A Monte Carlo technique using a fast repetitive analog computer for determining lowest eigenvalues of partial differential equations for various boundaries, with applications

Abstract

This paper describes a Monte Carlo technique for estimating the lowest eigenvalues of certain elliptic and hyperbolic partial differential equations with Dirichlet boundary conditions. (2) A stochastic process whose output conditional probability density distribution satisfies a partial differential equation similar to the partial differential equation under consideration is, along with the boundary conditions, implemented on ASTRAC II, a fast repetitive analog computer. Eight different boundaries on the scalar Helmholtz equation in one, two, and three-dimensional space were implemented. Each of the resulting eight lowest-eigenvalue estimates is then compared to its corresponding and expected true lowest eigenvalue. Computer hardware errors, along with the error resulting from the mathematical approximation employed in deriving the estimate, are indicated. Corrective measures are included when necessary. Applications result from a presentation of analogies relating the partial differential equations under consideration to partial differential equations modeling physical processes.