The authors study some aspects of a method, the so called quasi steady state approximation, that has been used to obtain approximate solutions to some systems of ordinary differential equations (ODEs) which appear e.g. in problems of transport theory and chemical kinetics.
The ODEs under consideration must have the form \( y'(t) = f(y(t),z(t)), z'(t) = g(y(t),z(t))\) where \(y\) and \(z\) are “fast” and “slow” variables respectively, so that the system after a possible transient tends to a steady state in which \( f( y(t), z(t)) \simeq 0 \). Then the quasi-steady-state approximation is obtained (under suitable assumptions) getting \( y = \phi (z) \) from \( f(y,z)= 0\) and then solving the system of slow variables \( z' = g ( \phi (z), z) \) with their initial conditions which is usually easily integrated.
The authors propose an additional algorithm which allows, with some extra work, to estimate the error in the approximate solution provided by the above approach. Further some numerical examples are presented to show that this estimate has the same order of the true error.
Finally several examples are presented to show the applicability and the performance of the method in linear and nonlinear problems. As remarked by the authors one advantage of this method is that due to the form of the solution it can be useful for simulations which require to estimate some parameters of the model by comparison of the solution with observational measurements.