u¯2∞+R1n∞+R2n∞.
Let the valueβ<1/2, and we can rewrite (22) as(23)u1−u¯1∞+u2−u¯2∞≤R1n∞+R2n∞1−2β,which completes the proof.
From the above analysis and [16], we can also obtain the following results.Corollary 1.
Assume thatu1x;u2x and u¯1x;u¯2x are as described in Lemma 2, kpq∈C2a,b×a,b, and ϕpq∈C2a,b×uq,min,uq,max, p,q=1,2. If (16) is satisfied and using the composite trapezoidal integration formula in (13), then we have(24)u1−u¯1∞+u2−u¯2∞=Oh2.Corollary 2.
Assume thatkpq∈C4a,b×a,b and ϕpq∈C4a,b×uq,min,uq,max, p,q=1,2. Using the Simpson integration formula in (13), thus(25)u1−u¯1∞+u2−u¯2∞=Oh4.
From (17), (24), and (25), when the value of n⟶∞, we obtain the following corollary.Corollary 3.
u1−u¯1∞+u2−u¯2∞ vanishes when h tends to zero.In the same manner as it has been carried out in the proofs of Lemma2 and Corollary 3, we can derive the estimate of ∑p=1nup−u¯p∞.Theorem 2.
Letupx, p=1,2,…,n, be the solutions of (1). Assume that kpq∈Ca,b×a,b and ϕpq∈C1a,b×uq,min,uq,max, p,q=1,2,…,n. If the following condition is satisfied β=∑p,q=1nMpq<1/n, then we have(I)
∑p=1nup−u¯p∞≤∑p=1nRpn∞/1−nβ(II)
∑p=1nup−u¯p∞ vanishes when h⟶0


## 2.1. The Existence and Uniqueness of Solution

In this section, we discuss the existence and uniqueness of solution for (1). For convenience, we rewrite (2) in the form(3)u1xu2x=f1xf2x+∫abλ11k11x,yϕ11y,u1y+λ12k12x,yϕ12y,u2yλ21k21x,yϕ21y,u1y+λ22k22x,yϕ22y,u2ydy.Using the well-known Banach fixed-point theorem, one can easily prove that the solution of (2) exists and is unique in the interval a,b. The uniqueness of a solution u1;u2 to (2) is provided in the following lemma.Lemma 1.
Assume thatkpq∈Ca,b×a,b and ϕpq∈C1a,b×uq,min,uq,max, p,q=1,2. If the following condition is satisfied,(4)γ=∫abλ11k11x,y∂ϕ11y,u1y∂u1λ12k12x,y∂ϕ12y,u2y∂u2λ21k21x,y∂ϕ21y,u1y∂u1λ22k22x,y∂ϕ22y,u2y∂u2dy∞<1,then (2) exists as an unique solution.Proof.
In a suitable Banach space, we define the operator:(5)Tu1u2=f1f2+∫abλ11k11ϕ11+λ12k12ϕ12λ21k21ϕ21+λ22k22ϕ22dy,for u1,u2∈Ca,b×Ca,b. Let u1,1;u2,1 and u1,2;u2,2 be two solutions of (2); using (5), we have(6)Tu1,1u2,1−Tu1,2u2,2=∫abλ11k11ϕ11y,u1,1−ϕ11y,u1,2+λ12k12ϕ12y,u2,1−ϕ12y,u2,2λ21k21ϕ21y,u1,1−ϕ21y,u1,2+λ22k22ϕ22y,u2,1−ϕ22y,u2,2dy.
Since(7)ϕpqy,uq,1−ϕpqy,uq,2=∂ϕpq∂uq|uq=ξpqyuq,1−uq,2,where ξpq, p,q=1,2, are some values between uq,1 and uq,2. Then,(8)Tu1,1u2,1−Tu1,2u2,2=∫abλ11k11∂ϕ11∂u1|u1=ξ11λ12k12∂ϕ12∂u2|u2=ξ12λ21k21∂ϕ21∂u1|u1=ξ21λ22k22∂ϕ22∂u2|u2=ξ22u1,1−u1,2u2,1−u2,2dy,and therefore(9)Tu1,1u2,1−Tu1,2u2,2∞≤∫abλ11k11∂ϕ11∂u1λ12k12∂ϕ12∂u2λ21k21∂ϕ21∂u1λ22k22∂ϕ22∂u2dy∞u1,1−u1,2u2,1−u2,2∞≤γu1,1u2,1−u1,2u2,2∞.
Note that0≤γ<1, so T is a contraction mapping; then, there exists a unique fixed point u1;u2 such that Tu1u2=u1u2.
The existence and uniqueness of a solutionUx=u1x,u2x,…,unxT to (1) is shown in the following theorem.Theorem 1.
A solutionUx to (1) exists and is unique.Proof.
Uniqueness follows directly from the proof of the previous lemma.


## 2.2. Description of the Method

Consider an intervala,b in which we select n+1 equidistant nodes a=x0<x1<⋯<xn=b, where h=xi−xi−1, i=1,2,…,n. Let(10)∫abψx,ydy=∑j=0nωjψx,xj+Rn,a≤x≤b,be a numerical integration formula, where ωj and xj=a+jh, h=b−a/n and j=0,1,…,n, are the coefficients and nodes of quadrature and Rn is the residual error for integration. Using (10) in (2) and neglecting the error Rn, we have(11)u1x=f1x+∑j=0nωjλ11k11x,xjϕ11xj,u1xj+λ12k12x,xjϕ12xj,u2xj,u2x=f2x+∑j=0nωjλ21k21x,xjϕ21xj,u1xj+λ22k22x,xjϕ22xj,u2xj.In the following, using the collocation pointsx=xi, i=0,1,…,n, we rewrite (11) as follows:(12)u1i=f1i+∑j=0nωjλ11k11ijϕ11ju1j+λ12k12ijϕ12ju2j,u2i=f2i+∑j=0nωjλ21k21ijϕ21ju1j+λ22k22ijϕ22ju2j,where upi=upxi, fpi=fpxi, kpqij=kpqxi,xj, and ϕpqjupj=ϕpqxj,uqxj, p,q=1,2 and i,j=0,1,…,n. Let upi=upi∗, p=1,2 and i=0,1,…,n, be the solutions of (12). Now, we substitute these solutions into (11), and there is(13)u¯1xi=f1x+∑j=0nwjλ11k11ijx,xjϕ11jxj,u1j∗+λ12k12ijx,xjϕ12jxj,u2j∗,u¯2xi=f1x+∑j=0nwjλ21k21ijx,xjϕ21jxj,u1j∗+λ22k22ijx,xjϕ22jxj,u2j∗.By (13), we can obtain the Nyström approximation solution u¯1x;u¯2x of (2).Remark 1.
(I)
Fori=0,1,…,n, the solution u¯1xi;u¯2xi of (13) satisfies(14)u¯1xi=u1i∗,u¯2xi=u2i∗.(II)
It is seen that when the composite trapezoidal and Simpson rules are used to solve (13), the given method is the special case of the Nyström-type method.


## 2.3. Error Analysis

In this section, the error estimation of (13) in the infinite norm sense and the results of convergence analysis are given. Let(15)Mpq=∑j=1nωjλpqkpq∂ϕpq∂uq∞,for p,q=1,2. Now, we prove the following lemma.Lemma 2.
Letu1x;u2x and u¯1x;u¯2x be the solutions of (2) and (13), respectively. Assume that kpq∈Ca,b×a,b and ϕpq∈C1a,b×uq,min,uq,max, p,q=1,2. If the following condition is satisfied(16)β=∑p,q=12Mpq<12,then we have(17)u1−u¯1∞+u2−u¯2∞≤R1n∞+R2n∞1−2β.Proof.
Substituting (10) into (2) yields(18)u1x=f1x+∑j=0nωjλ11k11x,xjϕ11xj,u1xj+λ12k12x,xjϕ12xj,u2xj+R1n,u2x=f2x+∑j=0nωjλ21k21x,xjϕ21xj,u1xj+λ22k22x,xjϕ22xj,u2xj+R2n.
It follows from (13) and (18) that(19)u1x−u¯1x=∑j=0nωjλ11k11x,xjϕ11xj,u1xj−ϕ11xj,u¯1xj+λ12k12x,xjϕ12xj,u2xj−ϕ12xj,u¯2xj+R1n=∑j=0nωjλ11k11x,xj∂ϕ11xj,u1∂u1|u1=ξ11xju1xj−u¯1xj+λ12k12x,xj∂ϕ12xj,u2∂u2|u2=ξ12xju2xj−u¯2xj+R1n,and therefore,(20)u1−u¯1∞≤M11max0≤j≤nu1xj−u¯1xj+M12max0≤j≤nu2xj−u¯2xj+R1n∞≤M11u1−u¯1∞+M12u2−u¯2∞+R1n∞≤M11+M12u1−u¯1∞+u2−u¯2∞+R1n∞≤βu1−u¯1∞+u2−u¯2∞+R1n∞,where β=∑p,q=12Mpq and Mpq (p,q=1,2) are as described in (15). Using the same argument as estimate (20), we deduce that there holds(21)u2−u¯2∞≤βu1−u¯1∞+u2−u¯2∞+R2n∞.
By (20) and (21), we have(22)u1−u¯1∞+u2−u¯2∞≤2βu1−u¯1∞+u2−u¯2∞+R1n∞+R2n∞.
Let the valueβ<1/2, and we can rewrite (22) as(23)u1−u¯1∞+u2−u¯2∞≤R1n∞+R2n∞1−2β,which completes the proof.
From the above analysis and [16], we can also obtain the following results.Corollary 1.
Assume thatu1x;u2x and u¯1x;u¯2x are as described in Lemma 2, kpq∈C2a,b×a,b, and ϕpq∈C2a,b×uq,min,uq,max, p,q=1,2. If (16) is satisfied and using the composite trapezoidal integration formula in (13), then we have(24)u1−u¯1∞+u2−u¯2∞=Oh2.Corollary 2.
Assume thatkpq∈C4a,b×a,b and ϕpq∈C4a,b×uq,min,uq,max, p,q=1,2. Using the Simpson integration formula in (13), thus(25)u1−u¯1∞+u2−u¯2∞=Oh4.
From (17), (24), and (25), when the value of n⟶∞, we obtain the following corollary.Corollary 3.
u1−u¯1∞+u2−u¯2∞ vanishes when h tends to zero.In the same manner as it has been carried out in the proofs of Lemma2 and Corollary 3, we can derive the estimate of ∑p=1nup−u¯p∞.Theorem 2.
Letupx, p=1,2,…,n, be the solutions of (1). Assume that kpq∈Ca,b×a,b and ϕpq∈C1a,b×uq,min,uq,max, p,q=1,2,…,n. If the following condition is satisfied β=∑p,q=1nMpq<1/n, then we have(I)
∑p=1nup−u¯p∞≤∑p=1nRpn∞/1−nβ(II)
∑p=1nup−u¯p∞ vanishes when h⟶0


## 3. The Iterative Algorithm and Convergence Analysis

In this section, we will consider a numerical method to solve the nonlinear system (13) by using Newton’s method. First, we define two vector sequences u¯1k and u¯2k and apply the iterative formula; thus, (13) becomes(26)u¯1k+1xi=f1xi+∑j=0nωjλ11k11ijϕ11xj,u¯1kxj+λ12k12ijϕ12xj,u¯2kxj,u¯2k+1xi=f2xi+∑j=0nωjλ21k21ijϕ21xj,u¯1kxj+λ22k22ijϕ22xj,u¯2kxj,where i=0,1,…,n and k=0,1,….In order to solve (26), we construct the iteration algorithm (Algorithm 1).Algorithm 1: Iteration algorithm.
(i)
Step 1. Choose n and calculate xi (i=0,1,…,n) along with h. Moreover, given the value of ε and the initial vectors u¯10=u¯20=0, set k=0.(ii)
Step 2. Solve the nonlinear system F1i=u¯1k+1xi−f1xi−∑j=0nωjλ11k11ijϕ11xj,u¯1kxj+λ12k12ijϕ12xj,u¯2kxj,F2i=u¯2k+1xi−f2xi−∑j=0nωjλ21k21ijϕ21xj,u¯1kxj+λ22k22ijϕ22xj,u¯2kxj, where i=0,1,…,n and k=0,1,….(iii)
Step 3. In Algorithm 1, using the composite trapezoidal rule ∑j=0nλpqkpqijωjϕpqxj,u¯qkxj=h/2∑j=0n−1λpqkpqijϕpqxj,u¯qkxj+kpqij+1ϕpqxj+1,u¯qkxj+1 or using the composite Simpson rule ∑j=0nλpqkpqijωjϕpqxj,u¯qkxj=h/6∑j=0n−1λpqkpqijϕpqxj,u¯qkxj+kpqij+1ϕpqxj+1,u¯qkxj+1+4kpqij+1/2ϕpqxj+1/2,u¯qkxj+1/2 where kpqij+1=kpqxi,xj+1 and kpqij+1/2=kpqxi,xj+1/2, p,q=1,2.(iv)
Step 4. Calculate the Jacobian matrix F′=∂F1i/∂xi;∂F2i/∂xi.(v)
Step 5. Calculate [u¯1ik+1; u¯2ik+1] by steps 2–4.(vi)
Step 6. Compute u¯1ik+1−u¯1i∞,u¯2ik+1−u¯2i∞. If u¯1ik+1−u¯1i∞<ε,u¯2ik+1−u¯2i∞<ε, stop; otherwise, set k=k+1 and go to step 5.Next, we can derive the following convergence result for the iteration method.Lemma 3.
Considering assumptions of Lemma2, the produced sequences u¯1k and u¯2k from the iteration process (26) tend to u¯1i;u¯2i of (13), for any arbitrary initial vector u¯10;u¯20.Proof.
By (12) and (26), we obtain(27)u¯1ik+1−u¯1i=∑j=0nωjλ11k11ijϕ11xj,u¯1jk−ϕ11xj,u¯1j+λ12k12ijϕ12xj,u¯2jk−ϕ12xj,u¯2j,u¯2ik+1−u¯2i=∑j=0nωjλ21k21ijϕ21xj,u¯1jk−ϕ21xj,u¯1j+λ22k22ijϕ22xj,u¯2jk−ϕ22xj,u¯2j,where i=0,1,…,n. According to the condition of Lemma 2, we have(28)∑j=0nωjλpqkpqijϕpqxj,u¯qjk−ϕpqxj,u¯qj=∑j=0nωjλpqkpqij∂ϕpq∂uqxj,ξpqjku¯qjk−u¯qj≤Mpqu¯qjk−u¯qj,where p,q=1,2 and ξpqjk, j=0,1,…,n, are some real numbers between u¯qjk and u¯qj. It follows from (27) and (28) that(29)u¯1ik+1−u¯1i∞≤βu¯1ik−u¯1i∞+u¯2ik−u¯2i∞≤2β2u¯1ik−1−u¯1i∞+u¯2ik−1−u¯2i∞≤⋯≤2kβk+1u¯1i0−u¯1i∞+u¯1i0−u¯2i∞,(30)u¯2ik+1−u¯2i∞≤2kβk+1u¯1i0−u¯1i∞+u¯2i0−u¯2i∞.
Combining estimates (29) and (30), we find that(31)u¯1ik+1−u¯1i∞+u¯2ik+1−u¯2i∞≤2βk+1u¯1i0−u¯1i∞+u¯2i0−u¯2i∞,for each k=0,1,…. Since β<1/2, then k⟶∞ implies that(32)limk⟶∞u¯1ik+1−u¯1i∞+u¯2ik+1−u¯2i∞=0.
Thus, the proof is completed.
From the above analysis, we can also extend the result of the iterative method to (1).Theorem 3.
Assume that the sequencesu¯pk and u¯p, p=1,2,…,n, are the iterative and Nyström approximation solutions, respectively. Then, we have limk⟶∞∑p=1nu¯pk−u¯p∞=0, where u¯pk=u¯p0k,u¯p1k,…,u¯pnk and u¯p=u¯p0,u¯p1,…,u¯pn.


## 4. Numerical Examples

In this section, we present two numerical examples to verify the above algorithm and the theoretical estimates obtained in the previous sections. In the numerical examples reported below, we define a discrete error functionE=up−u¯p∞=maxa≤xi≤bupxi−u¯pkxi, where xi=a+b−a/ni, i=0,1,…,n. In addition, the convergence rate is defined as log2E2h/Eh, and we select ε=1e−10 as the tolerance error of the iteration in real calculations. All the computations are carried out in Matlab 2012b.Example 1.
Consider the following problem:(33)u1x=f1x+1240∫011300+2x1+cosy54u14ydy+150∫011excosysinu2ydy,u2x=f2x+1100∫01xey40+x2y+u1y44dy+120∫01110+xsinycosu2ydy,where 0≤x≤1 and f1x and f2x are suitably chosen such that the exact solution u1x;u2x=sinx;2x. Conditions (4) and (16) hold; the solution of (33) will be approximated by the Nyström method, and the resulting nonlinear algebraic systems are solved by the Newton iteration method.
The numerical results of the iteration method withn=1,2,4,8,16 are demonstrated in Tables 1 and 2. From Tables 1 and 2, we see that the relative errors decrease as the mesh becomes fine, and the numerical results calculated by the composite Simpson rule have more accuracy. Moreover, we also observe from Tables 1 and 2 that the convergence rates are two-order and four-order for the composite trapezoidal and Simpson rules, respectively. These values also indicate agreement with predicted convergence rates. Table 3 represents the error estimates using the methods of [17, 18]. From Tables 1–3, compared with the methods of [17, 18], our method is very well in numerical solution of (1).Table 1
Numerical results for Example1 by using the composite trapezoidal rule.
nu1−u¯1∞Rateu2−u¯2∞Rate16.2974e−31.4862e−221.3921e−32.17763.2583e−32.189443.3884e−42.03857.9146e−42.041588.3829e−52.01511.9557e−42.0168162.0819e−52.00954.8422e−52.0140Table 2
Numerical results for Example1 by using the composite Simpson rule.
nu1−u¯1∞Rateu2−u¯2∞Rate12.5422e−46.4248e−421.2619e−54.33243.1949e−54.329847.0189e−74.16821.7816e−64.164583.8452e−84.19019.7781e−84.1875162.2774e−94.07765.8021e−94.0749Table 3
Error estimates for Example1 by using the methods of [17, 18].
nMethod of [17]Method of [18]u1−u¯1∞u2−u¯2∞u1−u¯1∞u2−u¯2∞13.5236e−2 9.7588e−21.3358e−2 4.3043e−229.4328e−3 2.7216e−27.8067e−3 1.1256e−242.4223e−3 7.0378e−34.5024e−4 6.6112e−486.2637e−4 1.9103e−33.6592e−5 4.7854e−5161.5973e−4 4.8047e−43.4139e−5 5.0901e−5Example 2.
Consider the one-dimensional nonlinear system of Fredholm integral equations:(34)u1x=f1x+1160∫01ex−1−sinycosu1ydy+1400∫01yey280+x2eu2ydy+150∫01sinxsiny1−logu3ydy,u2x=f2x+1160∫01ex−x22−cosyeu1ydy+1240∫01ycosxsiny210+2xcosu2ydy+150∫01cosxcosylogu3ydy=f3x+1160∫01ey5−11+xsinu1ydy+1100∫01x4+y641−u2y55dy+1200∫01sinxcosyy6+logu3ydy,where 0≤x≤1, the condition β<1/3 holds, and f1x, f2x, and f3x are determined by the exact solution u1x;u2x;u3x=x;x2;ex. The initial vectors u¯p0=0 (p=1,2,3) are considered for starting the algorithm. The results for different n are given in Tables 4 and 5. From the computed error results and convergence rates in Tables 4 and 5, we can derive the same performance as Example 1.Table 4
Numerical results for Example2 by using the composite trapezoidal rule.
nu1−u¯1∞Rateu2−u¯2∞Rateu3−u¯3∞Rate11.7357e−21.8882e−24.2653e−224.5935e−31.91785.1775e−31.86671.0519e−22.019541.1051e−32.05541.2734e−32.02352.6167e−32.007182.7036e−42.03123.1476e−42.01646.5183e−42.0052166.1391e−52.13887.5873e−52.05261.6041e−42.0227Table 5
Numerical results for Example2 by using the composite Simpson rule.
nu1−u¯1∞Rateu2−u¯2∞Rateu3−u¯3∞Rate11.4973e−44.9234e−48.3659e−429.5114e−63.97653.1388e−53.97725.8626e−53.834846.0999e−73.96282.0234e−63.95543.9241e−63.901183.9161e−83.96131.2866e−73.97512.5498e−73.9439162.4815e−93.98018.1007e−93.98941.6457e−83.9536


## 5. Conclusions

The present paper discussed an iteration method for the system of Fredholm integral equations. The method was based on the composite collocation method. Some theorems have been proved for convergence analysis. The efficiency and accuracy of the proposed method are shown in two numerical examples. From Examples1 and 2, our numerical results are in agreement with the obtained theoretical results. Moreover, our method is not difficult to extend to deal with integro-differential equations.



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