Algebraic Modification of the Method of Undetermined Coefficients For Solving Nonhomogeneous Linear Difference Equations

TL;DR

This paper introduces an algebraic modification to the classical method of undetermined coefficients, enhancing its applicability to complex nonhomogeneous linear difference equations with quasipolynomial right-hand sides.

Contribution

It proposes a novel algebraic modification that overcomes limitations of the classical method and enables solving entire difference equations simultaneously using a matrix superposition principle.

Findings

Eliminates major drawbacks of the classical method

Allows solving complex difference equations with quasipolynomial right-hand sides

Formulates superposition principle in matrix form

Abstract

In this paper, an algebraic modification of the method of undetermined coefficients for solving nonhomogeneous linear stationary difference equations for quasipolynomial right-hand sides is proposed. Although the classical method of undetermined coefficients is well-known in both differential equations and difference equations case, its application in the difference equations case is severely limited. For example, it is hard to apply for rather complex expressions that can arise in case of complex quasipolynomials and resonance. The novelty of the research is the proposition of an algebraic modification to the method. That modification eliminates major drawbacks of the primary method and also allows to modify the superposition principle to apply the method to the entire difference equation at once without dividing the problem into several less complicated ones. The superposition principle in matrix form is formulated.