## Integral Equations: Theory and Numerical TreatmentThe theory of integral equations has been an active research field for many years and is based on analysis, function theory, and functional analysis. On the other hand, integral equations are of practical interest because of the «boundary integral equation method», which transforms partial differential equations on a domain into integral equations over its boundary. This book grew out of a series of lectures given by the author at the Ruhr-Universitat Bochum and the Christian-Albrecht-Universitat zu Kiel to students of mathematics. The contents of the first six chapters correspond to an intensive lecture course of four hours per week for a semester. Readers of the book require background from analysis and the foundations of numeri cal mathematics. Knowledge of functional analysis is helpful, but to begin with some basic facts about Banach and Hilbert spaces are sufficient. The theoretical part of this book is reduced to a minimum; in Chapters 2, 4, and 5 more importance is attached to the numerical treatment of the integral equations than to their theory. Important parts of functional analysis (e. g. , the Riesz-Schauder theory) are presented without proof. We expect the reader either to be already familiar with functional analysis or to become motivated by the practical examples given here to read a book about this topic. We recall that also from a historical point of view, functional analysis was initially stimulated by the investigation of integral equations. |

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### Contents

Introduction | 1 |

12 Basics from Analysis | 3 |

123 Holder Continuous Functions | 4 |

13 Basics from Functional Analysis | 5 |

132 Banach Spaces CD CD CD | 6 |

133 Banach Spaces L1D L2D LiD | 7 |

134 Dense Subspaces | 8 |

136 Linear Operators | 9 |

526 Convergence | 167 |

53 The Hierarchy of Discrete Problems | 168 |

533 Relative Consistency | 172 |

534 Convergence | 173 |

54 TwoGrid Iteration | 174 |

542 Convergence Analysis | 175 |

543 Amount of Computational Work | 176 |

544 Variant for ApI | 178 |

137 Theorem of Uniform Boundedness | 10 |

138 Compact Sets and Compact Mappings | 11 |

139 RieszSchauder Theory | 14 |

14 Basics from Numerical Mathematics | 15 |

142 Quadrature | 19 |

143 Condition Number of a System of Equations | 23 |

Volterra Integral Equations | 25 |

212 Regularity of the Solution | 27 |

22 Numerical Solution by Quadrature Methods | 29 |

222 Error Estimate | 30 |

23 Further Numerical Methods | 37 |

24 Linear Volterra Integral Equations of Convolution Type | 39 |

25 Volterra Integral Equations of the First Kind | 41 |

Theory of Fredholm Integral Equations of the Second Kind | 42 |

32 Compactness of the Integral Operator X | 43 |

322 The Case XCD | 45 |

323 The Case XLD | 47 |

33 Finite Approximability of the Integral Operator 1C | 48 |

332 Degenerate Kernels | 49 |

34 The Image Space of 1C | 50 |

342 The Image Kf for | 52 |

343 Kernels with Integrable Singularity | 54 |

344 Compactness | 57 |

35 Solution of the Fredholm Integral Equation of the Second Kind | 58 |

Numerical Treatment of Fredholm Integral Equations of the Second Kind | 59 |

412 Consistency and Stability | 60 |

413 Convergence | 61 |

414 Stability and Convergence Theorem | 62 |

415 Error Estimates | 64 |

42 Discretisation by Kernel Approximation | 65 |

422 Setting Up the System of Equations | 66 |

423 Kernel Approximation by Interpolation | 67 |

424 Tensor Approximation of k | 68 |

425 Examples of Kernel Approximations | 69 |

426 A Variant of the Kernel Approximation | 70 |

428 Numerical Examples | 73 |

43 Projection Methods in General | 75 |

433 Lemmata | 76 |

434 Discretisation by means of a Projection | 77 |

435 Convergence Analysis | 79 |

44 Collocation Method | 81 |

442 Setting up the System of Linear Equations | 82 |

443 Examples for Interpolations | 84 |

444 Condition Number of the System of Equations | 86 |

445 Numerical Examples | 88 |

45 Galerkin Method | 91 |

452 Derivation of the System of Equations | 92 |

453 Convergence in LD and LeoD | 93 |

454 Error Estimates | 96 |

4SS Condition Number of the System of Equations | 97 |

Piecewise Constant Functions | 100 |

Piecewise Linear Functions | 104 |

458 General Analysis of Projection Errors | 105 |

Piecewise Linear Functions | 107 |

4510 Numerical Examples | 109 |

46 Additional Comments Concerning Projection Methods | 111 |

462 Estimates with Respect to Weaker Norms | 112 |

463 The Iterated Approximation | 116 |

464 Superconvergence | 118 |

465 More General Formulations of the Projection Method | 121 |

466 Numerical Quadrature | 123 |

467 Product Integration | 126 |

The Nystrom Method | 128 |

472 Convergence Analysis | 130 |

473 Stability | 132 |

474 Consistency Order | 136 |

475 Condition Number of the System of Equations | 137 |

476 Regularisation | 138 |

477 Numerical Examples | 139 |

48 Supplements | 141 |

4813 From the Collocation to the Nystrom Method | 142 |

4814 From the Collocation to the Galerkin Method | 143 |

483 Extrapolation Method | 144 |

484 Eigenvalue Problems | 147 |

485 Complementary Integral Equations | 152 |

Perturbation Theorem for Stability | 154 |

MultiGrid Methods for Solving Systems Arising from Integral Equations of the Second Kind | 155 |

512 Direct Solution of the System of Equations | 156 |

514 Conjugate Gradient Method | 158 |

52 Stability and Convergence Discrete Formulation | 160 |

522 The Banach Space Y and the Discrete Spaces Yn | 163 |

523 The Interpolation Error or Projection Error | 165 |

525 Stability | 166 |

545 Numerical Examples | 179 |

55 MultiGrid Iteration | 181 |

552 Amount of Computational Work | 183 |

553 Convergence | 184 |

554 Numerical Examples | 188 |

555 Variants of the MultiGrid Methods | 190 |

56 Nested Iteration | 195 |

562 Amount of Computational Work | 196 |

563 Convergence | 197 |

564 Numerical Examples | 199 |

6 Abels Integral Equation | 201 |

613 Improper Integrals | 203 |

62 A Necessary Condition for a Bounded Solution | 206 |

63 Eulers Integrals | 207 |

64 Inversion of Abels Integral Equation | 208 |

65 Reformulation for Kernels kxyxyfi | 213 |

66 Numerical Methods for Abels Integral Equation | 214 |

Singular Integral Equations | 216 |

712 Curvilinear Integrals | 220 |

713 Cauchys Principal Value for Curvilinear Integrals | 222 |

714 The Example ffCJ2fCzJ | 224 |

72 The Cauchy Kernel | 230 |

722 Regularity Properties | 234 |

723 Properties of the Generated Holomorphic Function | 236 |

724 Representation of K | 244 |

725 The Cauchy Integral on the Unit Circle | 245 |

73 The Singular Integral Equation | 247 |

733 General Singular Integral Equations | 248 |

734 Approximation of the Cauchy Integral on the Unit Circle | 249 |

73 S Approximation of the Cauchy Integral on an Arbitrary Curve F | 250 |

736 MultiGrid Methods for Equations of a Special Form | 251 |

74 Application to the Dirichlet Problem for Laplaces Equation | 253 |

743 Uniqueness and Representation Theorem | 257 |

744 The Case of a Smooth Boundary T | 259 |

745 The DoubleLayer Potential for Solving the Exterior Problem | 260 |

746 The Tangential Derivative of the SingleLayer Potential | 262 |

75 Hypersingular Integrals | 264 |

The Integral Equation Method | 266 |

812 Continuity of the SingleLayer Potential | 268 |

8123 Improper Integrals on Surfaces | 269 |

8124 Properties of the SingleLayer Potential | 271 |

8132 The Cauchy Principal Value for Surface Integrals | 275 |

8133 Other Directional Derivatives | 279 |

814 Formulation of the Di rich let Boundary Value Problem as Integral Equation of the First Kind for the SingleLayer Potential | 281 |

8142 The Integral Equation of the First Kind | 283 |

81S Formulation of the Neumann Boundary Value Problem as Integral Equation of the Second Kind for the SingleLayer Potential | 284 |

82 The DoubleLayer Potential | 287 |

822 Regularity Properties of the DoubleLayer Integral Operator | 289 |

823 Jump Properties of the DoubleLayer Potential | 292 |

824 Further Properties of the DoubleLayer Potential | 294 |

8242 The Potential close to the Jump Discontinuity of the Density | 295 |

8243 The DoubleLayer Potential of the Density f1 | 296 |

825 Derivatives of the DoubleLayer Potential | 299 |

826 Integral Equations with the DoubleLayer Operator | 303 |

8262 Formulation of the Neumann Boundary Value Problem as Integral Equation of the Second Kind with the Double Layer Operator | 305 |

827 NonSmooth Curves or Surfaces | 306 |

83 The Hypersingular Integral Equation | 309 |

Integral Equations for the Laplace Equation | 313 |

85 The Integral Equation Method for Other Differential Equations | 314 |

852 Equations of Higher Order | 315 |

853 Systems of Differential Equations | 316 |

The Boundary Element Method | 318 |

913 Collocation Method | 319 |

914 Convergence in the Compact Case | 320 |

92 The Boundary Elements | 323 |

922 Geometric Discretisation | 324 |

923 Elements in the ThreeDimensional Case | 325 |

924 Error Considerations | 326 |

93 MultiGrid Methods | 327 |

932 Equations of the First Kind | 328 |

94 Integration and Numerical Quadrature | 330 |

943 Nearly Singular Integrals | 331 |

944 Strongly Singular Integrals | 333 |

95 Solution of Inhomogeneous Equations | 335 |

96 Computation of the Potential | 336 |

964 Extrapolation | 337 |

972 Panels | 338 |

9733 Admissible Clusters and Admissible Coverings | 339 |

9734 The Algorithm for MatrixVector Multiplication | 340 |

974 The Additional Quadrature Error | 342 |

344 | |

353 | |

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### Common terms and phrases

algorithm ansatz applied Assume assumption Banach space boundary value problem Cauchy principal value closed curve coefficients coincides collectively compact collocation method collocation points computation condition number consistency const convergence defined definition denotes discrete discretisation double-layer potential eigenvalue error estimate example Exercise exists exterior f-fn Fredholm integral equation Galerkin method Hackbusch Hence Holder continuous holds implies improper integral improperly integrable inequality integral operator integrand interior domain interval kernel approximation Lagrange functions Laplace equation Lemma Lipschitz Lipschitz continuous matrix multi-grid method neighbourhood normal derivative Nystrom method obtains operator norm orthogonal projection parametrisation Picard iteration piecewise constant piecewise linear interpolation polynomial projection method Proof Prove the following quadrature error quadrature formula quadrature method Remark representation respect right-hand side satisfies second kind semidiscrete sequence single-layer potential singularity smooth solution f stability sufficiently support abscissae Theorem uniform boundedness unique vector Volterra integral equation yields