Estimations for the characteristic values of a Sturm-Liouville problem

Abstract

We consider a particular Sturm-Liouville problem, with boundary conditions containing the characteristic values. This problem governs the stability of the interfaces in oil recovery, by using the Hele-Shaw model of a homogeneous porous medium. We obtain estimations of the approximative characteristic values, by using a finite-difference approximation and the Gerschgorin's localization theorem. We obtain estimations of the exact characteristic values by using the Rayleigh's quotient of the initial problem. The upper limit of the approximative characteristic values is (in general) greater than the lower limit of the exact characteristic values. The same upper limit is given by the two methods, for a particular value of a parameter of the problem. This value gives us a large improvement of the stability. A previous convergence result is confirmed.