A Cauchy Problem for Fredholm Integral Equations with Kernels of the Form k1(/t-y/) + k2(t+y).

by H. H. Natsuyama, Robert E. Kalaba

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A method for converting Fredholm integral equations with "spectral" kernels into equivalent initial-value (Cauchy) problems that can be solved effectively by analog or digital computer. In this treatment the upper limit of integration, c, is viewed as an independent variable. An initial-value problem is derived for u(t, c), where u evaluated at a fixed point t is regarded as a function of c. The auxiliary functions R, e, and J, and the function u, satisfy differential-integral equations, subject to initial conditions. In the numerical method, the integrals in the differential equations are approximated by sums according to a quadrature formula. Then the system of differential-integral equations reduces to ordinary differential equations that can easily be solved by a computer. The formalism presented in this study opens the way to the treatment of many inverse or system identification problems. In particular, if the reflection function R(v, z, x) is measured experimentally, it is possible to estimate the specular reflector function r(v). 20 pp. Ref.

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