Computational conformal mapping

Computational Conformal Mapping provides a self-contained and systematic survey of the theory and computation of conformal mappings of simply and multiply-connected regions onto the unit disk or canonical region. It provides a comprehensive coverage of the concepts and related numerical computations with applications to relevant areas of applied math, physics, and engineering. The style and presentation are readily accessible to all graduates, practitioners, and professionals. Features and Topics: comprehensive coverage of theory, numerical aspects, and selected applications; discussion of a variety of methods for approximate conformal representations; presentation on applications of classical topics and modern boundary value problem topics; detailed numerical presentation of computation of approximate construction of Green's function; Meletiev's method of interior and exterior multiply-connected regions; more than 75 case studies and numerous end-of-chapter problems. This new book is a complete and accessible text/reference for all graduates, mathematicians, scientists, and engineers who need to effectively use a variety of methods for conformal mappling problems. The case studies and problems make it suitable for use as a self-study resource for practitioners and professionals. Contents: Preface Chapter 0. Introduction 0.1 Historical Background 0.2 Modern Developments Chapter 1. Basic Concepts 1.1 Notation and Definitions 1.2 Some Basic Theorems 1.3 Harmonic Functions 1.4 Conformal Mapping 1.5 Problems Chapter 2. Conformal Mapping 2.1 Polynomials 2.2 Bilinear Transformations 2.3 Schwarz-Christoffel Transformations 2.4 Catalog of Conformal Maps 2.5 Problems Chapter 3. Schwarz-Christoffel Integrals 3.1 Parameter Problem 3.2 Newton's Method 3.3 Numerical Computations 3.4 Kirchhoff Flow Problem 3.5 Problems Chapter 4. Polynomial Approximations 4.1 Minimum Area Problem 4.1.1 Bergman Kernel 4.2 Numerical Methods for Problem 4.2.1 Ritz Method 4.2.2 Bergman Kernel Method 4.3 Minimum Boundary Problem 4.4 Ritz Method for Problem II 4.5 Orthogonal Polynomials 4.5.1 Polynomials Orthogonal to the Boundary 4.5.2 Polynomials Orthogonal to a Region 4.6 Problems Chapter 5. Nearly Circular Regions 5.1 Small Parameter Expansions 5.2 Method of Infinite Systems 5.3 Three Special Cases 5.4 Exterior Regions 5.5 Problems Chapter 6. Green's Functions 6.1 Mean-Value Theorem 6.2 Dirichlet Problem 6.3 Numerical Computation 6.4 Schwarz Formula 6.5 Neumann Problem 6.6 Series Representations 6.7 Problems Chapter 7. Integral Equation Methods 7.1 Neumann Kernel 7.2 Interior Regions 7.2.1 Lichtenstein's Integral Equation 7.2.2 Gershgorin's Integral Equation 7.2.3 Carrier's Integral Equation 7.3 Exterior Regions 7.3.1 Banin's Integral Equation 7.3.2 Warschawski-Stiefel's Integral Equation 7.3.3 Interior and Exterior Maps 7.4 Iterative Method 7.5 Degenerate Kernel 7.6 Szego Kernel 7.7 Problems Chapter 8. Theodorsen's Integral Equation 8.1 Classical Iterative Method 8.2 Convergence 8.3 Proofs 8.4 Integral Representation 8.5 Starlike Regions 8.6 Exterior Regions 8.7 Trigonometric Interpolation 8.8 Wegmann's Method 8.9 Newton's Method 8.10 Problems Chapter 9. Symm's Integral Equation 9.1 Symm's Integral Equation 9.1.1 Interior Regions 9.1.2 Exterior Regions 9.2 Orthonormal Polynomial Method 9.3 Modified ONP Method 9.4 Lagrange Interpolation 9.5 Splilne Approximation 9.6 Problems Chapter 10. Airfoils 10.1 Joukowski Function 10.2 Generalized Joukowski Mappings 10.2.1 Glauert's Modifications 10.2.2 Symmetric Joukowski 10.3 Nearly Circular Approximations 10.4 James's Method 10.4.1 Single-Element Airfoils 10.4.2 von Karman-Trefftz Transformation 10.4.3 Two-Element Airfoils 10.4.4 Multi-Element Airfoils 10.5 Problems Chapter 11. Doubly Connected Regions 11.1 Conformal Modulus 11.1.1 Conformal Invariants 11.1.2 Area Theorem 11.2 Source Density 11.3 Dipole Distribution 11.4 Problems Chapter 12. Singularities 12.1 Arbenz's Integral Equation 12.2 Boundary Corner 12.3 Singularity Behavior 12.4 Pole-Type Singularities 12.5 Exterior Regions 12.6 Doubly Connected Regions 12.7 Problems Chapter 13. Multiply Connected Regions 13.1 Existence and Uniqueness 13.2 Dirichlet Problem 13.3 Mikhlin's Integral Equation 13.4 Mayo's Method 13.5 Fast Poisson Solver 13.6 Problems Chapter 14. Grid Generation 14.1 Computational Region 14.2 Inlet Configurations 14.3 Cascade Configuration 14.4 Problems Appendix A. Cauchy P.V. Integrals A.1 Numerical Evaluation Appendix B. Green's Identities Appendix C. RiemannHilbert Problem C.1 Homogeneous Hilbert Problem C.2 Nonhomogeneous Hilbert Problem C.3 Riemann-Hilbert Problem Appendix D. Successive Approximations D.1 Tables Appendix E. Catalog of Conformal Mappings Bibliography Notation Index