A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials

Abstract

The purpose of this study is to give a Taylor polynomial approximation for the solution of mth-order linear difference equations with variable coefficients under the mixed conditions about any point. For this purpose, the Taylor matrix method is introduced. This method is based on first taking the truncated Taylor expansions of the functions in the difference equation and then substituting their matrix forms into the given equation. Hence the resultant matrix equation can be solved and the unknown Taylor coefficients can be found approximately. In addition, examples that illustrate the pertinent features of the method are presented, and the results of the study are discussed.