A state-variable approach to the solution of Fredholm integral equations

A method of solving Fredholm integral equations by state-variable techniques is presented. A principal feature of this method is that it leads to efficient computer algorithms for calculating numerical solutions. The assumptions made are 1) the kernel of the integral equation is the covariance function of a random process, 2) this random process is the output of a linear system having a white-noise input, 3) this linear system has a finite-dimensional state-variable description. Both the homogeneous and inhomogeneous equations are reduced to two linear first-order differential equations and an associated set of boundary conditions. The coefficients of these differential equations and the boundary conditions are specified directly by the matrices describing the random process that generates the kernel. The eigenvalues of the homogeneous integral equation are found to be solutions of a transcendental equation involving the transition matrix of the vector differential equations. The eigenfunctions follow directly. By using this same transcendental equation, an effective method of calculating the Fredholm determinant is derived. For the inhomogeneous equation, the vector differential equations are identical to those obtained in the state-variable formulation of the optimal linear smoother. Several examples illustrating the methods developed are presented.