Oscillation of Fractional Nonlinear Partial Differential Equations with Continuous Distributed Deviating Arguments

Abstract

In this article, we establish some oscillation criteria for the fractional order partial differential equation with continuous distributed deviating arguments of the form \begin{align*} \dfrac{\partial}{\partial t}\left[r(t)D_{+,t}^{\alpha}\left(u(x,t)-\int_{\gamma}^{\delta}{q_0}(t,\zeta)u(x,\rho(t,\zeta))d\eta(\zeta)\right)\right]&=a(t)\Delta u(x,t)+\int_{c}^{d}p(t,\xi)\Delta u[x,\tau(t,\xi)]d\omega(\xi)\\ -\int_{c}^{d}q(x,t,\xi)g\left(u[x,\sigma(t,\xi)]\right)d\omega(\xi)+f(x,t),~~(x,t)\in G&=\Omega\times \mathbb{R}_+, \end{align*} with subject to the boundary conditions \begin{align*} \dfrac{\partial u(x,t)}{\partial \nu}+\mu(x,t) u(x,t)=\psi(x,t), \hspace{0.15in} (x,t)\in\partial\Omega \times \mathbb{R}_{+} \end{align*} and $u=\chi(x,t)$, $(x,t)\in\partial\Omega\times \mathbb{R}_{+}$. Using the generalized Riccati technique and integral averaging method, new oscillation criteria are obtained.