Oscillation for Fractional Partial Differential Equations

Abstract

In this paper, we develop the sufficient criteria for the oscillation of all solutions to the following fractional functional partial differential equation involving Riemann–Liouville fractional derivative equipped with initial and Neumann, Dirichlet and Robin boundary conditions:

$$\begin{aligned} \displaystyle \frac{\partial ^{\alpha } u(x, t)}{\partial t^{\alpha }}=C(t)\triangle u+\displaystyle \sum \limits _{i=1}^{n}P_i(x)u(x, t-\sigma _i)+R(x, t), \end{aligned}$$

(1.1)

where $0<\alpha <1$, $(x, t)\in \Omega \times (0, \infty )$, $\Omega$ is a bounded domain in Euclidean $n-$dimensional space $\mathbb {R}^n$ with a piecewise smooth boundary $\partial \Omega$; $C\in C((0,\infty ),(-\infty ,0]),$$\triangle$ is the Laplacian in $\mathbb {R}^\texttt {n}, P_i\in C(\Omega ,[0,\infty )), R(x,t)\in C(G, (-\infty ,\infty )), \sigma _i\in [0,\infty ), i=1,2,\ldots ,n$.