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A novel approach to solve nonlinear Fredholm integral equations of the second kind
SpringerPlusvolume 5, Article number: 154 (2016)
Abstract
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 past several years, the nonlinear integral equations have been solved numerically by several workers, utilizing various approximate methods (see Atkinson and Potra 1988; Atkinson and Flores 1993; Babolian and Shahsavaran 2009; Lepik and Tamme 2007; SaberiNadjafi and Heidari 2010; Aziz and Islam 2013; Maleknejad and Nedaiasl 2011).
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:
Theorem 1
If s(x) is continuous on the closed interval [a, b], there is a number \(c\in [a,b]\) so that
Let \(h=(ba)/n, n\in N\) be the mesh with \(x_k=a+kh, k=0,\ldots , n\). By (4), we can construct a sequence of quadrature formula as
where \(c_k,(k=0,\ldots ,n1)\) are constants.
We apply (5) to the integral operator K and get
where the unknown function \(c_k(x),(k=0,\ldots ,n1)\) 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 \(u(x), x\in [a,b]\) 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 \([a,b]\times [a,b]\) and g(x) is continuous on [a, b], they satisfy the following Lipschitz conditions
with the constants \(L_{1,2,3}>0\) , the sequence \((k_ng(u))(x)\) of quadrature formula is convergent. That is, we have
Proof
where \(0<c_k<1\) and \(0<c_k(x)<1\). We have \(\Vert (k_ng(u))(x)(kg(u))(x)\Vert _{\infty }\rightarrow 0, n\rightarrow \infty\), and the proof of the theorem is completed. \(\square\)
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 \(z=(z_0,\ldots ,z_{n1})^T=u_n\), and
with \(K_{i,j}=K(x_i+hc_i,x_j+hc_j)\). Then, (8) can be rewritten as
The Jaccobi matrix of \(\Psi (z)\) is
So New iteration is constructed
Lemma 1
[Ostrowski see Ortege and Kheinboldt (1970)] Suppose there is a fixed point \(z^{*}\in int(D)\) of the mapping: \(\omega : D\subset R^n\rightarrow R^n\) and the Fderivation of \(\omega\) at point \(z^{*}\) exists. If the spectral radius of \(\omega ^{'}(z^{*})\) satisfies
Then, there is an open ball \(S=S(z^{*},\delta _{0})\subset D\) that for \(z^{0}\in S\) , the iterative sequence (14) is stable and convergent to \(z^{*}\).
Lemma 2
[see Ortege and Kheinboldt (1970)] Suppose \(A, C\in L(R^n), \Vert A^{1}\Vert <\beta , \Vert AC\Vert <\alpha , \alpha \beta <1,\) then C is invertible and \(\Vert C^{1}\Vert <\beta /(1\alpha \beta )\).
Theorem 3
Suppose \(\Psi : D\subset R^n\rightarrow R^n\) is Fderivative, and \(z^{*}\) satisfies equation \(\Psi (z)=0\). \(A: S\subset D\rightarrow L(R^n)\) is continuous and invertible at \(z^{*}\), where S is the neighborhood of \(z^{*}\) . Then, there is a close ball \(\bar{S}=\bar{S}(z^{*},\delta )\subset S\) that \(\Omega\) is Fderivative at \(z^{*}\):
Proof
Let \(\beta =\Vert (A(z^{*}))^{1}\Vert >0\). Since \(A(z^{*})\) is invertible, and A(z) is continuous at \(z^{*}\), for \(0<\varepsilon <(2\beta )^{1}, \exists \delta >0,\) when \(z\in \bar{S}(z^{*},\delta ),\) there is \(\Vert A(z)A(z^{*})\Vert <\varepsilon.\) According to Lemma 2, \((A(z))^{1}\) exists and \(\Vert (A(z))^{1}\Vert \le \beta /(1\varepsilon \beta )\) for any \(z\in \bar{S}.\) So we construct the function
Since \(\Psi (z)\) is derivative at \(z^{*}, \exists \delta >0.\) When \(z\in \bar{S}(z^{*},\delta ),\) we obtain an inequality by the definition of the Fderivation:
Consider the derivation of \(\omega (z)\)
where \(c=2\beta (\beta \Vert \Psi ^{'}(z^{*})\Vert +1)\). According to the definition of the Fderivation, we obtain the the Fderivation of \(\omega\) at \(z^{*}\)
Using the definition of the matrix A in (13), we have \(\rho (\omega ^{'}(z^{*}))=0<1\). 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 1 Take n and Let \(x_k=a+hk,(k=0,\ldots ,n1)\) with \(h=(ba)/n.\)

Step 2 Let \(c_k=c, (k=0,\ldots ,n1)\) and randomly choose a series of \(\sigma _i\) so that \(0<c=\sigma _i<1, (i=0,1,\ldots ,m)\).

Step 3 Solve the nonlinear system by Newton iteration
$$u_n^jh\sum _{k=0}^{n1}K(x_j+h\sigma _i,x_k+h\sigma _i)g\left( u_n^k\right) =f(x_j+h\sigma _i).$$ 
Step 4 Get the approximate solutions
$$u_n(x,\sigma _i)=f(x)+h\sum _{k=0}^{n1}K(x,x_k+h\sigma _i)g\left( u_n^k\right) .$$ 
Step 5 Let the mean value of \(u_n(x,\sigma _i)\) be the last approximate solution
$$u_n(x)=\sum _{i=0}^m\frac{u_n(x,\sigma _i)}{m+1}.$$
\(\square\)
Numerical results
In this section, the theoretical results of the previous section are used for some numerical examples.
Example 1
The following nonlinear integral equation is considered
with \(0<x<1\) and the exact solution \(u(x)=2x^2.\)
For the sake of simplicity, we choose \(\sigma _i=i/10, (i=0,1,\ldots ,10)\). 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 2
The following nonlinear integral equation is considered
with \(0<x<1\) and the exact solution \(u(x)=\sin (\pi x)+\frac{1}{3}(20\sqrt{391})\cos (\pi x)\).
We take n = 25 along with h = 1/25 and get \(x_k=k/25, (k=0,1,\ldots ,24)\). For the sake of simplicity, \(\sigma _i\) is given as \(i/10, (i=0,1,\ldots ,10)\). Table 2 shows the four kinds results by using Newton–Kantorovichquadrature method in SaberiNadjafi and Heidari (2010), the SESinc method in Maleknejad and Nedaiasl (2011), the DESinc method in Maleknejad and Nedaiasl (2011), and the present method, respectively. Figure 3 shows the comparison of approximate and exact solutions with n = 25 and Fig. 4 presents the error curve on [0, 1] with n = 25.
Example 3
The following nonlinear integral equation is considered
with \(0<x<1\) and the exact solution \(u(x)=x.\)
In Table 3, we choose \(\sigma _i=i/10, (i=0,1,\ldots ,10)\). In Table 4, we choose \(\sigma _i=0,1/2,1,\) and \(r_i, (i=0,1,\ldots ,10)\), respectively. \(r_i, (i=0,1,\ldots ,10)\) are randomly selected. Table 3 shows the numerical results by using the present method, and Table 4 shows the choice of \(\sigma _i\) has a great influence on the accuracy of numerical solutions.
Conclusions
Based on the idea of the integral mean value theorem and Newton iteration, a novel algorithm is constructed to solve the nonlinear Fredholm integral equations of the second kind. The convergence and the error of numerical results have been analyzed. By the obtained numerical results, we know the algorithm is feasible and valuable.
References
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Authors’ contributions
HL and JH were involved in the study design and manuscript preparation. Both authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to the National Natural Science Foundation of China for financial funding under Grant number (11371079).
Competing interests
The authors declare that they have no competing interests.
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Keywords
 A novel approach
 Nonlinear Fredholm integral equations
 Integral mean value theorem
 Newton iteration