# Numerics for Common First-Passage Problem

*by*Polychronis-Thomas D. Spanos, (M.ASCE), Assoc. Prof. of Engrg. Mechanics; Univ. of Texas, Austin, Tex. 78712,

**Serial Information**:

*Journal of the Engineering Mechanics Division*, 1982, Vol. 108, Issue 5, Pg. 864-882

**Document Type:**Journal Paper

**Abstract:**

Numerical analysis aspects of the Kolmogorov backward partial differential equation associated with the first-passage problem of the response amplitude of a lightly damped linear structure are considered. It is assumed that the structure is excited by a stationary broad-band random process. A formula is presented for the analytical estimation of the eigenvalues of the boundary value problem constructed by a separation of variables procedure on the Kolmogorov equation. The analytical estimates are used as initial values in an iterative scheme which determines the eigenvalues numerically, for several values of the circular barrier of the first-passage problem. An efficient algorithm for the numerical computation of the corresponding eigenfunctions is presented. This algorithm is used, as well, to compute the constant coefficients of a solution of the Kolmogorov equation in the form of an eigenvalues-eigenfunctions series expansion. The numerical data obtained are examined in context with the physics of the problem.

**Subject Headings:**Numerical methods | Numerical analysis | Linear analysis | Eigenvalues | Algorithms | Damping | Stationary processes | Boundary value problem

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