Национальный исследовательский университет «Высшая школа экономики» (НИУ ВШЭ) сообщил, что математик Иван Ремизов из Нижнего Новгорода нашёл возможность для условно простого решения дифференциальных уравнений второго порядка с переменными коэффициентами. На протяжении почти двух столетий такие уравнения считались нерешаемыми. Между тем, они играют ключевую роль в математике и естественных науках, поскольку используются для описания динамических процессов.
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Исторически ограничение было связано с результатами французского математика Жозефа Лиувилля, который ещё в 1834 году показал, что решения подобных уравнений нельзя выразить через конечное число стандартных операций и элементарных функций. Из-за этого математики были вынуждены либо искать частные решения, либо использовать приближения, что исключало универсальную методику и очень сильно усложняло расчёты. Иными словами, общей формулы, в которую можно просто подставить «циферки» и получить решение, не существовало.
Иван Ремизов предложил новый подход, расширив класс допустимых математических операций. Он не стал спорить с Лиувиллем, а просто добавил в уравнения ещё один математический инструмент — нахождение предела последовательности. Для этого математик воспользовался теорией Чернова и преобразованием Лапласа. Это позволило ему выстроить универсальную формулу, которая формально даёт решение любого уравнения «нерешаемого» класса, обходя классические ограничения теории.
«Суть идеи в том, что сложный, постоянно меняющийся процесс разбивается на бесконечное множество простых шагов. Для каждого такого участка строится свое приближение — элементарный фрагмент, который описывает поведение системы в конкретной точке. По отдельности эти кусочки дают лишь упрощенную картину, но, когда их число устремляется к бесконечности, они бесшовно соединяются в идеально точный график решения», — поясняется в пресс-релизе НИУ ВШЭ.
«Дифференциальные уравнения второго порядка используются не только для моделирования событий реального мира, но и для определения новых функций, которые нельзя задать иным образом. К ним относятся, например, так называемые специальные функции Матье и Хилла, они критически важны для понимания того, как движутся спутники на орбите или протоны в Большом адронном коллайдере».
Чуть более сложным математическим языком об открытии можно прочитать на сайте НИУ ВШЭ. На английском языке работа опубликована полностью во «Владикавказском математическом журнале».
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