A general analysis of approximate nonlinear lumping in chemical kinetics. I. Unconstrained lumping

A general analysis of approximate nonlinear lumping for a chemical kinetic system described by an n-dimensional first order ordinary differential equation system dy/dt=f(y) is presented. There is a one-to-one relation between the differential equation system and the linear partial differential operator A = Σ_i=1ⁿf_i(y)(∂/∂y_i). The algebraic method in nonlinear perturbation theory is utilized to approximately transform A into some canonical forms in which the new dependent variables are partly separated. These canonical forms of A will give the generalized eigenfunctions or other higher dimensional unconstrained nonlinear lumping schemes of the original system approximately. Unconstrained nonlinear lumping gives a reduced differential equation system describing new variables which are nonlinear functions of the original ones. This approach may supply some purely fast variables. The solutions of original dependent variables can be obtained by the inverse transformation from the lumped variables and the approximate analytical solutions of the purely fast variables. The theoretical basis of this approach is presented. A simple example is used for illustration.