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.
1. Introduction
In this paper, a system of nonlinear Fredholm integral equations can be written as follows:where , , , , and , , are known functions, and 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 and , 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 Section 2, 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 :
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
Using the well-known Banach fixed-point theorem, one can easily prove that the solution of (2) exists and is unique in the interval . The uniqueness of a solution to (2) is provided in the following lemma.
Lemma 1. Assume that and , . If the following condition is satisfied,then (2) exists as an unique solution.
Proof. In a suitable Banach space, we define the operator:for . Let and be two solutions of (2); using (5), we haveSincewhere , , are some values between and . Then,and thereforeNote that , so is a contraction mapping; then, there exists a unique fixed point such that .
The existence and uniqueness of a solution to (1) is shown in the following theorem.
Theorem 1. A solution 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 interval in which we select equidistant nodes , where , . Letbe a numerical integration formula, where and , and , are the coefficients and nodes of quadrature and is the residual error for integration. Using (10) in (2) and neglecting the error , we have
In the following, using the collocation points , , we rewrite (11) as follows:where , , , and , and . Let , and , be the solutions of (12). Now, we substitute these solutions into (11), and there is
By (13), we can obtain the Nyström approximation solution of (2).
Remark 1. (I)For , the solution of (13) satisfies(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. Letfor . Now, we prove the following lemma.
Lemma 2. Let and be the solutions of (2) and (13), respectively. Assume that and , . If the following condition is satisfiedthen we have
Proof. Substituting (10) into (2) yieldsIt follows from (13) and (18) thatand therefore,where and () are as described in (15). Using the same argument as estimate (20), we deduce that there holdsBy (20) and (21), we haveLet the value , and we can rewrite (22) aswhich completes the proof.
From the above analysis and [16], we can also obtain the following results.
Corollary 1. Assume that and are as described in Lemma 2, , and , . If (16) is satisfied and using the composite trapezoidal integration formula in (13), then we have
Corollary 2. Assume that and , . Using the Simpson integration formula in (13), thusFrom (17), (24), and (25), when the value of , we obtain the following corollary.
Corollary 3. vanishes when tends to zero.
In the same manner as it has been carried out in the proofs of Lemma 2 and Corollary 3, we can derive the estimate of .
Theorem 2. Let , , be the solutions of (1). Assume that and , . If the following condition is satisfied , then we have(I)(II) vanishes when
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 and and apply the iterative formula; thus, (13) becomeswhere and .
In order to solve (26), we construct the iteration algorithm (Algorithm 1).
| ||||||||||||||
Lemma 3. Considering assumptions of Lemma 2, the produced sequences and from the iteration process (26) tend to of (13), for any arbitrary initial vector .
Proof. By (12) and (26), we obtainwhere . According to the condition of Lemma 2, we havewhere and , , are some real numbers between and . It follows from (27) and (28) thatCombining estimates (29) and (30), we find thatfor each . Since , then implies thatThus, 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 sequences and , , are the iterative and Nyström approximation solutions, respectively. Then, we have , where and .
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 function , where , . In addition, the convergence rate is defined as , and we select 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:where and and are suitably chosen such that the exact solution . 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 with 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).
Example 2. Consider the one-dimensional nonlinear system of Fredholm integral equations:where , the condition holds, and , , and are determined by the exact solution . The initial vectors () are considered for starting the algorithm. The results for different 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.
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 Examples 1 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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the Science and Technology Plan Project of Hunan Province (Grant No. 2016TP1020) and the Doctoral Scientific Research Foundation of Hengyang Normal University (Grant No. 18D12).