Steady-state, one-dimensional Fokker-Planck equation with an absorbing boundary: A half-range treatment

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Abstract
An approximate solution to the steady-state version of the one-dimensional Fokker-Planck equation is obtained by (extending an approach due originally to Harris [J. Chem. Phys. 75, 3103 (1981)] and) using the ansatz that the solution, which has an explicit half-range character, can be approximated in each half of the velocity space by a truncated expansion in Hermite polynomials. The convergence of the expansion is studied and found to be remarkably rapid. With only a modicum of effort, the ninth-order approximation (i.e., the value based on an expansion containing nine terms) to the cardinal quantity in this context, viz., the extrapolated end point, is found to be 1.459 88, whereas the exact value is 1.460 35. . . .