Author | Hackbusch, Wolfgang. author |
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Title | Integral Equations [electronic resource] : Theory and Numerical Treatment / by Wolfgang Hackbusch |
Imprint | Basel : Birkhรคuser Basel, 1995 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-9215-5 |
Descript | XIV, 362 p. online resource |
1 Introduction -- 1.1 Integral Equations -- 1.2 Basics from Analysis -- 1.3 Basics from Functional Analysis -- 1.4 Basics from Numerical Mathematics -- 2 Volterra Integral Equations -- 2.1 Theory of Volterra Integral Equations of the Second Kind -- 2.2 Numerical Solution by Quadrature Methods -- 2.3 Further Numerical Methods -- 2.4 Linear Volterra Integral Equations of Convolution Type -- 2.5 The Volterra Integral Equations of the First Kind -- 3 Theory of Fredholm Integral Equations of the Second Kind -- 3.1 The Fredholm Integral Equation of the Second Kind -- 3.2 Compactness of the Integral Operator K -- 3.3 Finite Approximability of the Integral Operator K -- 3.4 The Image Space of K -- 3.5 Solution of the Fredholm Integral Equation of the Second Kind -- 4 Numerical Treatment of Fredholm Integral Equations of the Second Kind -- 4.1 General Considerations -- 4.2 Discretisation by Kernel Approximation -- 4.3 Projection Methods in General -- 4.4 Collocation Method -- 4.5 Galerkin Method -- 4.6 Additional Comments Concerning Projection Methods -- 4.7 Discretisation by Quadrature: The Nystrรถm Method -- 4.8 Supplements -- 5 Multi-Grid Methods for Solving Systems Arising from Integral Equations of the Second Kind -- 5.1 Preliminaries -- 5.2 Stability and Convergence (Discrete Formulation) -- 5.3 The Hierarchy of Discrete Problems -- 5.4 Two-Grid Iteration -- 5.5 Multi-Grid Iteration -- 5.6 Nested Iteration -- 6 Abelโs Integral Equation -- 6.1 Notations and Examples -- 6.2 A Necessary Condition for a Bounded Solution -- 6.3 Eulerโs Integrals -- 6.4 Inversion of Abelโs Integral Equation -- 6.5 Reformulation for Kernels k(x,y)/(x-y)? -- 6.6 Numerical Methods for Abelโs Integral Equation -- 7 Singular Integral Equations -- 7.1 The Cauchy Principal Value -- 7.2 The Cauchy Kernel -- 7.3 The Singular Integral Equation -- 7.4 Application to the Dirichlet Problem for Laplaceโs Equation -- 7.5 Hypersingular Integrals -- 8 The Integral Equation Method -- 8.1 The Single-Layer Potential -- 8.2 The Double-Layer Potential -- 8.3 The Hypersingular Integral Equation -- 8.4 Synopsis: Integral Equations for the Laplace Equation -- 8.5 The Integral Equation Method for Other Differential Equations -- 9 The Boundary Element Method -- 9.1 Construction of the Boundary Element Method -- 9.2 The Boundary Elements -- 9.3 Multi-Grid Methods -- 9.4 Integration and Numerical Quadrature -- 9.5 Solution of Inhomogeneous Equations -- 9.6 Computation of the Potential -- 9.7 The Panel Clustering Algorithm