Remark on stability of fractional order partial differential equation

Abstract

In this paper, using a fractional order partial derivative with non-singular kernel we investigate, the stability and its generalization on semi-closed and semi-open interval for the solution of a fractional order partial differential equation with the help of an inequality.

In this paper, we will consider the following fractional order partial differential equation

\begin{equation}

\frac{\partial_{\beta,\psi}^{3\alpha}u}{\partial_{\beta,\psi}x^{\alpha}\,\partial_{\beta,\psi}y^{\alpha}\,\partial_{\beta,\psi}z^{\alpha}}= f(x,y,z,u(x,y,z),\frac{\partial^{\alpha}_{\beta,\psi}}{\partial_{\beta,\psi}x^{\alpha}}u(x,y,z),\frac{\partial^{\alpha}_{\beta,\psi}}{\partial_{\beta,\psi}y^{\alpha}}u(x,y,z),\frac{\partial^{\alpha}_{\beta,\psi}}{\partial_{\beta,\psi}x^{\alpha}}u(x,y,z))

\end{equation}

where, $\frac{\partial^{\alpha}_{\beta,\psi}}{\partial_{\beta,\psi}x^{\alpha}}u(x,y,z)$ is the $\psi $ - Hilfer fractional partial derivative [1], with parameter $ 0<\alpha<1 $ and $ 0\leq \beta\leq 1$, $ 0 \leq x \leq a $, \, $ 0 \leq y \leq b $, \,$0 \leq x \leq c $ and $f\in C ([0,a)\times[0,b)\times[0,c)\times\mathbb{B}^{4},\mathbb{B})$ and $(\mathbb{B},\mid.\mid)$ a real or complex Banach space.