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## Asymptotic%20Methods:%20Introduction%20to%20Boundary

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### Based of the book The Boundary Function Method for Singular Perturbation ... dent variables. Phase space. Invariant manifold. Subspace of a. phase space ... – PowerPoint PPT presentation

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Title: Asymptotic%20Methods:%20Introduction%20to%20Boundary

1
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.
Vasileva, V.F. Butuzov and L.V. Kalachev, SIAM,
2
Lectures 10, 11 Reaction-Diffusion Equations
with Fast Diffusion. A Method to Determine the
Dimension of Long-Time Dynamics in Multi-Scale
Systems.
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Reaction-Diffusion Equations with Fast Diffusion
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A Method to Determine the Dimension of Long-Time
Dynamics in Multi-Scale Systems
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
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Quasi-equilibrium assumption
Sensitivity analysis, etc.
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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
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Space of depen- dent variables Phase
space
Invariant manifold Subspace of a
phase space
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Existence of Integral Manifold
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Assumptions
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Equivalent operator equation
Solve using successive approximations
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In the limit
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If operator is contracting
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Construction of operator
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Local State Space Reduction
Steps of the algorithm
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Example 1.
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Example 2.
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REFERENCES
• A.B.Vasileva, V.F.Butuzov, and L.V.Kalachev, The
• Boundary Function Method for Singular
Perturbation
• 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. 457 478.
• L.V.Kalachev, Asymptotic methods application to
• reduction of models, Natural Resource Modeling
(2000),
• V.13, No. 3, pp. 305 338.

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• 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.

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