The distribution function of the solution of the random eigenvalue problem for differential equations (Q1101874)

This paper considers the random eigenvalue problem: \[ - d/dx[p(x,\omega)du/dx]+q(x,\omega)u=\lambda \rho (x,\omega)u, \] \[ \alpha (\omega)u(0)+\alpha '(\omega)u'(0)=0,\quad \beta (\omega)u(1)+\beta '(\omega)u'(1)=0,\quad 0<x<1. \] For every deterministic \(\lambda^*\) the nth eigenvalue \(\lambda_ n\) has the property: \[ \Pr ob\{\omega | \quad \lambda_ n(\omega)\leq \lambda^*\} = \Pr ob\{\omega | \quad \theta (1,\lambda^*)\geq \delta +n\pi \}, \] where \(\theta (x,\lambda^*)\) is the Prüfer function, and satisfies the initial value problem of a random differential equation. If the random factors contains only random variables, the Liouville equation can help us to derive a deterministic partial differential equation and then the density of \(\theta\) and the probability of \(\lambda_ n\) can be evaluated efficiently. This procedure can also be used in general case for finding the approximate solution. One example is given in the last section.