We consider the Cauchy problem for a system of partial differential equations of fractional order \({D}_{t}^{\mathcal{B}}\) U(t, x) + \({\mathbb{A}}\) (D)U(t, x) = H(t, x), where U and H are vector-valued functions and the m × m-matrix of differential operators \({\mathbb{A}}\) (D) is triangular with elliptic operators on the diagonal. The main feature of this system is that the vector order \(\mathcal{B}\) has different components βj ∈ (0, 1] which are not necessarily rational. We find sufficient (and necessary in some cases) conditions on the initial function and right-hand side of the equation that guarantee the existence of the classical solution.
