An analog computer method to solve Fredholm integral equations of the first kind

Many problems of contemporary interest are characterized by Fredholm type integral equations of the first kind. These equations are inherently ill-posed and difficult to solve. It is customary to convert the equation into a set of m algebraic equations Af = g in n unknowns with m not necessarily equal to n. Then one can solve these m equations in a least square sense. Among the class of vectors f that minimize the Euclidean norm of the error, there exists a unique vector A⁺g which is of least norm where A⁺ is the generalized inverse of A. One method of finding the generalized inverse of A is to reformulate the problem into an equivalent system of first order ordinary differential equations with specified initial conditions. The steady state solution of this system is A⁺g, the required value of f. This procedure was implemented on an analog computer and the results presented.