Invariant Imbedding and the Solution of Fredholm Integral Equations with Displacement Kernels.

by J. L. Casti

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A new theory for solving integral equations of a type that arises in many practical problems of radiative transfer through the atmosphere and in nuclear reactors. By regarding the interval length as a basic imbedding parameter rather than as a constant, two separate initial value problems are obtained for the unknown function. If the two Cauchy systems have a unique solution, that solution satisfies the original equation; if not, there is no unique solution to the original Fredholm equation. An appended FORTRAN program turns the solution of Fredholm integral equations with displacement kernels into a routine computing task. Compared with 3 standard methods--successive approximations, linear algebraic equations, and averaging functional corrections--the new program was much faster in almost all cases tried. Moreover, it automatically provides solutions for all interval lengths less than the specified length. Finally, the report shows how to extend the concept to other types of integral equations in many important cases. 163 pp. Ref. (MW)

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