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В работе исследуется дифференциальное уравнение второго порядка с дробной производной в младшем члене, при этом порядок производной находится в диапазоне от нуля до двух. Данная модель используется для описания колебательных процессов в вязкой среде. Разработан принципиально новый метод для приближенного решения первой краевой задачи для уравнения колебания струны с учетом трения в среде с фрактальной геометрией., The paper investigates a second-order differential equation with a fractional derivative in the lower term, in which the order of the fractional derivative is in the range from zero to two and is not known in advance. This model is used to describe oscillatory processes in a viscous medium. A fundamentally new method has been developed for the approximate solution of the first boundary-value problem for the equation of string vibration taking into account friction in a medium with fractal geometry., №4(47) (2020)

In this paper, we study the difference method of order (2-α) of the accuracy of the solution of the Cauchy problem for a differential equation with the Riemann-Liouville fractional differentiation operator. A difference approximation is obtained for the fractional Riemann-Liouville derivative of order (2-α) of accuracy. It is shown that the difference scheme approximates the differential equation (2-α) by the order of accuracy. It is shown that the difference scheme approximates the differential equation (2-α) by the order of accuracy, and the sum of the coefficients is less than or equal to one. An error estimate is obtained for the proposed numerical method for solving the Cauchy problem. For the error of the method under the conditions that if the right-hand side of the equation satisfies the Lipschitz condi-tion with constant А in the second argument and , an estimate is obtained from which the conver-gence of the proposed numerical method for solving the Cauchy problem for the ordinary differential equation with the Riemann-Liouville fractional differentiation operator follows. The proposed numerical method for solving the Cauchy problem has a higher accuracy order than the numerical methods proposed earlier.

Рассматривается задача расчёта температурных полей в составном полубесконечном теле. В качестве модели принято уравнение теплопроводности с дробной производной, которая учитывает нелокальность тепловых процессов по времени. На границе полосы и полупространства предполагается идеальный тепловой контакт. Задача решена с помощью преобразования Лапласа. На основании тауберовых теорем получено асимтотическое решение для малых времён. Решение записано в виде обобщённых рядов. Temperature fields in extended solids and the solids that have symmetry can be described with a high degree of accuracy by one-dimensional heat conduction equation. In this scientific paper the half-space with the layer is simulated using one-dimensional compound bar and consideration is given to the problem of calculation of temperature fields in the compound semi-infinite bar the finite and semi-infinite parts of which have different thermal physical parameters. The heat conduction equation with the fractional derivative in time that takes into account the nonlocality of thermal processes in time is taken as a mathematical model.