On the Stability of a Hyperbolic Fractional Partial Differential Equation

In this paper, the ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}-Riemann–Liouville fractional partial integral and the ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}-Hilfer fractional partial derivative are introduced and some of its particular cases are recovered. Using the Gronwall inequality and these results, we investigate the Ulam–Hyers and Ulam–Hyers–Rassias stabilities of the solutions of a fractional partial differential equation of hyperbolic type in a Banach space (B,·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathbb {B}}, \left| \cdot \right| )$$\end{document}, real or complex. Finally, we present an example in order to elucidate the results obtained.