[Paper Review] Transformation of the linear difference equation into a system of the first order difference equations

M. I. Ayzatsky|arXiv (Cornell University)|Jun 12, 2018

Optical Network Technologies15 references3 citations

TL;DR

This paper presents a novel transformation method to convert an Nth-order linear difference equation into an equivalent system of first-order difference equations. The approach enables new analytical forms, including a nonlinear second-order equation analogous to the Riccati equation for third-order cases, and facilitates WKB approximation for slowly varying coefficients.

ABSTRACT

The transformation of the Nth- order linear difference equation into a system of the first order difference equations is presented. The proposed transformation gives possibility to get new forms of the N-dimensional system of the first order equations that can be useful for analysis of the solutions of the Nth- order difference equations. In particular, for the third-order linear difference equation the nonlinear second-order difference equation that plays the same role as the Riccati equation for second-order linear difference equation is obtained. The new form of the N-dimensional system of first order equations can be also used for finding the WKB solutions of the linear difference equation with coefficients that vary sufficiently slowly with index.

Motivation & Objective

• To develop a systematic method for converting higher-order linear difference equations into systems of first-order equations.
• To explore new analytical forms of the resulting first-order systems that may simplify solution analysis.
• To extend the applicability of WKB methods to linear difference equations with slowly varying coefficients.
• To derive a nonlinear second-order difference equation analogous to the Riccati equation for third-order linear equations.
• To provide a framework for studying solutions of Nth-order linear difference equations through first-order system formulations.

Proposed method

• The transformation uses a change of variables to express the Nth-order difference equation as a system of N first-order difference equations.
• The method introduces auxiliary variables to represent successive differences of the original sequence.
• The resulting system is represented in matrix form, enabling standard linear algebra techniques for analysis.
• For third-order equations, the method yields a nonlinear second-order difference equation through elimination of variables.
• The approach is extended to analyze slowly varying coefficient cases using WKB-type asymptotic approximations.
• The transformation preserves the solution space of the original equation while enabling new analytical tools.

Experimental results

Research questions

• RQ1 Can an Nth-order linear difference equation be systematically transformed into a system of first-order equations?
• RQ2 What new analytical forms emerge from this transformation, particularly for third-order equations?
• RQ3 Does the transformation enable the derivation of a nonlinear second-order equation analogous to the Riccati equation?
• RQ4 Can the transformed system support WKB approximation methods for slowly varying coefficients?
• RQ5 What are the structural and analytical advantages of the first-order system representation over the original higher-order equation?

Key findings

• The transformation successfully converts any Nth-order linear difference equation into an equivalent system of N first-order difference equations.
• For third-order linear difference equations, the method yields a nonlinear second-order difference equation that plays a role analogous to the Riccati equation.
• The resulting first-order system allows for the application of WKB approximation techniques to equations with slowly varying coefficients.
• The transformation preserves the solution space, ensuring equivalence between the original and transformed systems.
• The new system formulation provides alternative pathways for analyzing stability, asymptotic behavior, and solution structure.
• The method offers a systematic framework for extending techniques from first-order systems to higher-order linear difference equations.