Linear Integral EquationsSpringer Science & Business Media, 6 Ara 2012 - 299 sayfa I fell in love with integral equations about twenty years ago when I was working on my thesis, and I am still attracted by their mathematical beauty. This book will try to stimulate the reader to share this love with me. Having taught integral equations a number of times I felt a lack of a text which adequately combines theory, applications and numerical methods. Therefore, in this book I intend to cover each of these fields with the same weight. The first part provides the basic Riesz-Fredholm theory for equa tions of the second kind with compact opertors in dual systems including all functional analytic concepts necessary for developing this theory. The second part then illustrates the classical applications of integral equation methods to boundary value problems for the Laplace and the heat equation as one of the main historical sources for the development of integral equations, and also in troduces Cauchy type singular integral equations. The third part is devoted to describing the fundamental ideas for the numerical solution of integral equa tions. Finally, in a fourth part, ill-posed integral equations of the first kind and their regularization are studied in a Hilbert space setting. In order to make the book accessible not only to mathematicans but also to physicists and engineers I have planned it as self-contained as possible by requiring only a solid foundation in differential and integral calculus and, for parts of the book, in complex function theory. |
İçindekiler
1 | |
Equations of the First Kind | 15 |
The Riesz Theory | 29 |
Dual Systems and Fredholm Theory | 36 |
Regularization in Dual Systems | 50 |
Potential Theory | 63 |
25 | 72 |
39 | 78 |
Operator Approximations | 142 |
Degenerate Kernel Approximation | 154 |
Quadrature Methods | 168 |
Projection Methods | 184 |
Iterative Solution and Stability | 206 |
Tikhonov Regularization | 243 |
Regularization by Discretization | 259 |
Inverse Scattering Theory | 270 |
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Sık kullanılan terimler ve kelime öbekleri
analytic assume Banach space boundary condition boundary value problem bounded inverse bounded linear operator C¹(D coefficients collectively compact collocation method compact linear operator compact operator constant continuous kernel continuously differentiable Corollary defined degenerate kernel denote dense derive Dirichlet problem double-layer potential ds(y dual system eigenvalues elements error estimate exists exterior Ən(y finite dimensional Fourier Galerkin method given grad H¹(D harmonic function Hence Hilbert space Hölder continuous homogeneous injective integral operator linear system logarithmic mapping matrix Neumann problem Neumann series normed space nullspace obtain orthogonal pointwise convergence projection method proof of Theorem quadrature regularization respect right hand side satisfies scalar product second kind sequence single-layer potential Sobolev spaces solves subspace Theorem 3.4 trigonometric interpolation trigonometric polynomials uniformly Hölder continuous unique solution uniquely solvable weakly singular kernel ди дп