TY - JOUR

T1 - A general analysis of exact nonlinear lumping in chemical kinetics

AU - Li, Genyuan

AU - Rabitz, Herschel

AU - Tóth, János

N1 - Funding Information:
Acknowledgments-The authors acknowledge support from the Denartment of Enerev and Exxon Cornoration. One of the authors (J.T.) has Go been support& by the Soros Foundation and by the National Scientific Research Fund, Hungary, No. 328s.

PY - 1994

Y1 - 1994

N2 - A general analysis of exact nonlinear lumping is presented. This analysis can be applied to the kinetics of any reaction system with n species described by a set of first-order ordinary differential equations dy/dt = f(y), where y is an n-dimensional vector and f(y) is an arbitrary n-dimensional function vector. We consider lumping by means of n̂ (n̂ ≤ n)-dimensional arbitrary transformation ŷ = h(y). The lumped differential equation system is d y ̂ Dt = y(h(ŷ))f(h(ŷ)), where hy(y) is teh Jacobian matrix of h(y), h is a generalized inverse transformation of h satisfying the relation h(h) = In̂. Three necessary and sufficient conditions of the existence of exact nonlinear lumping schemes have been determined. The geometric and algebraic interpretations of these conditions are discussed. It is found that a system is exactly lumpable by h only if h(y) = 0 is its invariant manifold. A linear partial differential operator A = Σni=1 fi(y)θ{symbol}/θ{symbol}yi corresponding to dy dt = f(y) is defined. Using the eigenfunctions and the generalized eigenfunctions of A, the operator can be transformed to Jordan or diagonal canonical forms which give the lumped differential equation systems without determination of h. These approaches are illustrated by a simple example. The results of this analysis serve as a theoretical basis for the development of approaches for approximate nonlinear lumping.

AB - A general analysis of exact nonlinear lumping is presented. This analysis can be applied to the kinetics of any reaction system with n species described by a set of first-order ordinary differential equations dy/dt = f(y), where y is an n-dimensional vector and f(y) is an arbitrary n-dimensional function vector. We consider lumping by means of n̂ (n̂ ≤ n)-dimensional arbitrary transformation ŷ = h(y). The lumped differential equation system is d y ̂ Dt = y(h(ŷ))f(h(ŷ)), where hy(y) is teh Jacobian matrix of h(y), h is a generalized inverse transformation of h satisfying the relation h(h) = In̂. Three necessary and sufficient conditions of the existence of exact nonlinear lumping schemes have been determined. The geometric and algebraic interpretations of these conditions are discussed. It is found that a system is exactly lumpable by h only if h(y) = 0 is its invariant manifold. A linear partial differential operator A = Σni=1 fi(y)θ{symbol}/θ{symbol}yi corresponding to dy dt = f(y) is defined. Using the eigenfunctions and the generalized eigenfunctions of A, the operator can be transformed to Jordan or diagonal canonical forms which give the lumped differential equation systems without determination of h. These approaches are illustrated by a simple example. The results of this analysis serve as a theoretical basis for the development of approaches for approximate nonlinear lumping.

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U2 - 10.1016/0009-2509(94)87006-3

DO - 10.1016/0009-2509(94)87006-3

M3 - Article

AN - SCOPUS:0028378405

SN - 0009-2509

VL - 49

SP - 343

EP - 361

JO - Chemical Engineering Science

JF - Chemical Engineering Science

IS - 3

ER -