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. Vasil’eva, V.F. Butuzov and L.V. Kalachev, SIAM, 1995 (with additional material included)

Lectures 7 - 9: Simple Boundary Value Problems, Method of Vishik and Lyusternik for Partial Differential Equations. Applied Chemical Engineering Example. Leonid V. Kalachev 2003 UM

Simple Boundary Value Problems Leonid V. Kalachev 2003 UM

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IMPORTANT ! Leonid V. Kalachev 2003 UM

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Illustration of Condition 3 ΄ : one point of intersection Leonid V. Kalachev 2003 UM

Illustration of Condition 3 ΄ : two points of intersection Leonid V. Kalachev 2003 UM

No points of intersection: Condition 3 ΄ is not satisfied Leonid V. Kalachev 2003 UM

!!! This concludes the construction of the leading order approximation! Leonid V. Kalachev 2003 UM

All the terms of the leading order approximation have now been determined: Leonid V. Kalachev 2003 UM

Generalizations: Leonid V. Kalachev 2003 UM

Singularly Perturbed Partial Differential Equations The Method of Vishik-Lyusternik Leonid V. Kalachev 2003 UM

IMPORTANT ! Leonid V. Kalachev 2003 UM

Generalization: Corner layer boundary functions. Natural applications include singularly perturbed parabolic equations and, e.g, singularly perturbed elliptic equations in rectangular domains. Leonid V. Kalachev 2003 UM

Chemical Engineering Example: Model reductions for multiphase phenomena (study of a catalytic reaction in a three phase continuously stirred tank reactor [CSTR]) Leonid V. Kalachev 2003 UM

Series process consisting of the following stages: Leonid V. Kalachev 2003 UM

Reaction scheme: Leonid V. Kalachev 2003 UM

Some notation (for a detailed notation list See Haario and Kalachev [2]): Leonid V. Kalachev 2003 UM

Initial and boundary conditions: Leonid V. Kalachev 2003 UM

Micro-model Leonid V. Kalachev 2003 UM

The Limiting Cases Leonid V. Kalachev 2003 UM

Uniform asymptotic approximation in the form: Leonid V. Kalachev 2003 UM

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Similar for higher order terms! Leonid V. Kalachev 2003 UM

We look for asymptotic expansion in the same form! Leonid V. Kalachev 2003 UM

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Higher order terms can be constructed in a similar way! Leonid V. Kalachev 2003 UM

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. Corresponding initial condition: And Similar analysis for higher order terms! Leonid V. Kalachev 2003 UM

We apply the same asymptotic procedure! Corresponding initial condition: Leonid V. Kalachev 2003 UM

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! Leonid V. Kalachev 2003 UM

The task then is to design an experimental setup that allows one to discriminate between Case 2 and Case 4: Changing input gas concentration! With typical experimental noise in the data, the discrepancy between Cases 2 and 4 might not exceed the error level! 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