Non-regular differential equations and calculations of electromagnetic fields

1. Dirichlet type problems for non-regular differential equations in the half-plane. 1.1. Description of the problem and main results. 1.2. Some auxiliary propositions. 1.3. Cauchy problem for equation (1.1) in the class of analytic functions. 1.4. Dirichlet type problem for equation (1.1). 1.5. Examples -- 2. Riemann-Hilbert problem for a class of non-regular elliptic equations. 2.1. Description of the problem. 2.2. The general solution of equation (2.1). 2.3. Cauchy problem for equation (2.1). 2.4. Riemann-Hilbert problem for analytic functions. 2.5. Investigation of the problem (2.1), (2.2). 2.6. General boundary value problems for equation (2.1). 2.7. Exceptional cases for problem (2.1), (2.2). 2.8. Correct boundary value problems for equation (2.1) in general. 2.9. Conclusions -- 3. Dirichlet type problem for the product of first order differential operators. 3.1. Description of the problem and main results. 3.2. The general solution of non-homogeneous elliptic equation. 3.3. Cauchy problem for equation [symbol] = 0. 3.4. Riemann-Hilbert problem for equation (3.2). 3.5. The existence of solution of the problem (3.2), (3.6), (3.7) 52. 3.6. The uniqueness of solution of the problem (3.2), (3.6), (3.7) 54. 3.7. Dirichlet problem for second order elliptic equations. 3.8. Dirichlet problem for 2m-th order elliptic equations . 3.9. Riemann-Hilbert problem for paired elliptic equation. 3.10. Dirichlet problem for generalized analytic functions. 3.11. Correct boundary value problems for products of first order differential operators. 3.12. Normal solvability of dirichlet-type problem for products of first order differential operators. 3.13. A method to solve cauchy problem for elliptic and hyperbolic equations

4. Propagation of plane periodic electromagnetic waves in stratified medium. 4.1. Introduction. 4.2. Boundary value problem for the system (4.4) - (4.6) in Homogeneous medium. 4.3. A general boundary value problem for the system (4.10) in homogeneous conducting medium. 4.4. Boundary value problems for the system (4.10) in [symbol] consisting of two homogeneous strata. 4.5. On Fredholmity of general boundary value problem for equation (4.10) in two strata medium [symbol]. Harmonic oscillations of electromagnetic waves in multi- strata medium [symbol]. 4.7. Harmonic oscillations of electromagnetic waves in non- homogeneous media -- 5. Calculation of capacitances of cylindrical and spherical capacitors. 5.1. Introduction. 5.2. Invariance of capacitances under conformal mappings. 5.3. Formulae of capacitances with cross-sections bounded by analytic curves. 5.4. Some tests of equality for capacitances. 5.5. A method of definition of capacitance for spherical capacitors. 5.6. Approximative formulae for capacitance. 5.7. Optimal choice of cylindrical capacitor's shape -- 6. Efficient methods of solution of boundary value problems for improperly elliptic equations. 6.1. Introduction. 6.2. Some auxiliary propositions. 6.3. Riemann-Hilbert type problem for a class of improperly elliptic equations. 6.4. Dirichlet type problem for improperly elliptic equations. 6.5. Riemann-Hilbert problem for second-order improperly elliptic equations in simple connected domains. 6.6. Dirichlet type problem for third-order improperly elliptic equations. 6.7. Riemann-Hilbert problem for second-order improperly elliptic equations in the circle. 6.8. Riemann-Hilbert problem for high order improperly elliptic equations. 6.9. Dirichlet type problem for high order improperly elliptic equations. 6.10. Neumann type problem for improperly elliptic equations. 6.11. Poincare problem for Bitzadze equation -- 7. Some classes of improperly elliptic equations. 7.1. Improperly elliptic equations in a class of real-valued functions. 7.2. Two elliptic equations with one unknown function

This book deals with the boundary value problems for non-regular partial differential equations in the half-plane. Efficient methods are developed for resolution of boundary value problems for improperly elliptic equations - based on the theory of analytic functions and functional analysis - having great theoretical and practical importance. In canonical domains (circle, ellipse, half-plane, etc.) explicit formulas for solutions to Riemann-Hilbert and Dirichlet type problems for improperly elliptic equations are obtained. A new approach is proposed for the investigation of the harmonic oscillations of electromagnetic fields in stratified and non-homogeneous media and for the calculation of capacitors bounded by analytic surfaces. The book can be used by professionals in a wide range of specialities, including mathematics, mechanics, physics and many realms of engineering