2. 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)

3. 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

4. Simple Boundary Value Problems Leonid V. Kalachev 2003 UM

9. !!! Leonid V. Kalachev 2003 UM

10. ---------------------------------------- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ============================ Leonid V. Kalachev 2003 UM

13. IMPORTANT ! Leonid V. Kalachev 2003 UM

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

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

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

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

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

30. Generalizations: Leonid V. Kalachev 2003 UM

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

38. IMPORTANT ! Leonid V. Kalachev 2003 UM

46. 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

47. 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

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

50. Reaction scheme: Leonid V. Kalachev 2003 UM

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

54. Initial and boundary conditions: Leonid V. Kalachev 2003 UM

55. Micro-model Leonid V. Kalachev 2003 UM

56. The Limiting Cases Leonid V. Kalachev 2003 UM

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

58. !!! Leonid V. Kalachev 2003 UM

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

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

63. !!! Leonid V. Kalachev 2003 UM

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

67. 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

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

70. 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

72. 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

74. REFERENCES: 1.A.B.Vasil’eva, V.F.Butuzov, and L.V.Kalachev, The Boundary Function Method for Singular Perturbation Problems, Philadelphia: SIAM, 1995. 2.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. 3.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