Optimality of the change-of-variable y = f(x) used to transform the boundary at infinity

Ascertain whether the specific change of variables y = f(x), with f^{-1}(·) = ∫_{·}^{∞} dη/σ(η) as defined in equation (3.5), is optimal for numerically solving the alternative Cauchy problem (2.5) by transforming the boundary at infinity to zero, and, if not, characterize preferable transformations in terms of numerical performance (e.g., truncation error reduction and coefficient regularity).

Background

The paper introduces the mapping y = f(x) to relocate the boundary at infinity to x = 0, aiming to reduce spatial truncation errors compared to imposing a boundary condition at a large finite point. However, the transformation may produce singular coefficients in the transformed PDE, raising questions about its numerical desirability.

The authors explicitly note that the optimality of this particular f in the context of numerical performance is not established, suggesting that alternative transformations might offer better trade-offs between accuracy and stability.