Numerical solution for multi-term fractional delay differential equations

In this paper, a multi-term nonlinear delay differential equation (DDE) of arbitrary order is studied.Adomian decomposition method (ADM) is used to solve these types of equations. Then the existence andstability of a unique solution will be proved. Convergence analysis of ADM is discussed. Moreover, themaximum absolute truncated error of Adomian’s series solution is estimated. The stability of the solutionis also discussed.

The local formula of representation of a solution for a functional differential equation with the mixed initial condition considering perturbations of delays containing in the phase coordinates and in controls

For the perturbed controlled nonlinear delay functional differential equation with the mixed initial condition a formula of the analytic representation of solution is proved in the left neighborhood of the endpoint of main interval. In the formula the effects of perturbations of the delay parameters containing in the phase coordinates and controls, the initial vector, the initial and control functions are detected.

STUDY OF FRACTIONAL ORDER DELAY CAUCHY NON-AUTONOMOUS EVOLUTION PROBLEMS VIA DEGREE THEORY

This work is devoted to derive some existence and uniqueness (EU) conditions for the solution to a class of nonlinear delay non-autonomous integro-differential Cauchy evolution problems (CEPs) under Caputo derivative of fractional order. The required results are derived via topological degree method (TDM). TDM is a powerful tool which relaxes strong compact conditions by some weaker ones. Hence, we establish the EU under the situation that the nonlinear function satisfies some appropriate local growth condition as well as of non-compactness measure condition. Furthermore, some results are established for Hyers–Ulam (HU) and generalized HU (GHU) stability. Our results generalize some previous results. At the end, by a pertinent example, the results are verified.

Asymptotic Stability of Solutions to a Class of Second-Order Delay Differential Equations

We consider a class of second-order nonlinear delay differential equations with periodic coefficients in linear terms. We obtain conditions under which the zero solution is asymptotically stable. Estimates for attraction sets and decay rates of solutions at infinity are established. This class of equations includes the equation of vibrations of the inverted pendulum, the suspension point of which performs arbitrary periodic oscillations along the vertical line.