On a mixed partial Caputo derivative and its applications to a hyperbolic partial fractional differential equation

We propose an alternative definition of a mixed partial derivative in the Caputo sense for functions of two variables defined on the rectangle \(P=[0,a]\times [0,b]\) $P=[0,a]\times [0,b]$ ( \(a>0, b>0\) $a>0,b>0$ ). We give an integral representation of functions possessing such a derivative. Moreover, we study the existence and uniqueness of a solution, as well as the Ulam–Hyers type stability of a fractional counterpart of a nonlinear continuous Goursat-Darboux system described by the introduced Caputo derivative. This paper is a continuation of our paper [R. Kamocki, C. Obczyński, On the single partial Caputo derivatives for functions of two variables, Periodica Mathematica Hungarica 87(2), (2023), 324–339].