A novel approach to solve nonlinear Fredholm integral equations of the second kind
 Hu Li^{1}Email author and
 Jin Huang^{1}
Received: 7 December 2015
Accepted: 12 February 2016
Published: 24 February 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.
Keywords
A novel approach Nonlinear Fredholm integral equations Integral mean value theorem Newton iterationBackground
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).
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
Convergence and error analysis
We give the convergence analysis of (8) and have a theorem as follows:
Theorem 2
Proof
From Theorem 2, we can get a corollary as follows:
Description of Newton iteration and a novel algorithm
Lemma 1
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
Proof
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 \(z^{*}\).

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$$u_n(x)=\sum _{i=0}^m\frac{u_n(x,\sigma _i)}{m+1}.$$
Numerical results
In this section, the theoretical results of the previous section are used for some numerical examples.
Example 1
Absolute errors for Example 1
x  n = 8  n = 16  n = 32  n = 64  n = 128 

0.2  1.63e−3  4.09e−4  9.94e−5  2.48e−6  6.21e−7 
0.4  3.27e−3  8.18e−4  1.99e−4  4.97e−6  1.24e−6 
0.6  4.90e−3  1.23e−3  2.98e−4  7.45e−6  1.86e−6 
0.8  6.54e−3  1.64e−3  3.97e−4  9.94e−6  2.48e−6 
Results in Aziz and Islam (2013)  1.0e−3  2.6e−4  6.6e−5  1.7e−5  4.2e−6 
Results in Lepik and Tamme (2007)  2.7e−3  1.1e−3  3.7e−4  1.1e−4  3.1e−5 
Example 2
Absolute errors for Example 2
Example 3
Absolute errors for Example 3
x  n = 4  n = 8  n = 16  n = 32  n = 64 

0.2  1.21e−3  3.02e−4  7.53e−5  1.88e−5  4.70e−6 
0.4  2.43e−3  6.03e−4  1.51e−4  3.76e−5  9.41e−6 
0.6  3.64e−3  9.05e−4  2.26e−4  5.65e−5  1.41e−5 
0.8  4.86e−3  1.21e−3  3.01e−4  7.52e−5  1.88e−5 
Absolute errors for Example 3
x  \(n=8,\sigma _i=0\)  \(n=8,\sigma _i=1/2\)  \(n=8,\sigma _i=1\)  \(n=8,\sigma _i=r_i\) 

0.2  1.14e−2  1.12e−3  1.02e−2  5.88e−4 
0.4  2.27e−2  2.24e−3  2.04e−2  1.18e−3 
0.6  3.41e−2  3.36e−3  3.06e−2  1.76e−3 
0.8  4.54e−2  4.48e−3  4.07e−2  2.35e−3 
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.
Declarations
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.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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