By Topic

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

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$31 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

1 Author(s)

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.

Published in:

Information Theory, IEEE Transactions on  (Volume:15 ,  Issue: 5 )