Asymptotic Methods: Introduction to Boundary

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Asymptotic Methods Introduction to Boundary
Function Method (Lectures 7 - 9)
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)
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Lectures 7 - 9 Simple Boundary Value Problems,
Method of Vishik and Lyusternik for Partial
Differential Equations. Applied Chemical
Engineering Example.
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Simple Boundary Value Problems
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!!!
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IMPORTANT !
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!!!
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Illustration of Condition 3? one point of
intersection
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Illustration of Condition 3? two points of
intersection
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No points of intersection Condition 3? is not
satisfied
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This concludes the construction of the leading
order approximation!
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All the terms of the leading order approximation
have now been determined
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Generalizations
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Singularly Perturbed Partial Differential
Equations
The Method of Vishik-Lyusternik
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IMPORTANT !
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Generalization
Corner layer boundary functions. Natural
applications include singularly perturbed
parabolic equations and, e.g, singularly
perturbed elliptic equations in rectangular
domains.
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Chemical Engineering Example Model reductions
for multiphase phenomena (study of a
catalytic reaction in a three phase continuously
stirred tank reactor CSTR)
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Series process consisting of the following stages
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Some notation (for a detailed notation list See
Haario and Kalachev 2)
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Initial and boundary conditions
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Micro-model
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The Limiting Cases
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Uniform asymptotic approximation in the form
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Similar for higher order terms!
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We look for asymptotic expansion in the same form!
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Higher order terms can be constructed in a
similar way!
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Asymptotic approximation in the same form!
This case is a combination of Cases 1 and 2.
Omitting the details, let us write down the
formulae for the leading order approximation.
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Similar analysis for higher order terms!
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We apply the same asymptotic procedure!
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Comparison of the solutions for a full model
(with typical numerical values of parameters)
and the limiting cases Cases 2 and 4 both
approximate the full model considerably well!
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With typical experimental noise in the data, the
discrepancy between Cases 2 and 4 might not
exceed the error level!
The task then is to design an experimental setup
that allows one to discriminate between Case 2
and Case 4
Changing input gas concentration!
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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.

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