This book presents the modeling and scaling of physical problems, which result in normalized perturbation equations. This is followed by solving perturbation problems and evaluating the results. The author refines perturbation methods into simple, understandable elements and avoids unnecessary theorems and proofs. In addition, the results are consolidated and interpreted, and the presented examples are succinct to illustrate the essential techniques. This book is ideal and beneficial for practicing scientists and engineers who need to understand and apply perturbation methods to difficult problems with applications in mathematics, engineering, and biology. Discussions on new perspectives, simpler presentations on convergence, and the expansion of integrals are included.
Author(s): C.Y. Wang
Series: Synthesis Lectures on Engineering, Science, and Technology
Publisher: Springer
Year: 2023
Language: English
Pages: 155
City: Cham
Preface
Contents
About the Author
1 Nondimensionalization and Scaling
2 Asymptotic Expansion
2.1 Ordering and Asymptotic Series
2.2 Inversion Formula
2.3 Lagrange Expansion
3 Basic Theory
3.1 Direct Expansion
3.2 Integral Form for Weakly Nonlinear Second Order Differential Equation
4 Eigenvalue Problems
4.1 Using Differential Equations
4.2 Using Integral Equation
4.3 Using Eigenfunction Expansion
5 Perturbation of Integrals
5.1 Basic Simple Techniques
5.1.1 Can the Integrand Be Expanded?
5.1.2 Is the Range Small?
5.1.3 Can We Integrate by Parts?
5.2 The Gamma Function
5.3 Effect of Exponentially Small Term in the Integrand
5.4 The Moving Maximum
5.5 The Oscillatory Integral
5.6 Method of Stationary Phase
6 The Phase Plane
6.1 First Order Singularities
6.1.1 λ1,λ2 Are Real and Distinct
6.1.2 λ1, λ2 Are Real and Equal
6.1.3 λ1, λ2 Are Complex Conjugates
6.2 Higher Order Singularities
6.2.1 The Basic Power Slope
6.3 Periodic Solutions
7 Periodic and Almost Periodic Solutions
7.1 Lindstedt Normalization for Periodic Solutions
7.2 Almost Periodic Solutions
7.2.1 Averaging Method
7.2.2 Method of Multiple Scales
7.3 Forced Oscillations
7.3.1 The Linear Problem
7.3.2 The Weakly Nonlinear Problem not Near Resonance
7.3.3 The Lightly Damped Case with Forcing Frequency Near Resonance
7.4 Parametric Excitation
7.4.1 The Solution When Not at “Resonance”
7.4.2 Ω Close to 0
7.4.3 Ω Close to 2
8 Singular Perturbation and Boundary Layers
8.1 An Algebraic Example
8.2 Boundary Layer Solution
8.3 Matched Asymptotic Expansions
9 Other Singular Problems
9.1 Method of Exponential Approximation
9.2 Turning Point Problems
9.2.1 Airy Functions
9.3 Singular Line Problems
9.3.1 Lighthill’s Method
9.3.2 Using Matched Asymptotic Expansions
10 Improving Asymptotic Series
10.1 Computer Extension
10.2 Improvement of Series
10.2.1 Peeling off Method
10.2.2 Shanks Method
10.2.3 Pade Method
11 Examples in Mechanics
11.1 Effect of Surface Roughness on Heat Conduction Through a Barrier
11.2 Buckling and Post Buckling of an Elastica Column
11.2.1 Small θ Perturbation
11.2.2 Large P Perturbation
11.3 Flow Due to Spreading of Material on the Surface
11.3.1 Small R Perturbation
11.3.2 Large R Perturbation
A Taylor Series
Index
