An Initial Value Method for Dual Integral Equations with Bessel Function Kernels.

A report prepared for the National Environmental Satellite Service, National Oceanic and Atmospheric Administration, as a step toward solving the temperature sounding problem for a cloud-free atmosphere. Traditional methods of calculation lead to integral equations of the first kind, which are extremely difficult to solve numerically. In this report, a system of two such equations, which arises in mixed boundary value problems of potential theory, is reduced to one Fredholm integral equation of the second kind, the kernel of which does not involve the order [n] of the Bessel function. This is further reduced to a Cauchy system, i.e., an initial value problem, suitable for solving on high-speed digital computers. To apply this result to the atmospheric temperature problem would involve finding an appropriate representation for the actual kernel in terms of an expansion of Bessel functions. The numerical method may involve a quadrature formula and the method of lines.