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Mathematics - Analysis | Introduction to Perturbation Methods

Introduction to Perturbation Methods

Series: Texts in Applied Mathematics, Vol. 20

Holmes, Mark H.

1995, XIII, 356p. 88 illus..

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  • About this textbook

This book is an introductory graduate text dealing with many of the perturbation methods currently used by applied mathematicians, scientists, and engineers. The author has based his book on a graduate course he has taught several times over the last ten years to students in applied mathematics, engineering sciences, and physics. The only prerequisite for the course is a background in differential equations. Each chapter begins with an introductory development involving ordinary differential equations. The book covers traditional topics, such as boundary layers and multiple scales. However, it also contains material arising from current research interest. This includes homogenization, slender body theory, symbolic computing, and discrete equations. One of the more important features of this book is contained in the exercises. Many are derived from problems of up- to-date research and are from a wide range of application areas.

Content Level » Graduate

Keywords » Perturbation Methods

Related subjects » Analysis

Table of contents 

1: Introduction to Asymptotic Approximations.- 1.1 Introduction.- 1.2 Taylor’s Theorem and l‘Hospital’s Rule.- 1.3 Order Symbols.- 1.4 Asymptotic Approximations.- 1.4.1 Asymptotic Expansions.- 1.4.2 Accuracy versus Convergence of an Asymptotic Series.- 1.4.3 Manipulating Asymptotic Expansions.- 1.5 Asymptotic Solution of Algebraic and Transcendental Equations.- 1.6 Introduction to the Asymptotic Solution of Differential Equations.- 1.7 Uniformity.- 1.8 Symbolic Computing.- 2: Matched Asymptotic Expansions.- 2.1 Introduction.- 2.2 Introductory Example.- 2.3 Examples with Multiple Boundary Layers.- 2.4 Interior Layers.- 2.5 Corner Layers.- 2.6 Partial Differential Equations.- 2.7 Difference Equations.- 3: Multiple Scales.- 3.1 Introduction.- 3.2 Introductory Example.- 3.3 Slowly Varying Coefficients.- 3.4 Forced Motion Near Resonance.- 3.5 Boundary Layers.- 3.6 Introduction to Partial Differential Equations.- 3.7 Linear Wave Propagation.- 3.8 Nonlinear Waves.- 3.9 Difference Equations.- 4: The WKB and Related Methods.- 4.1 Introduction.- 4.2 Introductory Example.- 4.3 Turning Points.- 4.4 Wave Propagation and Energy Methods.- 4.5 Wave Propagation and Slender Body Approximations.- 4.6 Ray Methods.- 4.7 Parabolic Approximations.- 4.8 Discrete WKB Method.- 5: The Method of Homogenization.- 5.1 Introduction.- 5.2 Introductory Example.- 5.3 Multidimensional Problem: Periodic Substructure.- 5.4 Porous Flow.- 6: Introduction to Bifurcation and Stability.- 6.1 Introduction.- 6.2 Introductory Example.- 6.3 Analysis of a Bifurcation Point.- 6.4 Linearized Stability.- 6.5 Relaxation Dynamics.- 6.6 An Example Involving a Nonlinear Partial Differential Equation.- 6.7 Bifurcation of Periodic Solutions.- 6.8 Systems of Ordinary Differential Equations.- Appendix AI: Solution and Properties of Transition Layer Equations.- A1.1 Airy Functions.- A1.2 Confluent Hypergeometric Functions.- A1.3 Higher-Order Turning Points.- Appendix A2: Asymptotic Approximations of Integrals.- A2.1 Introduction.- A2.2 Watson’s Lemma.- A2.3 Laplace’s Approximation.- A2.4 Stationary Phase Approximation.- Appendix A3: Numerical Solution of Nonlinear Boundary-Value Problems.- A3.1 Introduction.- A3.2 Examples.- A3.3 Computer Code.- References.

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