ЗАСТОСУВАННЯ МЕТОДУ ПАРАМЕТРИЧНИХ ПОДАНЬ СИНГУЛЯРНИХ ТА ГІПЕРСИНГУЛЯРНИХ ІНТЕГРАЛЬНИХ ОПЕРАТОРІВ ДЛЯ ОТРИМАННЯ МАТЕМАТИЧНИХ МОДЕЛЕЙ ЗАДАЧ ЕЛЕКТРОДИНАМІКИ
DOI:
https://doi.org/10.20998/2222-0631.2024.02(7).04Ключові слова:
параметричні подання, сингулярні та гіперсингулярні інтегральні оператори, системи граничних інтегральних рів- нянь, метод дискретних особливостей, математичні моделі, задачі електродинамікиАнотація
У роботі дається аналіз практики застосування методу параметричних подань сингулярних та гіперсингулярних інтегральних операторів (PR-методу) для розв’язання задач електродинаміки. Цей метод був розроблений у дев’яностих роках двадцятого сторіччя Ю. В. Ганделем і є узагальненням методу парних суматорних та інтегральних рівнянь. Він ґрунтується на застосуванні властивостей псевдодиференціальних сингулярних та гіперсингулярних інтегральних операторів. Застосування цього підходу дозволяє зводити розв’язання крайових задач для рівняння Гельмгольця з граничними умовами Дирихле, Неймана або Робена до систем граничних інтегральних рівнянь. В залежності від по- становки початкових крайових задач отримані системи граничних рівнянь можуть містити інтегральні оператори, які потрібно розглядати у сенсі головного значення за Коші або скінченної частини за Адамаром. Також граничні інтегральні рівняння можуть містити інтеграли зі змінною верхньою межею, а підінтегральні вирази цих рівнянь, окрім регулярної частини, можуть містити доданки зі слабкою логарифміч- ною особливістю, доданки з неусувною особливістю першого роду. Для знаходження наближених розв’язків систем граничних інтегральних рівнянь використовувались алгоритми методу дискретних особливостей. Метод параметричних подань інтегральних операторів дозволив побудувати математичні моделі процесів збудження надрозмірних неперіодичних гребінок структур потоком електронів, розподілу поля власних мод у прямокутному хвилеводному каналі з гребінчастими канавками, електродинамічних процесів у гіротронах, поширення елект- ромагнітних хвиль в аксіально-симетричних хвилеводах, дифракції довільної електромагнітної хвилі на довільній поверхні обертання. Сут- тєві переваги цього підходу полягають у відсутності необхідності проведення додаткової аналітичної роботи при збільшенні подібних еле- ментів структури та практичної нечутливості комп’ютерної реалізації моделі від розмірів елементів структури та їх взаємного розташування. Це робить перспективним застосування PR-методу для створення математичних моделей багатошарових квазіфрактальних структур.
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