Topic
Integral equation
About: Integral equation is a research topic. Over the lifetime, 41607 publications have been published within this topic receiving 768093 citations.
Papers published on a yearly basis
Papers
More filters
Book•
01 Jan 1989
TL;DR: Inverse Boundary Value Problems (IBV) as discussed by the authors, the heat equation is replaced by the Tikhonov regularization and regularization by Discretization (TBD) method.
Abstract: Normed Spaces.- Bounded and Compact Operators.- Riesz Theory.- Dual Systems and Fredholm Alternative.- Regularization in Dual Systems.- Potential Theory.- Singular Integral Equations.- Sobolev Spaces.- The Heat Equation.- Operator Approximations .-Degenerate Kernel Approximation.- Quadrature Methods.- Projection Methods.- Iterative Solution and Stability.- Equations of the First Kind.- Tikhonov Regularization.- Regularization by Discretization.- Inverse Boundary Value Problems.- References.- Index.
2,323 citations
Book•
01 Jan 1983
TL;DR: In this article, the Riesz-Fredholm theory for compact operators is applied to boundary-value problems for the scalar Helmholtz equation and the time-harmonic Maxwell equations.
Abstract: Preface to the Classics Edition Preface Symbols 1. The Riesz-Fredholm theory for compact operators 2. Regularity properties of surface potentials 3. Boundary-value problems for the scalar Helmholtz equation 4. Boundary-value problems for the time-harmonic Maxwell equations and the vector Helmholtz equation 5. Low frequency behavior of solutions to boundary-value problems in scattering theory 6. The inverse scattering problem: exact data 7. Improperly posed problems and compact families 8. The determination of the shape of an obstacle from inexact far-field data 9. Optimal control problems in radiation and scattering theory References Index.
2,135 citations
TL;DR: A continuous surface charge (CSC) approach is introduced that leads to a smooth and robust formalism for the PCM models and achieves a clear separation between "model" and "cavity" which, together with simple generalizations of modern integral codes, is all that is required for an extensible and efficient implementation of thePCM models.
Abstract: Continuum solvation models are appealing because of the simplified yet accurate description they provide of the solvent effect on a solute, described either by quantum mechanical or classical methods. The polarizable continuum model (PCM) family of solvation models is among the most widely used, although their application has been hampered by discontinuities and singularities arising from the discretization of the integral equations at the solute-solvent interface. In this contribution we introduce a continuous surface charge (CSC) approach that leads to a smooth and robust formalism for the PCM models. We start from the scheme proposed over ten years ago by York and Karplus and we generalize it in various ways, including the extension to analytic second derivatives with respect to atomic positions. We propose an optimal discrete representation of the integral operators required for the determination of the apparent surface charge. We achieve a clear separation between “model” and “cavity” which, together with simple generalizations of modern integral codes, is all that is required for an extensible and efficient implementation of the PCM models. Following this approach we are now able to introduce solvent effects on energies, structures, and vibrational frequencies (analytical first and second derivatives with respect to atomic coordinates), magnetic properties (derivatives with respect of magnetic field using GIAOs), and in the calculation more complex properties like frequency-dependent Raman activities, vibrational circular dichroism, and Raman optical activity.
2,033 citations
TL;DR: In this article, an equation of state is proposed for the mixture of hard spheres based on an averaging process over the two results of the solution of the Percus-Yevick integral equation.
Abstract: An equation of state is proposed for the mixture of hard spheres based on an averaging process over the two results of the solution of the Percus–Yevick integral equation for the mixture of hard spheres. Compressibility and other equilibrium properties of the binary mixtures of hard spheres are calculated and they are compared with the related machine‐calculated (Monte Carlo and molecular dynamics) data. The comparison shows excellent agreement between the proposed equation of state and the machine‐calculated data.
1,894 citations
TL;DR: Here the authors will consider only nonsingular linear integral equations of the first kind, where the known functions h(x), K(x, y) and g(x) are assumed to be bounded and usually to be continuous.
Abstract: where the known functions h(x) , K(x, y) and g(x) are assumed to be bounded and usually to be continuous. If h(x) ~0 the equation is of first kind; if h(x) ~ 0 for a -<_ x ~ b, the equation is of second kind; if h(x) vanishes somewhere but not identically, the equation is of third kind. If the range of integration is infinite or if the kernel K(x, y) is not bounded, the equation is singular. Here we will consider only nonsingular linear integral equations of the first kind:
1,879 citations