УСЛОВИЕ УСТОЙЧИВОСТИ КОНЕЧНО-РАЗНОСТНОЙ СХЕМЫ ДЛЯ УРАВНЕНИЙ ПАРАБОЛИЧЕСКОГО И ГИПЕРБОЛИЧЕСКОГО ТИПОВ: СРАВНЕНИЕ МЕТОДА КОНЕЧНЫХ ОБЪЕМОВ ДЛЯ УРАВНЕНИЙ ДИФФУЗИИ ДРОБНОГО ПОРЯДКА

В данной работе проведен сравнительный анализ методов конечных разностей и объемов. Данные методы широко известны для уравнений диффузии с целым порядком, но тем не менее недостаточно исследована эффективность данных методов для уравнений диффузии с дробным порядком по времени. Для аппроксимации уравнения использовано определение дробной производной Грюнвальда-Летникова. Для метода конечных разностей получена явная разностная схема и выведено условие устойчивости для разностной схемы с дробным порядком по времени, что также является обобщением для параболических и гиперболических типов уравнений, что ранее было неизвестно для схем с дробным порядком по времени. Представлена явная дискретная форма для решения уравнений субдиффузии в двумерном пространстве с дробным порядком по времени методом конечных объемов. Результаты показывают, что метод конечных разностей демонстрирует высокую точность, тогда как метод конечных объемов лучше подходит для сложных геометрических форм. Эти результаты открывают возможности для дальнейшего развития численных методов в задачах, связанных с моделированием процессов аномальной диффузии.

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