CONTENTS -- Preface -- PART I MINI-COURSES -- Ismail, Mourad E. H. Lectures on Orthogonal Polynomials -- 1. Construction of Orthogonal Polynomials -- 2. Some Properties of Orthogonal Polynomials -- 3. Differential Equations -- 4. Electrostatic Equilibrium Problems -- 5. Generating Functions and Asymptotics -- 6. Applications -- 7. Zeros of Orthogonal Polynomials and Eigenvalues -- Acknowledgments. -- References -- Ramis, Jean-Pierre Gevrey Asymptotics and Applications to Holomorphic Ordinary Differential Equations -- 0. Introduction -- 1. Asymptotic expansions -- 1.1. Classical asymptotics (or Poincare' asymptotics) -- 1.2. Gevrey asyrnptotics -- 1.3. k-summability -- 1.4. Laplace transform -- 1.5. Ramis-Sibuya theorem and k-summability -- 2. Applications to ordinary differential equations -- 2.1. Linear ordinary differential equations, index theorem and Newton polygons -- 2.2. Fundamental existence theorems -- 3. Multisummability -- 4. Gevrey asymptotic and singular perturbations -- 4.1. Delay in bifurcations -- References -- Ward, Michael J. Spikes for Singularly Perturbed Reaction-Diffusion Systems and Carrier's Problem -- 1. Introduction -- 2. Spike Equilibria for Scalar Problems -- 2.1. Interior Spike Solutions: No Boundary Layers -- 2.2. Interior Spike Solutions: Boundary Layers -- 2.3. Exponential Sensitivity to the Data -- 2.4. Related Problems: One Space Dimension -- 2.5. Spikes for Quasilinear Elliptic PDE -- 3. Spikes for Nonlocal Scalar Problems -- 3.1. The Shadow Gierer-Meinhardt Model -- 3.2. Hopf Bifurcations for the Shadow GM Model -- 3.3. A Microwave Heating Model -- 3.4. A Flame-Front Evolution Model -- 4. Dynamics and Equilibria of a Spike in an R-D System -- 4.1. The Near-Shadow System -- 4.2. Pinning of a Spike -- 4.3. Sudden Oscillatory Instabilities -- 5 . Multi-Spike Solutions in Reaction-Diffusion Systems.

5.1. Gray-Scott Model (Low Feed-Rate): Equilibrium -- 5.2. Gray-Scott Model (High Feed-Rate): Equilibrium -- 5.3. Stability of Multi-Spike Equilibria: One Dimension -- 6. Concluding Remarks -- Acknowledgments -- References -- Wong, Roderick S. C. Five Lectures on Asymptotic Theory -- LECTURE I STATIONARY PHASE APPROXIMATION -- 1. Introduction -- 2. Stationary-Phase Approximation -- 3. Coalescing Stationary Points -- 4. Two-dimensional Approximation -- 5. A Uniform Asymptotic Formula -- References -- LECTURE II METHOD OF STEEPEST DESCENT -- 1. Introduction -- 2. The Airy Integral -- 3. Stokes' Phenomenon -- 4. Adjacent Saddle and Adjacent Contour -- 5. Exponential Asymptotics -- 6. Hyperasymptotics -- References -- LECTURE III WKB METHOD AND TURNING POINT -- 1. Introduction -- 2. Successive Approximations -- 3. Turning Point Problem -- 4. Simple Pole -- 5. Examples -- References -- LECTURE IV BOUNDARY LAYER THEORY -- 1. Introduction -- 2. Derivation of (1.3) -- 3. Internal Layers -- 4. Carrier-Pearson Equation -- References -- LECTURE V DIFFERENCE EQUATIONS AND ORTHOGONAL POLYNOMIALS -- 1. Introduction -- 2. Normal and Subnormal Series -- 3. Equation (1.2) with p and q 0. -- 4. Airy-type Expansion -- 5. Bessel-type Expansion -- References -- PART II INVITED PAPERS -- Bohun, C. S., Frigaard, I., Huang, H.X. and Liang S. Q. A Perturbation Model for the Growth of Type III-V Compound Crystals -- 1. Introduction -- 2. Mathematical Model and Dimensional Analysis -- 2.1. Non- dimensionalization -- 3. Perturbation Solution -- 3.1. Resolution of the zeroth order model -- 3.1.1. Constant radius crystals -- 3.1.2. Conical crystals -- 3.1.3. Comments -- 3.2. Radial variations: resolution of the first order model -- 3.2.1. Constant radius crystals -- 3.2.2. Conical crystals -- 4. Thermal Stress -- 4.1. Plain strain assumption -- 4.2. Results -- 5. Conclusion.

Acknowledgement. -- References -- Chen, H. and Yu, C. Asymptotic Behaviour of the Trace for Schrodinger Operator on Irregular Domains -- 1. Main Result -- 2. Preparation -- 3. Proof of the Main Results -- 4. Further Results -- References -- Jiang, L. S. and Ren, X. M. Limitations and Modifications of Black-Scholes Model -- 1. Introduction -- 1.1. Motivation of Black-Scholes model -- 1.2. Foundations of Black-Scholes model -- 1.3. Modifications of the model -- 2. Jump-diffusion process -- 2.1. Movement of asset price -- 2.2. Black-Scholes model with jump-diffusion -- 2.3. Some results -- 3. Stochastic interest rate -- 3.1. The models of stochastic interest rate -- 3.2 Black-Scholes model with stochastic interest rate -- 3.2. American style option-valuation of mortgage -- 4. Variable volatility & implied volatility -- 4.1. Implied volatility surface -- 4.2. Related Optimal Control Problem -- 4.3. Numerical results -- References -- Li, T.T. (Li, Daqian) Exact Boundary Controllability of Unsteady Flows in a Network of Open Canals -- 1. Introduction -- 2. General consideration -- 3. Statement on the exact boundary controllability of unsteady flows in a network of open canals -- 4. Preliminaries for quasilinear hyperbolic systems of diagonal form -- 5. Corresponding consideration for Saint-Venant system -- 6. Exact boundary controllability of unsteady flows in a tree-like network of open canals -- 7. Some remarks -- References -- Miyake, M. and Ichinobe, K. Hierarchy of Partial Differential Equations and Fundamental Solutions Associated with Summable Formal Solutions of a Partial Differential Equation of non Kowalevski Type -- 1. Introduction -- 2. (0)-Gevrey summability (Review) -- 3. The case of entire function Cauchy data -- 4. Proof of Proposition 1 -- 5. Hierarchy of differential equations and fundamental solutions -- 6. Singular perturbation.

References -- Tahara, H. On the Singularities of Solutions of Nonlinear Partial Differential Equations in the Complex Domain, II -- 1. Introduction -- 2. Non-existence of singularities -- 3. Criteria for the existence of singularities -- 4. Sufficient conditions for the existence of singularities -- References -- Tan, Y . J. and Chen, X. X. Identifying Corrosion Boundary by Perturbation Method -- 1. Introduction -- 2. Perturbation Solution for Direct Problem -- 3. Reconstruct Free Boundary -- 4. Numerical Examples -- 5. Some Theoretical Results -- References -- Wei, J. C. Existence and Stability of Lamellar and Wriggled Lamellar Solutions in the Diblock Copolymer Problem -- 1. Introduction -- 2. One-dimensional Local Minimizers -- 3. Stability of the Perfect Lamellar Solutions in 2D -- 4. Existence of Wriggled Lamellar Solutions -- 5. Stability of the Bifurcating Solutions -- Acknowledgments. -- References.