Application of Interval Integration to the Solution of Integral Equations.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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A method of numerical solution is presented for a class of integral equations which includes linear and nonlinear equations encountered in applications. A brief sketch is given of methods from classical analysis inversion of power series and functional analysis functional and monotone iteration, together with some of their shortcomings difficulty of implementation, error estimation, and special conditions on operators and initial approximations. Interval functions, which may be considered to be sets consisting of all functions bounded above and below by given end-point functions, are defined, and the recently developed theory of interval integration is used to construct interval extensions of real integral operators. These interval operators are used to define an interval iteration process which converges if the initial interval contains a solution of the integral equation. Furthermore, the endpoint functions of the iterated interval functions provide upper and lower bounds for the solution at each stage of the iteration, and the interval iteration operator can be constructed so that the results of each transformation can be represented exactly. A numerical example is given of an interval iteration which gives a numerical solution of a nonlinear integral equation outside the limits of convergence of classical and function methods. Some problems connected with the use of interval iteration are also discussed. Author
- Theoretical Mathematics