An Initial-value Method for Fredholm Integral Equations with Displacement Kernels : Reflection Functions.

by H. H. Natsuyama, Robert E. Kalaba, Alan Schumitzky

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A method is given for reducing Fredholm integral equations of the second kind with displacement kernels to initial-value problems, using an approach that involves introducing certain auxiliary functions, one of which is analogous to the reflection function of transport theory. An extension of the method of invariant imbedding is used to develop a complete system of differential equations suitable for numerical solution. The theory is then applied to radiative transfer. Good results are obtained with a Gaussian quadrature formula of order 7 and an integration step length of 0.01, using a fourth-order predictor-corrector formula for the integration of ordinary differential equations. Such results agree to about five significant figures with results obtained using a higher-order formula and a shorter step length. The equations are numerically stable, which means that solutions may be obtained to any degree of accuracy by the proper choices of the quadrature formula order and the step length, and for media of arbitrarily great thicknesses. 21 pp. Ref. (See also RM-5186, RM-5258, RM-5307.)

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