Linear Integral EquationsSpringer Science & Business Media, 06.12.2012 - 367 Seiten In the ten years since the first edition of this book appeared, integral equations and integral operators have revealed more of their mathematical beauty and power to me. Therefore, I am pleased to have the opportunity to share some of these new insights with the readers of this book. As in the first edition, the main motivation is to present the fundamental theory of integral equations, some of their main applications, and the basic concepts of their numerical solution in a single volume. This is done from my own perspective of integral equations; I have made no attempt to include all of the recent developments. In addition to making corrections and adjustments throughout the text and updating the references, the following topics have been added: In Sec tion 4.3 the presentation of the Fredholm alternative in dual systems has been slightly simplified and in Section 5.3 the short presentation on the index of operators has been extended. The treatment of boundary value problems in potential theory now includes proofs of the jump relations for single-and double-layer potentials in Section 6.3 and the solution of the Dirichlet problem for the exterior of an arc in two dimensions (Section 7.6). The numerical analysis of the boundary integral equations in Sobolev space settings has been extended for both integral equations of the first kind in Section 13.4 and integral equations of the second kind in Section 12.4. |
Inhalt
1 | |
4 | |
Bounded and Compact Operators | 15 |
Riesz Theory | 28 |
Regularization in Dual Systems | 55 |
Potential Theory | 75 |
Singular Integral Equations | 94 |
Problems | 120 |
Degenerate Kernel Approximation | 175 |
Problems | 192 |
Projection Methods | 223 |
Iterative Solution and Stability | 247 |
Problems | 260 |
Equations of the First Kind | 269 |
Tikhonov Regularization | 291 |
Regularization by Discretization | 308 |
Sobolev Spaces | 129 |
The Heat Equation | 153 |
Operator Approximations | 163 |
Inverse Boundary Value Problems | 320 |
347 | |
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adjoint assume Banach space best approximation boundary condition boundary value problem bounded inverse bounded linear operator Cauchy-Schwarz inequality collocation method compact linear operator compact operator constant continuous kernel continuously differentiable Corollary defined Definition degenerate kernel denote dense density Dirichlet problem double-layer potential ds(y dual system eigenvalues element error estimate exists finite finite-dimensional following theorem ƒ³ given grad harmonic function Hence Hilbert space Hölder continuous homogeneous ill-posed implies injective integral equation integral operator interpolation iteration Lemma linear space linear system mapping matrix maximum norm Neumann problem normed space nullspace numerical Nyström method obtain orthogonal pointwise convergence projection method projection operator proof of Theorem quadrature regularization respect Riesz right-hand side satisfies scalar product second kind sin² single-layer potential Sobolev spaces solvable solving subset subspace unique solution weakly singular kernels ди