Asymptotic%20Methods:%20Introduction%20to%20Boundary - PowerPoint PPT Presentation

View by Category
About This Presentation
Title:

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

Description:

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

Number of Views:32
Avg rating:3.0/5.0
Slides: 65
Provided by: lk772
Learn more at: http://www.math.umt.edu
Category:

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

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,
1995 (with additional material included)
2
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
3
Reaction-Diffusion Equations with Fast Diffusion
Leonid V. Kalachev 2003 UM
4
Leonid V. Kalachev 2003 UM
5
Leading order approximation ?
Leonid V. Kalachev 2003 UM
6
Leonid V. Kalachev 2003 UM
7
Leonid V. Kalachev 2003 UM
8
Leonid V. Kalachev 2003 UM
9
Leonid V. Kalachev 2003 UM
10
Leonid V. Kalachev 2003 UM
11
Leonid V. Kalachev 2003 UM
12
Leonid V. Kalachev 2003 UM
13
Leonid V. Kalachev 2003 UM
14
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
Quasi-steady state assumption
Leonid V. Kalachev 2003 UM
Quasi-equilibrium assumption
Sensitivity analysis, etc.
15
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
16
Space of depen- dent variables Phase
space
Invariant manifold Subspace of a
phase space
Leonid V. Kalachev 2003 UM
17
Leonid V. Kalachev 2003 UM
18
Leonid V. Kalachev 2003 UM
19
Leonid V. Kalachev 2003 UM
20
Existence of Integral Manifold
Leonid V. Kalachev 2003 UM
21
Assumptions
Leonid V. Kalachev 2003 UM
22
Leonid V. Kalachev 2003 UM
23
Equivalent operator equation
Solve using successive approximations
Leonid V. Kalachev 2003 UM

24
In the limit
Leonid V. Kalachev 2003 UM
If operator is contracting
25
Leonid V. Kalachev 2003 UM
26
Construction of operator
Leonid V. Kalachev 2003 UM
27
Leonid V. Kalachev 2003 UM
28
Leonid V. Kalachev 2003 UM
29
Leonid V. Kalachev 2003 UM
30
Leonid V. Kalachev 2003 UM
31
Leonid V. Kalachev 2003 UM
32
Leonid V. Kalachev 2003 UM
33
Leonid V. Kalachev 2003 UM
34
Leonid V. Kalachev 2003 UM
35
Leonid V. Kalachev 2003 UM
36
Leonid V. Kalachev 2003 UM
37
Local State Space Reduction
Steps of the algorithm
Leonid V. Kalachev 2003 UM
38
Leonid V. Kalachev 2003 UM
39
Leonid V. Kalachev 2003 UM
40
Leonid V. Kalachev 2003 UM
41
Leonid V. Kalachev 2003 UM
42
Leonid V. Kalachev 2003 UM
43
Leonid V. Kalachev 2003 UM
44
Leonid V. Kalachev 2003 UM
45
Leonid V. Kalachev 2003 UM
46
Leonid V. Kalachev 2003 UM
47
Example 1.
Leonid V. Kalachev 2003 UM
48
Leonid V. Kalachev 2003 UM
49
Leonid V. Kalachev 2003 UM
50
Leonid V. Kalachev 2003 UM
51
Leonid V. Kalachev 2003 UM
52
Leonid V. Kalachev 2003 UM
53
Leonid V. Kalachev 2003 UM
54
Leonid V. Kalachev 2003 UM
55
Example 2.
Leonid V. Kalachev 2003 UM
56
Leonid V. Kalachev 2003 UM
57
Leonid V. Kalachev 2003 UM
58
Leonid V. Kalachev 2003 UM
59
Leonid V. Kalachev 2003 UM
60
Leonid V. Kalachev 2003 UM
61
Leonid V. Kalachev 2003 UM
62
Leonid V. Kalachev 2003 UM
63
REFERENCES
  • A.B.Vasileva, V.F.Butuzov, and L.V.Kalachev, The
  • Boundary Function Method for Singular
    Perturbation
  • Problems, Philadelphia SIAM, 1995.
  • 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.

Leonid V. Kalachev 2003 UM
64
  • 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
About PowerShow.com