Resumen de Ecuaciones en diferencias lineales de segundo orden y su aplicación al análisis matricial
M. José Jiménez Jiménez
It is well known the important role that difference equations play in several problems of the engineering or of the science. However, whereas the expression of the solutions when the coefficients of the equations are constant is widely known, the same does not happen for variable coefficients, except for the simplest case of first order equations. This work presents an analysis of the second order linear difference equations equations in the infinite, semi-infinite and finite cases. Moreover, for second order difference equations in the finite case, we study its associated boundary value problem. Our goal is to develop techniques which are analogous to those used in the analysis of the differential equations, and for the finite case, those used for boundary value problems. Although our techniques, with appropriate modifications, are also in accordance with the complex case, in this work we mainly deal the real case, i.e. the case in which all functions involved take values in the field of real numbers. We study first the case of constant coefficients, showing initially the known results about second order linear diffrence equations with constant coefficients, to finish describing the multiple relationships of these equations with Chebyshev polynomials. These relationships allow us to calculate their solutions more directly and simply, without imposing conditions on the coefficients values of the equation . Another advantage of solving equations with constant coefficients using their relationship with Chebyshev polynomials is that you can extend the same approach to equations with variable coefficients and express as well their solutions in terms of functions with two arguments that we named Chebyshev functions. In this way, we get a closed formula to obtain the solutions for second order linear difference equations with variable coefficients. Given the close relationship between second order difference equations in the finite case and the inverse of tridiagonal matrices, in the last chapter we present the application of the above results to determine the invertibility of those matrices and, in that case, explicitly get its inverse.
