The Simplified-Fredholm Integral Equation Solver and Its Use in Thermal Radiation

Abstract

The application of a new Fredholm integral equation solver to problems in thermal radiation is explored. The new method provides a simplified version of Fredholm’s own 1903 solution which, while being highly important from a theoretical point of view, had been considered too complex to provide a practical tool for solving integral equations. The method does not involve solving large arrays of simultaneous equations; rather, the simplified-Fredholm method provides an explicit solution. The solution consists of an infinite series with each term containing multiple integrals. It has been found, however, that the series can be safely truncated after about a dozen terms, and the multiple integrals can be resolved through repeated matrix multiplications, all of this leading to a practical methodology. Implicit in the method and highly useful in radiant analyses is the idea of the resolvent kernel, which permits generalized solutions to be obtained, independent of the forcing function. The method also adapts itself to a simple technique for establishing the possible error in any result. It is illustrated here on some enclosure problems that can be reduced to solving Fredholm’s equation in a single variable.