Basis functions for solution of non-homogeneous wave equation

TL;DR

This paper extends the Differential Transfer Matrix Method to complex domains, introducing new basis functions that improve solution accuracy for the non-homogeneous wave equation, especially near turning points, and removes previous singularity issues.

Contribution

It develops a complex-plane extension of DTMM, derives new basis functions for real-valued problems, and proposes an analytical matrix exponential transformation to enhance solution accuracy.

Findings

New basis functions satisfy initial conditions exactly

Basis functions approach turning points without divergence

Analytical transformation improves solution accuracy

Abstract

In this note we extend the Differential Transfer Matrix Method (DTMM) for a second-order linear ordinary differential equation to the complex plane. This is achieved by separation of real and imaginary parts, and then forming a system of equations having a rank twice the size of the real-valued problem. The method discussed in this paper also successfully removes the problem of dealing with essential singularities, which was present in the earlier formulations. Then we simplify the result for real-valued problems and obtain a new set of basis functions, which may be used instead of the WKB solutions. These basis functions not only satisfy the initial conditions perfectly, but also, may approach the turning points without the divergent behavior, which is observed in WKB solutions. Finally, an analytical transformation in the form of a matrix exponential is presented for improving the accuracy of solutions.