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Zbl 1092.65117
A different approach for solving the nonlinear Fredholm integral equations of the second kind.
(English)
[J] Appl. Math. Comput. 173, No. 2, 724-735 (2006). ISSN 0096-3003

The authors discuss a method for finding an approximate solution of the nonlinear Fredholm integral equation of the second kind $$x(t)=y(t)+\int_a^b k(t,s,x(s))\,ds,\quad\text{a.e. on }[a,b]$$ First, the problem under consideration is converted to an optimal control problem by introducing an artificial control function. Then a linear programming problem is formulated by using some concepts of measure theory. The solution of this linear programming problem provides the approximate solution of the Fredholm integral equation. It is shown that the nonlinearity of the kernel has no serious effect on the convergence of the solution. Some examples are given to illustrate the applicability of the method.
[Narahari Parhi (Bhubaneswar)]
MSC 2000:
*65R20 Integral equations (numerical methods)
45G10 Nonsingular nonlinear integral equations
49J22 Optimal control problems with integral equations (existence)
90C05 Linear programming

Keywords: nonlinear Fredholm integral equation; optimal control problem; linear programming; measure theory; numerical examples; convergence

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