Applied asymptotic analysis / Peter D. Miller.

Format:

Book

Author/Creator:

Miller, Peter D. (Peter David), 1967- author.

Series:

Graduate studies in mathematics ; v. 75.

Graduate studies in mathematics, 1065-7339 ; volume 75

Language:

English

Subjects (All):

Asymptotic expansions.

Differential equations--Asymptotic theory.

Differential equations.

Approximation theory.

Integral equations--Asymptotic theory.

Integral equations.

Physical Description:

1 online resource (488 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2006]

Language Note:

English

Summary:

This book is a survey of asymptotic methods set in the current applied research context of wave propagation. It stresses rigorous analysis in addition to formal manipulations. Asymptotic expansions developed in the text are justified rigorously, and students are shown how to obtain solid error estimates for asymptotic formulae. The book relates examples and exercises to subjects of current research interest, such as the problem of locating the zeros of Taylor polynomials of entirenonvanishing functions and the problem of counting integer lattice points in subsets of the plane with various geometrical properties of the boundary. The book is intended for a beginning graduate course on asymptotic analysis in applied mathematics and is aimed at students of pure and appliedmathematics as well as science and engineering. The basic prerequisite is a background in differential equations, linear algebra, advanced calculus, and complex variables at the level of introductory undergraduate courses on these subjects. The book is ideally suited to the needs of a graduate student who, on the one hand, wants to learn basic applied mathematics, and on the other, wants to understand what is needed to make the various arguments rigorous. Down here in the Village, this is knownas the Courant point of view!! --Percy Deift, Courant Institute, New York Peter D. Miller is an associate professor of mathematics at the University of Michigan at Ann Arbor. He earned a Ph.D. in Applied Mathematics from the University of Arizona and has held positions at the Australian NationalUniversity (Canberra) and Monash University (Melbourne). His current research interests lie in singular limits for integrable systems.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Preface""; ""Part 1. Fundamentals""; ""Chapter 0. Themes of Asymptotic Analysis""; ""Â0.1. Theme: Asymptotics, Convergent and DivergentAsymptotic Series""; ""Â0.2. Theme: Other Parameters and Nonuniformity""; ""0.2.1. First example. Oscillations""; ""0.2.2. Second example. Boundary layers""; ""Â0.3. Theme: Differential Equations""; ""Â0.4. Theme: Universal Partial Differential Equations and Canonical Physical Models""; ""Chapter 1. The Nature of Asymptotic Approximations""; ""Â1.1. Asymptotic Approximations and Errors""

""1.1.1. Order relations among functions""""1.1.2. Statements following from the order relations""; ""1.1.3. Absolute and relative errors""; ""Â1.2. Convergent versus Asymptotic Series: Concepts""; ""1.2.1. Convergent power series""; ""1.2.2. Introduction to asymptotic series""; ""Â1.3. Asymptotic Sequences and Series: General Definitions""; ""Â1.4. How to ""Sum"" an Asymptotic Series""; ""Â1.5. Asymptotic Root Finding""; ""1.5.1. A regular perturbation problem""; ""1.5.2. A singular perturbation problem. Rescaling and the principle of dominant balance""; ""Â1.6. Notes and References""

""Part 2. Asymptotic Analysis of Exponential Integrals""""Chapter 2. Fundamental Techniques for Integrals""; ""Â2.1. Review of Basic Methods""; ""Â2.2. Exponential Integrals and Watson's Lemma""; ""Â2.3. Elementary Generalizations of Watson's Lemma""; ""Chapter 3. Laplace's Method for Asymptotic Expansions of Integrals""; ""Â3.1. Introduction""; ""Â3.2. Nonlocal Contributions""; ""Â3.3. Contributions from Endpoints""; ""Â3.4. Contributions from Interior Maxima""; ""Â3.5. Summary of Generic Leading-order Behavior""; ""Â3.6. Application: Weakly Diffusive Regularization of Shock Waves""

""3.6.1. The method of characteristics""""3.6.2. Regularization of shocks by diffusion. Burgers' equation""; ""3.6.3. The Cole-Hopf transformation and the solution of the initial-value problem for Burgers' equation""; ""3.6.4. Analysis of the solution in the limit of vanishing diffusion""; ""Â3.7. Multidimensional Integrals""; ""Â3.8. Notes and References""; ""Chapter 4. The Method of Steepest Descents for Asymptotic Expansions of Integrals""; ""Â4.1. Introduction""; ""Â4.2. Contour Deformation""; ""Â4.3. Paths of Steepest Descent""; ""Â4.4. Saddle Points""

""Â4.5. Parametrization-independent Local Contributions""""Â4.6. Application: Long-time Asymptotic Behavior of Diffusion Processes""; ""4.6.1. A derivation of the diffusion equation""; ""4.6.2. Solution of the diffusion equation and the corresponding initial-value problem""; ""4.6.3. Long-time asymptotics via the method of steepest descents""; ""Â4.7. Application: Asymptotic Behavior of Special Functions, Airy Functions and the Stokes Phenomenon""; ""4.7.1. Integral representations for Airy functions""

""4.7.2. Preliminary transformations necessary for asymptotic analysis of Ai(x) for large x""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 453-454) and index.

Description based on print version record.

ISBN:

1-4704-1154-7