A novel approach to solve nonlinear Fredholm integral equations of the second kind

In this paper, we present a novel approach to solve nonlinear Fredholm integral equations of the second kind. This algorithm is constructed by the integral mean value theorem and Newton iteration. Convergence and error analysis of the numerical solutions are given. Moreover, Numerical examples show the algorithm is very effective and simple.


Background
Integral equations have several applications in Physics and Engineering. However, these occur nonlinearly. In particular, nonlinear integral equations arise in fluid mechanics, biological models, solid state physics, kinetics in chemistry etc. In most cases, it is difficult to solve them, especially analytically.
In the present work, we have developed a novel approach to solve nonlinear Fredholm integral equations of the second. This algorithm is obtained by integral mean value theorem and Newton iteration. We consider the nonlinear Fredholm integral equations, given as follows: where f(x) is a known continuous function defined on [a, b] and g(u(y)) is a nonlinear function defined on [a, b]. The nonlinear integral operator k is defined as follows: and k is compact on C [a, b] into C [a, b] with continuous kernel K(x, y). Then (1) is equivalent to the operator form as follows: This paper is organized as follows: In section "A novel numerical method", based on the idea of the integral mean value theorem, a novel numerical method is given. In section "Convergence and error analysis", we address the convergence and error analysis of the numerical solutions. In section "Description of Newton iteration and a novel algorithm", Newton iteration is introduced and a novel algorithm is given. In section "Numerical results", numerical examples are carried out.

A novel numerical method
In order to obtain a novel numerical method, we firstly introduce the integral mean value theorem, is given as follows: Let h = (b − a)/n, n ∈ N be the mesh with x k = a + kh, k = 0, . . . , n. By (4), we can construct a sequence of quadrature formula as where c k , (k = 0, . . . , n − 1) are constants.
We apply (5) to the integral operator K and get where the unknown function c k (x), (k = 0, . . . , n − 1) are dependent on the variable x and 0 < c k (x) < 1. Especially, Let c k (x) = c k be constants. We can obtain Nyström approximation with a high accuracy, is given as follows: Thus we obtain the numerical approximate form of (3) Obviously, Eq. (8) is a nonlinear equations system. Once u n is get, we obtain u(x), x ∈ [a, b] by (3). (

Convergence and error analysis
We give the convergence analysis of (8) and have a theorem as follows:

Theorem 2 If the function K(x,y) is continuous on [a, b] × [a, b] and g(x) is continuous on [a, b], they satisfy the following Lipschitz conditions
with the constants L 1,2,3 > 0, the sequence (k n g(u))(x) of quadrature formula is convergent. That is, we have Proof By (6) and (7), we easily get where 0 < c k < 1 and 0 < c k (x) < 1. We have �(k n g(u))(x) − (kg(u))(x)� ∞ → 0, n → ∞ , and the proof of the theorem is completed.
From Theorem 2, we can get a corollary as follows: Corollary 1 Under the assumption of Theorem 2, the error of the approximate solutions in (8) can be estimated, is given as follows:

Description of Newton iteration and a novel algorithm
We shall give Newton iteration to solve nonlinear equations. For convenience, we denote where z = (z 0 , . . . , z n−1 ) T = u n , and (8) can be rewritten as The Jaccobi matrix of �(z) is

So New iteration is constructed
Lemma 1 [Ostrowski see Ortege and Kheinboldt (1970)] Suppose there is a fixed point z * ∈ int(D) of the mapping: ω : D ⊂ R n → R n and the F-derivation of ω at point z * exists.
If the spectral radius of ω ′ (z * ) satisfies Then, there is an open ball S = S(z * , δ 0 ) ⊂ D that for z 0 ∈ S, the iterative sequence (14) is stable and convergent to z * .

Consider the derivation of ω(z)
where c = 2β(β�� ′ (z * )� + 1). According to the definition of the F-derivation, we obtain the the F-derivation of ω at z * Using the definition of the matrix A in (13), we have ρ(ω ′ (z * )) = 0 < 1. According to Lemma 1, the iterative sequence is stable and convergent to z * .
In what follows, in order to give the numerical solutions with more stability, we provide a novel algorithm (see Zhong 2013).

Step 3 Solve the nonlinear system by Newton iteration
Step 4 Get the approximate solutions Step 5 Let the mean value of u n (x, σ i ) be the last approximate solution

Numerical results
In this section, the theoretical results of the previous section are used for some numerical examples. Li and Huang SpringerPlus (2016) 5:154 Example 1 The following nonlinear integral equation is considered with 0 < x < 1 and the exact solution u(x) = 2 − x 2 . For the sake of simplicity, we choose σ i = i/10, (i = 0, 1, . . . , 10). Table 1 shows the three kinds results by using the methods in Lepik and Tamme (2007), Aziz and Islam (2013), and the present method, respectively. Figure 1 shows the comparison of approximate and exact solutions with n = 128 and Fig. 2 presents the error curve on [0, 1] with n = 128.
Example 3 The following nonlinear integral equation is considered with 0 < x < 1 and the exact solution u(x) = x.