A Riemann-Hilbert approach to the computation of transform pairs

Abstract

We develop a unified methodology that integrates spectral theory, Riemann-Hilbert problems, and inverse scattering theory for the construction and numerical evaluation of transform pairs associated with linear variable-coefficient partial differential equations. The approach combines analytical formulae with numerical methods for ordinary differential equations and Riemann-Hilbert problems, yielding a hybrid analytical-numerical strategy for working with these transforms. Results are presented for transforms arising in the Dirac equation, demonstrating accurate computations, even in the presence of discontinuous coefficients.