Forced oscillation of conformable fractional partial delay differential equations with impulses

S H Saker,V Sadhasivam,K Logaarasi

doi:10.17951/a.2020.74.2.61-80

Abstract

In this paper, we establish some interval oscillation criteria for impulsive conformable fractional partial delay differential equations with a forced term. The main results will be obtained by employing Riccati technique. Our results extend and improve some results reported in the literature for the classical differential equations without impulses. An example is provided to illustrate the relevance of the new theorems.

Highlights

Many researchers found that fractional differential equations are more accurate in describing the mathematical modeling of systems and processes in the field of chemical processes, electrodynamics of computer medium, polymer rheology, mathematical biology, etc
The applications of fractional calculus to biomedical problems are done in the areas of membrane biophysics and polymer viscoelasticity, where the experimentally observed power law dynamics for current-voltage and stress-strain relationships are concisely captured by fractional order differential equations
We mention to the problem of anomalous diffusion [7, 8], the nonlinear oscillation of earthquake which can be modeled with fractional derivative [6], and fluid-dynamic traffic model with fractional derivatives [10] can be used to eliminate the deficiency arising from the assumption to continuum traffic flow and many other, see [18, 23] and the references they are cited for recent developments in the description of anomalous transport by fractional dynamics

Summary

Introduction

Many researchers found that fractional differential equations are more accurate in describing the mathematical modeling of systems and processes in the field of chemical processes, electrodynamics of computer medium, polymer rheology, mathematical biology, etc. We mention to the problem of anomalous diffusion [7, 8], the nonlinear oscillation of earthquake which can be modeled with fractional derivative [6], and fluid-dynamic traffic model with fractional derivatives [10] can be used to eliminate the deficiency arising from the assumption to continuum traffic flow and many other, see [18, 23] and the references they are cited for recent developments in the description of anomalous transport by fractional dynamics Following this trend, our aim in this paper is to study oscillation properties of partial differential equation of fractional order of the form (1.1). (x, t) ∈ Ω × R+ ≡ G, where Ω is a bounded domain in RN with a piecewise smooth boundary ∂Ω,

RN and

Multiplying both sides by

We consider several cases to estimate

We obtain

Thus we have