Invariant Imbedding and the Variational Treatment of Fredholm Integral Equations with Displacement Kernels
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A new and computationally efficient method of solving Fredholm integral equations with displacement kernels, such as those arising in radiative transfer and optimal filtering theory. Frequently, studies of these equations are based on the fact that their solutions minimize certain quadratic functionals, which opens the way to the employment of the Rayleigh-Ritz method. The aim of the present study is radically different: It is shown that the minimizer of the quadratic functional satisfies a Cauchy problem. The numerical integration of the initial-value problem is carried out by replacing the integrals by finite sums using gaussian quadrature formulas. This reduces the differential-integral equations to a system of ordinary differential equations.
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