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This introductory graduate text is based on a graduate course the author has taught repeatedly over the last ten years to students in applied mathematics, engineering sciences, and physics. Each chapter begins with an introductory development involving ordinary differential equations, and goes on to cover such traditional topics as boundary layers and multiple scales. However, it also contains material arising from current research interest, including homogenisation, slender body theory, symbolic computing, and discrete equations. Many of the excellent exercises are derived from problems of up-to-date research and are drawn from a wide range of application areas. One hundred new pages added including new material on transcedentally small terms, Kummer's function, weakly coupled oscillators and wave interactions.
Taylor's Theorem and l'Hospital's Rule.
Order Symbols.
Asymptotic Approximations.
Asymptotic Expansions.
Accuracy Versus Convergence of an Asymptotic Series.
Manipulating Asymptotic Expansions.
Asymptotic Solution of Algebraic and Transcendental Equations.
Introduction to the Asymptotic Solution of Differential Equations.
Uniformity.
Symbolic Computing.
Introductory Example.
Step : Outer Solution.
Step : Boundary Layer.
Step : Matching.
Matching Revisited.
Second Term.
Discussion.
Steps and : Boundary Layers and Matching.
Step : Composite Expansion.
Transcendentally Small Terms.
Step : Locating the Layer.
Steps and : Interior Layer and Matching.
Step : Missing Equation.
Step : Composite Expansion.
Kummer Functions.
Corner Layers.
Step : Corner Layer.
Step : Composite Expansion.
Elliptic Problem.
Outer Expansion.
Boundary-Layer Expansion.
Composite Expansion.
Parabolic Boundary Layer.
Parabolic Problem.
Outer Expansion.
Inner Expansion.
Difference Equations.
Boundary-Layer Approximation.
Numerical Solution of Differential Equations.
Regular Expansion.
Multiple-Scale Expansion.
Labor-Saving Observations.
Discussion.
Introductory Example (continued).
Three Time Scales.
Uniqueness and Minimum Error.
Forced Motion Near Resonance.
Weakly Coupled Oscillators.
Slowly Varying Coefficients.
Boundary Layers.
Introduction to Partial Differential Equations.
Linear Wave Propagation.
Nonlinear Wave Equation.
Wave–Wave Interactions.
Nonlinear Diffusion.
Example: Fisher's Equation.
Weakly Nonlinear Difference Equation.
Chain of Oscillators.
Example: Exact Solution.
Example: Plane Wave Solution.
Introductory Example.
Second Term of Expansion.
General Discussion.
The Case Where q'(xt)&gt.
Solution in Transition Layer.
Matching for x &gt x t.
The Case Where q'(xt)