Theory and Applications of Some New Classes of Integral Equations

This book is intended for &tudents, research engineers, and mathematicians interested in applications or numerical analysis. Pure analysts will also find some new problems to tackle. Most of the material can be understood by a reader with a relatively modest knowledge of differential and inte­ gral equations and functional analysis. Readers interested in stochastic optimization will find a new theory of prac­ tical . importance. Readers interested in problems of static and quasi-static electrodynamics, wave scattering by small bodies of arbitrary shape, and corresponding applications in geophysics, optics, and radiophysics will find explicit analytical formulas for the scattering matrix, polarizability tensor, electrical capacitance of bodies of an arbitrary shape; numerical examples showing the practical utility of these formulas; two-sided variational estimates for the pol­ arizability tensor; and some open problems such as working out a standard program for calculating the capacitance and polarizability of bodies of arbitrary shape and numerical calculation of multiple integrals with weak singularities. Readers interested in nonlinear vibration theory will find a new method for qualitative study of stationary regimes in the general one-loop passive nonlinear network, including stabil­ ity in the large, convergence, and an iterative process for calculation the stationary regime. No assumptions concerning the smallness of the nonlinearity or the filter property of the linear one-port are made. New results in the theory of nonlinear operator equations form the basis for the study.

I. Investigation of a New Class of Integral Equations and Applications to Estimation Problems (Filtering, Prediction, System Identification).- 1. Statement of the Problems and Main Results.- 2. Investigation of the Scalar Equations.- 3. Investigation of the Vector Equations.- 4. Investigation of the Multidimensional Equations.- 5. Approximate Solution of the Integral Equations in the Space of Distributions.- 6. Asymptotics of the Spectrum of the Investigated Integral Equations.- 7. General Theorems about Perturbations Preserving the Asymptotics of a Spectrum.- 8. Remarks and Examples.- 9. Research Problems.- 10. Bibliographical Note.- II. Investigation of Integral Equations of the Static and Quasi-Static Fields and Applications to the Scattering from Small Bodies of Arbitrary Shape.- 0. Introduction.- 1. Statement of the Problems and Main Results.- 2. Investigation of a Class of Linear Operator Equations.- 3. Integral Equations of Static Field Theory for a Single Body and Their Applications. Explicit Formulas for the Scattering Matrix in the Problem of Wave Scattering from a Small Body of Arbitrary Shape.- 4. Variational Principles for Calculation of the Electrical Capacitance and Polarizability Tensors for Bodies of Arbitrary Shape and Two-Sided Estimates of the Tensors.- 5. Inverse Problem of Radiation Theory.- 6. Wave Scattering by a System of Small Bodies; Formulas for the Scattering Amplitude; and Determination of the Medium Properties from the Scattering Data.- 7. Research Problems.- 8. Bibliographical Note.- III. Investigation of a Class of Nonlinear Integral Equations and Applications to Nonlinear Network Theory.- 0. Introduction.- 1. Statement of the Problems and Main Results.- 2. Existence, Uniqueness and Stability of Solutions of Some Nonlinear Operator Equations and an Iterative Process to Solve the Equations.- 3. Existence, Uniqueness, and Stability of the Stationary Regimes in Some Nonlinear Networks. Stability in the Large and Convergence in the Nonlinear Networks.- 4. Stationary Regime in a Nonlinear Feedback Amplifier.- 5. Research Problems.- 6. Bibliographical Note.- IV. Integral Equations Arising in the Open System Theory.- 1. Calculation of the Complex Poles of Green’s Function in Scattering and Diffraction Problems.- 2. Calculation of Diffraction Losses in Some Open Resonators.- 3. Some Spectral Properties of Nonselfadjoint Integral Operators of Diffraction Theory.- 4. Research Problems.- 5. Bibliographical Note.- V. Investigation of Some Integral Equations Arising in Antenna Synthesis.- 1. A Method for Stable Solution of an Equation of the First Kind.- 2. Some Results Concerning the General Antenna Synthesis Problem.- 3. Formula for Approximation by Entire Functions.- 4. Nonlinear Synthesis Problems.- 5. Inverse Diffraction Problems.- 6. Optimal Solution to the Antenna Synthesis Problem.- 7. Research Problems.- 8. Bibliographical Note.- Appendix 1. Stable Solution of the Integral Equation of the Inverse Problem of Potential Theory.- Appendix 2. Iterative Processes for Solving Boundary Value Problems.- Appendix 3. Electromagnetic Wave Scattering by Small Bodies.- Appendix 4. Two-Sided Estimates of the Scattering Amplitude for Low Energies.- Appendix 5. Variational Principles, for Eigenvalues of Compact Nonselfadjoint Operators.- Appendix 6. Boundary-Value Problems with Discontinuous Boundary Conditions.- Appendix 7. Poles of Green’s Function.- Appendix 8. A Uniqueness Theorem for Schrödinger Equation.- Appendix 9. Stable Solution of Integral Equations of the First Kind with Logarithmic Kernels.- Appendix 10. Nonselfadjoint Operators in Diffraction and Scattering.- Appendix 11. On the Basis Property for the Root Vectors of Some Nonselfadjoint Operators.- Bibliographical Notes for Appendices.- List of Symbols.- Author Index.