Abstract
This paper is devoted to the oscillatory problem in the fractional-order delay differential equations. First, we prove the convergence of the Laplace transform of a fractional operator by the generalized Gronwall inequality with singularity and fractional calculus technique. Then we show that it exhibits oscillation dynamics if the corresponding characteristic equation has no real roots. We further provide other direct and effective criteria depending on the system parameters and fractional exponent. Finally, we carry out some numerical simulations to illustrate our results.
Highlights
Fractional calculus has drawn much attention in various fields of science and engineering over the past few decades [1,2,3]
Delay differential equations are adopted to represent systems with time delay. Such effects arise in many processes, such as chemical processes, technical processes, biosciences, economics, and other branches
Proof Our goal is to prove that Eq (3.1) has no real roots due to Theorem 3.1
Summary
Fractional calculus has drawn much attention in various fields of science and engineering over the past few decades [1,2,3]. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This main advantage makes them useful to model some neglected effects with classical integer-order models. Delay differential equations are adopted to represent systems with time delay. Such effects arise in many processes, such as chemical processes (behaviors in chemical kinetics), technical processes (electric, pneumatic, and hydraulic networks), biosciences (heredity in population dynamics), economics (dynamics of business cycles), and other branches. The basic qualitative theory of these delay differential equations is well established, especially in the linear case (for general references, see [6,7,8,9])
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