A method for the algebraic solution of a class of non-stationary, discrete-time, two-point boundary-value problems

Abstract

Considerable attention has been given, even recently, to the solution of two-point boundary-value problems, by means of algorithms that cannot always be considered satisfactory. The present work suggests, for the discrete case, a method which, with reference to certain hypotheses, allows the algebraic solution of a large class of these problems. Specifically, by applying Casorati'a linear operator, it is possible to pass from a single system of 2n complete linear difference equations with time-varying coefficients to two distinct systems, each of n second-order difference equations. Thus after appropriate transformation of the boundary values it is possible to solve independently each of the two above systems. Also the technique proposed can be usefully employed both to solve systems of non-linear, two-point boundary-value difference equations (using the quasi-linearization method) and when it is desired, or it is only possible, to determine just a part of the unknown vector of a system of difference equations.