Asymptotic Methods: Introduction to Boundary Function Method (Lectures 10, 11) Leonid V. Kalachev Department of Mathematical Sciences University of Montana Based of the book The Boundary Function Method for Singular Perturbation Problems by A.B. Vasil’eva, V.F. Butuzov and L.V. Kalachev, SIAM, 1995 (with additional material included)

Lectures 10, 11: Reaction-Diffusion Equations with Fast Diffusion. A Method to Determine the Dimension of Long-Time Dynamics in Multi-Scale Systems. Leonid V. Kalachev 2003 UM

Reaction-Diffusion Equations with Fast Diffusion Leonid V. Kalachev 2003 UM

Leading order approximation = ? Leonid V. Kalachev 2003 UM

Important problem: determination of the minimal number of phase variables needed to describe the characteristic behavior of large scale systems. Different approaches are based on the presence of a wide range of characteristic time-scales in a chemical system: Quasi-steady state assumption Quasi-equilibrium assumption Sensitivity analysis, etc. A Method to Determine the Dimension of Long-Time Dynamics in Multi-Scale Systems Leonid V. Kalachev 2003 UM

In most cases NO MATHEMATICAL JUSTIFICATION! Here we estimate the dimension of the underlying long-time dynamics in a multi-scale systems using approach based on the method of integral manifolds. Some notions Leonid V. Kalachev 2003 UM

Space of depen- dent variables = Phase space Invariant manifold = Subspace of a phase space Leonid V. Kalachev 2003 UM

Existence of Integral Manifold Leonid V. Kalachev 2003 UM

Assumptions: Leonid V. Kalachev 2003 UM

Equivalent operator equation: Solve using successive approximations: … Leonid V. Kalachev 2003 UM

In the limit: If operator is contracting: Leonid V. Kalachev 2003 UM

Construction of operator Leonid V. Kalachev 2003 UM

Local State Space Reduction Steps of the algorithm Leonid V. Kalachev 2003 UM

Example 1. Leonid V. Kalachev 2003 UM

Example 2. Leonid V. Kalachev 2003 UM

REFERENCES: 1.A.B.Vasil’eva, V.F.Butuzov, and L.V.Kalachev, The Boundary Function Method for Singular Perturbation Problems, Philadelphia: SIAM, H.Haario and L.Kalachev, Model reductions for multi-phase phenomena, Intl. J.of Math. Engineering with Industrial Applications (1999), V.7, No.4, pp. 457 – L.V.Kalachev, Asymptotic methods: application to reduction of models, Natural Resource Modeling (2000), V.13, No. 3, pp. 305 – 338. Leonid V. Kalachev 2003 UM

4.S. Handrock-Meyer, L.V.Kalachev and K.R. Schneider, A method to determine the dimension of long-time dynamics in multi-scale systms, J. Math. Chem. (2001), Vol. 30, No. 2, pp. 133 – 160. Leonid V. Kalachev 2003 UM