Asymptotic Methods: Introduction to Boundary Function Method (Lectures 7 - 9)

Leonid V. KalachevDepartment 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 BoundaryValue 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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IMPORTANT !L

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Illustration of Condition 3΄: one point of intersectionL

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Illustration of Condition 3΄: two points of intersectionL

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No points of intersection: Condition 3΄ is not satisfied L

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This concludes the construction of the leading order approximation!

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All the terms of theleading order approximation havenow 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 applicationsinclude singularly perturbed parabolic equations and, e.g, singularly perturbed elliptic equations in rectangular domains.

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Chemical EngineeringExample:

Model reductions for multiphase phenomena(study of a catalyticreaction in a three phase continuously stirred tank reactor[CSTR])

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Series process consisting of the following stages:

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Reaction scheme:

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Some notation (for a detailed notation listSee 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 thedetails, let us write down the formulae for the leading orderapproximation.

Corresponding initial condition:

And

Similar analysis for higher order terms!

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We apply the same asymptotic procedure!

Corresponding initial condition:

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Comparison of the solutions for a fullmodel (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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The task then is to design anexperimental setup that allows one to

discriminate between Case 2 and Case 4:

Changing input gas concentration!

With typical experimental noise in thedata, the discrepancy between Cases 2 and 4

might not exceed the error level!

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REFERENCES:

1. A.B.Vasil’eva, V.F.Butuzov, and L.V.Kalachev, TheBoundary Function Method for Singular PerturbationProblems, Philadelphia: SIAM, 1995.

2. H.Haario and L.Kalachev, Model reductions for multi-phase phenomena, Intl. J.of Math. Engineeringwith Industrial Applications (1999), V.7, No.4,pp. 457 – 478.

3. 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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