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Math 515 Essential Perturbation Theory and Asymptotic Analysis

Math 515 Essential Perturbation Theory and Asymptotic Analysis (Spring 2024). Instructor: Prof. Steven Wise, Department of Mathematics, University of Tennessee. Analysis of advanced techniques in modern context for applied problems: dimensional analysis and scaling, perturbation theory, variational approaches, transform theory, wave phenomena and conservation laws, stability and bifurcation, distributions, integral equations.

Lecture 03 - Boundary Layer Problems

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Lecture 01 - Stirling's Approximation and Integrals with a Large Parameter

Lecture 02 - Regular and Singular Algebraic Perturbation Problems

Lecture 03 - Boundary Layer Problems

Lecture 04 - The van der Waals-Cahn-Hilliard Diffuse Interface

Lecture 05 - Order Symbols

Lecture 06 - Asymptotic Expansions

Lecture 07 - A General Principle for Matching

Lecture 08 - Matching in Action

Lecture 09 - Integration by Parts

Lecture 10 - Watson's Lemma

Lecture 11 - Stirling's Approximation and More Laplace Integrals

Lecture 12 - Method of Stationary Phase

Lecture 13 - A Rigorous Proof for the Method of Stationary Phase

Lecture 14 - Real and Complex Contour Integration

Lecture 15 - Cauchy's Integral Theorem and Formulae

Lecture 16 - The Method of Steepest Descent

Lecture 17 - The Saddle Point Method

Lecture 18 - Matching in Linear, Singularly Perturbed BVP

Lecture 19 - Existence Theory for a Class of Linear, Singularly Perturbed BVPs

Lecture 20 - Location of the Boundary Layer(s)

Lecture 21 - Higher Order Matching in a Non-constant Coefficient BVP

Lecture 22 - A Corner Layer Problem and an Intro to Nonlinear Boundary Layer Problems

Lecture 23 - Nonlinear Boundary Layer Problems

Lecture 24 - The WKB Method

Lecture 25 - WKB Approximation of Sturm-Liouville Problems

Lecture 26 - WKB Approximation of One-Turning-Point Problems

Lecture 27 - The Quantum Harmonic Oscillator and Two-Turning-Point Problems