Generalized Functions Method in Boundary Value Problems for Wave Equations

This monograph presents the method of generalized functions and the method of boundary integral equations for solving nonstationary and stationary boundary value problems for classical hyperbolic equations of mathematical physics and electrodynamics: the wave equation, the KleinGordon equation, the Schrödinger equation and the system of Maxwell equations in spaces of dimension 1, 2, 3. It also discusses the theory of generalized functions for solving hyperbolic equations and systems described by pseudo-differential operators. The monograph studies the processes of shock waves, which is often simply impossible within the framework of the classical theory of differential equations. Generalized solutions of the considered boundary value problems, their regular integral representations and resolving singular boundary integral equations have been constructed, which belong to a new class of boundary integral equations, which can become the subject of a separate study in the field of functional analysis and function theory.

Explores groundbreaking methods for solving boundary value problems in hyperbolic equations using generalized functions Discusses unique approaches to studying shock waves and discontinuous solutions in mathematical physics Advances the knowledge by dedicating the study entirely to hyperbolic equations and boundary value problems

Autorentext

Lyudmila Alexeyeva is a Doctor of Physical and Mathematical Sciences, Professor, and now she is Chief Researcher at the Department of Mathematical Physics and Modeling at the Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan. For over 20 years, she worked as a professor at the Faculty of Mechanics and Mathematics at Al-Farabi Kazakh National University, and taught at the graduate and doctoral levels. She was born in Kaliningrad (1947, USSR, Russian Federation). In 1965, she graduated from high school in Murom, Vladimir Region, with a gold medal and entered the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University. From 1970 to 1973, she studied at a the postgraduate level of Faculty of Mechanics and Mathematics at this university, in the Department of Theoretical Mechanics.

Aigulim Bayegizova is Candidate of Physical and Mathematical Sciences (PhD), Senior Lecturer of the Department of Information Security of the Faculty of Information Technologies of the Eurasian National University named after L.N. Gumilyov, Astana, Kazakhstan, since 2024. She was born in the city Satpayev, Zhezkazgan region, Kazakhstan. In 1977, she graduated from 10th grade of Secondary School No. 19 in Satpayev. Then she entered the Faculty of Mechanics and Applied Mathematics of the Kazakh State University named after S.M. Kirov (Alma-Ata). After graduating from the university in 1982, she worked as an assistant in the Department of Higher Mathematics of the Zhezkazgan branch of the Karaganda Polytechnic Institute (1982-1990). Then she worked as the Head of the Department of Informatics and Computer Engineering of the Zhezkazgan Regional Department of Public Education (1990-1992). From 1998 to 2001, she studied in graduate school at the Department of Computer Engineering and Applied Mathematics of the Karaganda State University named after E.A. Buketov.

Inhalt

Basic Concepts of Generalized Functions Theory: Fundamental Solutions and their Properties.- Integral Equations of Boundary Value Problems for SchrodingerGordonFock Equation: Scattering Amplitude.- Nonstationary Boundary Value Problems for d'Alembert Wave Equation and their Solutions.- Boundary Integral Equations of Nonstationary Boundary Value Problems for the KleinGordonFock Equation.- Generalized Solutions of Transport Boundary Value Problems for Wave Equation.- Fundamental and Generalized Solutions of Maxwell Equations and their Properties.- Generalized Solutions of Stationary Boundary Value Problems for Maxwell Equations.- Generalized Solutions of Nonstationary Boundary Value Problems for Maxwell Equations.