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Random search algorithm for solving the nonlinear Fredholm integral equations of the second kind.

Hong Z, Yan Z, Yan J - PLoS ONE (2014)

Bottom Line: The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation.And then we convert the problem to an optimal control problem by introducing an artificial control function.Finally, some examples are given to show the efficiency of the proposed approach.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Science College of Inner Mongolia University of Technology, Hohhot, P.R. China.

ABSTRACT
In this paper, a randomized numerical approach is used to obtain approximate solutions for a class of nonlinear Fredholm integral equations of the second kind. The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation. And then we convert the problem to an optimal control problem by introducing an artificial control function. Following that, in the next step, solution of the discretized form is approximated by a kind of Monte Carlo (MC) random search algorithm. Finally, some examples are given to show the efficiency of the proposed approach.

Show MeSH
Approximation process of MC random search with  in Example 2 for solving .
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pone-0103068-g003: Approximation process of MC random search with in Example 2 for solving .


Random search algorithm for solving the nonlinear Fredholm integral equations of the second kind.

Hong Z, Yan Z, Yan J - PLoS ONE (2014)

Approximation process of MC random search with  in Example 2 for solving .
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4114472&req=5

pone-0103068-g003: Approximation process of MC random search with in Example 2 for solving .
Bottom Line: The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation.And then we convert the problem to an optimal control problem by introducing an artificial control function.Finally, some examples are given to show the efficiency of the proposed approach.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Science College of Inner Mongolia University of Technology, Hohhot, P.R. China.

ABSTRACT
In this paper, a randomized numerical approach is used to obtain approximate solutions for a class of nonlinear Fredholm integral equations of the second kind. The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation. And then we convert the problem to an optimal control problem by introducing an artificial control function. Following that, in the next step, solution of the discretized form is approximated by a kind of Monte Carlo (MC) random search algorithm. Finally, some examples are given to show the efficiency of the proposed approach.

Show MeSH