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We present a numerical spectral method to solve systems of differentialequations on an infinite interval $y\in (-\infty, \infty)$ in presence oflinear differential operators of the form $Q(y)\left(\partial/\partial_y\right)^b$ (where $Q(y)$ is a rational fraction and$b$ a positive integer). Even when these operators are not parity-preserving,we demonstrate how a mixed expansion in interleaved Chebyshev rationalfunctions $TB_n(y)$ and $SB_n(y)$ preserves the sparsity of theirdiscretization. This paves the way for fast $O(N\ln N)$ and spectrally accuratemixed implicit-explicit time-marching of sets of linear and nonlinear equationsin unbounded geometries.

Hybrid Chebyshev function bases for sparse spectral methods in parity-mixed PDEs on an infinite domain