Ebook: Asymptotics beyond All Orders

An asymptotic expansion is a series that provides a sequence of increasingly accurate approximations to a function in a particular limit. The formal definition, given by Poincare (1886, Acta Math. 8:295), is as follows. Given a function,

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Content:
Front Matter....Pages i-xiii
Asymptotics, Superasymptotics, Hyperasymptotics.......Pages 1-14
Computation of Transcendental Effects in Growth Problems: Linear Solvability Conditions and Nonlinear Methods-The Example of the Geometric Model....Pages 15-28
The Geometric Model of Crystal Growth — An Overview....Pages 29-35
Numerical Analysis of the Geometric Model for Dendritic Growth of Crystals....Pages 37-66
Dendritic Crystal Growth: Overview....Pages 67-73
An Experimental Assessment of Continuum Models of Dendritic Growth....Pages 75-86
A New Formulation for Dendritic Crystal Growth in two Dimensions....Pages 87-104
Directional Growth of Dilute Mixtures and Lamellar Eutectics....Pages 105-118
A Flat Interface and its Unfolding Bifurcations....Pages 119-129
Viscous Displacement in a Hele-Shaw Cell....Pages 131-153
Saffman — Taylor Viscous Fingering in a Wedge....Pages 155-173
Saffman-Taylor Problem in Sector Geometry....Pages 175-186
Exponentially Small Estimates for Separatrix Splittings....Pages 187-195
Exponentially Small Phenomena in the Rapidly Forced Pendulum....Pages 197-211
Proof of an Asymptotic Symmetry of the Rapidly Forced Pendulum....Pages 213-221
Exponentially Small Oscillations in the Solution of an Ordinary Differential Equation....Pages 223-240
Singular Perturbation of Solitons....Pages 241-246
Reflection Coefficient Beyond All Orders for Singular Problems....Pages 247-253
Laminar Flow in a Porous Channel....Pages 255-266
Existence and Stability of Particle Channeling in Crystals on Timescales Beyond all Orders....Pages 267-273
Gravity-Capillary Free Surface Flows....Pages 275-291
Solitary Water Waves with Ripples Beyond All Orders....Pages 293-298
Generalized Solitary Waves in a Stratified Fluid....Pages 299-307
Bending Losses in Optical Fibers....Pages 309-316
Exponential Asymptotics and Spectral Theory for Curved Optical Waveguides....Pages 317-326
Solitary-Waves in Self-Induced Transparency....Pages 327-336
Exponential Asymptotics for Partial Differential Equations....Pages 337-356
Problems of Existence of Nontopological Solitons (Breathers) for Nonlinear Klein-Gordon Equations....Pages 357-363
Exponentially Small Residues Near Analytic Invariant Circles....Pages 365-373
Asymptotics of Partial Differential Equations and the Renormalisation Group....Pages 375-383
Back Matter....Pages 385-389

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Content:
Front Matter....Pages i-xiii
Asymptotics, Superasymptotics, Hyperasymptotics.......Pages 1-14
Computation of Transcendental Effects in Growth Problems: Linear Solvability Conditions and Nonlinear Methods-The Example of the Geometric Model....Pages 15-28
The Geometric Model of Crystal Growth — An Overview....Pages 29-35
Numerical Analysis of the Geometric Model for Dendritic Growth of Crystals....Pages 37-66
Dendritic Crystal Growth: Overview....Pages 67-73
An Experimental Assessment of Continuum Models of Dendritic Growth....Pages 75-86
A New Formulation for Dendritic Crystal Growth in two Dimensions....Pages 87-104
Directional Growth of Dilute Mixtures and Lamellar Eutectics....Pages 105-118
A Flat Interface and its Unfolding Bifurcations....Pages 119-129
Viscous Displacement in a Hele-Shaw Cell....Pages 131-153
Saffman — Taylor Viscous Fingering in a Wedge....Pages 155-173
Saffman-Taylor Problem in Sector Geometry....Pages 175-186
Exponentially Small Estimates for Separatrix Splittings....Pages 187-195
Exponentially Small Phenomena in the Rapidly Forced Pendulum....Pages 197-211
Proof of an Asymptotic Symmetry of the Rapidly Forced Pendulum....Pages 213-221
Exponentially Small Oscillations in the Solution of an Ordinary Differential Equation....Pages 223-240
Singular Perturbation of Solitons....Pages 241-246
Reflection Coefficient Beyond All Orders for Singular Problems....Pages 247-253
Laminar Flow in a Porous Channel....Pages 255-266
Existence and Stability of Particle Channeling in Crystals on Timescales Beyond all Orders....Pages 267-273
Gravity-Capillary Free Surface Flows....Pages 275-291
Solitary Water Waves with Ripples Beyond All Orders....Pages 293-298
Generalized Solitary Waves in a Stratified Fluid....Pages 299-307
Bending Losses in Optical Fibers....Pages 309-316
Exponential Asymptotics and Spectral Theory for Curved Optical Waveguides....Pages 317-326
Solitary-Waves in Self-Induced Transparency....Pages 327-336
Exponential Asymptotics for Partial Differential Equations....Pages 337-356
Problems of Existence of Nontopological Solitons (Breathers) for Nonlinear Klein-Gordon Equations....Pages 357-363
Exponentially Small Residues Near Analytic Invariant Circles....Pages 365-373
Asymptotics of Partial Differential Equations and the Renormalisation Group....Pages 375-383
Back Matter....Pages 385-389
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