Resonating delay equation

We propose here a delay differential equation that exhibits a new type of resonating oscillatory dynamics. The oscillatory transient dynamics appear and disappear as the delay is increased between zero to asymptotically large delay. The optimal height of the power spectrum of the dynamical trajectory is observed with the suitably tuned delay. This resonant behavior contrasts itself against the general behaviors where an increase of delay parameter leads to the persistence of oscillations or more complex dynamics.

A Specific Method for Solving Fractional Delay Differential Equation via Fraction Taylor’s Series

It is well known that the appearance of the delay in the fractional delay differential equation (FDDE) makes the convergence analysis very difficult. Dealing with the problem with the traditional reproducing kernel method (RKM) is very tricky. The feature of this paper is to gain a more credible approximate solution via fractional Taylor’s series (FTS). We use the FTS to deal with the delay for improving the accuracy of the approximate solutions. Compared with other methods, the five numerical examples demonstrate the accuracy and efficiency of the proposed method in this paper.

Steklov problem of the first class for a fractional order delay differential equation

Методом функции Грина получено решение задачи Стеклова первого класса для линейного уравнения с дробной производной Герасимова-Капуто с запаздывающим аргументом. Доказана теорема существования и единственности задачи. The solution to the Steklov problem with conditions of the first class for a linear delay differential equation with a Gerasimov-Caputo fractional derivative is obtained by Green function method. The existence and uniqueness theorem to the problem is proved.

Asymptotic and Oscillatory Properties of Noncanonical Delay Differential Equations

In this work, by establishing new asymptotic properties of non-oscillatory solutions of the even-order delay differential equation, we obtain new criteria for oscillation. The new criteria provide better results when determining the values of coefficients that correspond to oscillatory solutions. To explain the significance of our results, we apply them to delay differential equation of Euler-type.