E.H. Volhov

¥ Solving the laplace Equation and for Constructing Conformal Iflappings

CRC Press Boca Raton Ann Arbor London Tokyo

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Contents

C h a p t e r 1. A p p r o x i m a t e Block M e t h o d for Solv ing t h e Laplace Equat ion on P o l y g o n s 1

1. Setting up a Mixed Boundary-Value Problem for the Laplace Equation on a Polygon 1

1.1. Description of the Boundary of a Polygon 1

1.2. Setting up a Boundary-Value Problem 7

2. A Finite Covering of a Polygon by Blocks of Three Types 13

3. Representation of the Solution of a Boundary-Value Problem on Blocks 19

3.1. Carrier Functions 19

3.2. Preliminary Lemmas 24

3.3. Representation of the Solution on Blocks 30

3.4. Additional Lemmas 32

4. An Algebraic Problem 34

5. The Main Result 40

5.1. Theorem on the Convergence of the Block

Method 40

5.2. Discussion of the Theorem 42

6. Proofs of Theorem 5.1 and of Lemmas 4.1-4.4 48

6.1. Preliminary Lemmas 48

6.2. Theorem on the Solvability of an Algebraic Problem 54

6.3. Fundamental Lemmas 58

VÜi Contents

6.4. Proof of Theorem 5.1 66 6.5. Proofs of Lemmas 4.2-4.4 67

7. The Stability and the Labor Content of Computations Required by the Block Method 73 7.1. Preliminary Lemmas 73 7.2. Verification of the Criterion of Solvability of an

Algebraic Problem 77 7.3. The Stability of the Block Method in a Uniform

Metrie 79 7.4. The Labor of Computations Required by the Block

Method 83 8. Approximation of a Conjugate Harmonie Function on

Blocks 85 9. Neumann's Problem 92

9.1. The Solvability of Neumann's Problem on a Polygon 92

9.2. Approximate Solving of Neumann's Problem by Block Method 96

10. The Case of Arbitrary Analytic Mixed Boundary Conditions 98

Chapter 2. Approximate Block Method of Conformal Mapping of Polygons onto Canonical Domains . . . . 108 11. Approximate Conformal Mapping of a Simply- Connected

Polygon onto a Disk 108 12. Basic Harmonie Functions 111 13. Approximate Conformal Mapping of a Multiply-

Connected Polygon onto a Plane with Cuts along Parallel Line Segments 113 13.1. Structure of a Mapping 113 13.2. Constructing an Approximate Mapping 115

14. Approximate Conformal Mapping of a Multiply-Connected Polygon onto a Ring with Cuts along the Ares of Concentric Circles 119 14.1. Structure of a Mapping 119 14.2. Constructing an Approximate Mapping 122

Contents IX

Chapter 3 . D e v e l o p m e n t and Appl i ca t ion of t h e A p p r o x i m a t e Block M e t h o d for Conformal M a p p i n g of S i m p l y - C o n n e c t e d and D o u b l y - C o n n e c t e d D o m a i n s 124 15. Approximate Conformal Mapping of Some Polygons

onto a Strip 124 15.1. Scheme of Constructing a Mapping 124

15.2. Mapping of a Rectangle 125

15.3. Mapping of Two Octagons onto a Strip and of a T-shaped Domain onto a Half-Strip 133

16. Scheme of Constructing a Conformal Mapping of a Doubly-Connected Domain onto a Ring 141

17. Mapping a Square Frame onto a Ring 143 18. Mapping a Square with a Circular Hole Using Circular

Lune Block 149 19. Representation of a Harmonie Function on a Ring . . . 155 20. Using a Block-Ring for Mapping Domain (18.1) onto a

Ring 157 21. A Block-Bridge 161

22. Limit Cases 167 23. Mapping a Disk with an Elliptic Hole or with a

Retrosection onto a Ring 169

24. Mapping a Disk with a Regulär Polygonal Hole 173 25. Mapping the Exterior of a Parabola with a Hole onto a

Ring 181

C h a p t e r 4. A p p r o x i m a t e Conformal M a p p i n g of D o m a i n s w i t h a Per iodic S tructure by t h e Block M e t h o d 188 26. Mapping a Domain of the Type of Half-Plane with a

Periodic Structure onto a Half-Plane 188

26.1. Construction of a Conformal Mapping of a Domain of the Type of Half-Plane with a Periodic Boundary onto a Half-Plane 188

26.2. Constructing Blocks 191 26.3. The Algorithm 194

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26.4. Practical Results 196 26.5. The General Case 198

27. Mapping a Domain of the Type of Strip with a Periodic Structure onto a Strip 202 27.1. Contruction of a Conformal Mapping of a Domain

of the Type of Strip with a Periodic Boundary onto a Strip 202

27.2. Constructing Blocks 203 27.3. The Algorithm 205 27.4. Practical Results 207

27.5. The General Case 208 28. Mapping the Exterior of a Lattice of Ellipses onto the

Exterior of a Lattice of Plates 210 28.1. Scheme of Mapping 210 28.2. Constructing Blocks 211 28.3. The Algorithm 213 28.4. Practical Results 216

References 220 Index 225