A method for solving stochastic eigenvalue problems

We present a novel approach for calculating stochastic eigenvalues of differential and integral equations as well as for random matrices. Five examples based on very different types of problem have been analysed and detailed numerical results obtained. It would seem that the method has considerable promise. The essence of the method is to replace the stochastic eigenvalue problem λ(ξ)ϕ(ξ)=A(ξ)ϕ(ξ)λ(ξ)ϕ(ξ)=A(ξ)ϕ(ξ), where ξξ is a set of random variables, by the introduction of an auxiliary equation in which λϕ=dϕ/dt≡ϕ˙. This changes the problem from an eigenvalue one to an initial value problem in the new pseudo-time variable t. The new linear time-dependent equation may then be solved by a polynomial chaos expansion (PCE) and the stochastic eigenvalue and its moments recovered by a limiting process. This technique has the advantage of avoiding the non-linear terms in the conventional method of stochastic eigenvalue calculation by PCE, but it does introduce an additional, ‘pseudo-time’, independent variable t. The paper illustrates the viability of this approach by application to several examples based on realistic problems.