Darboux problem for a differential equation with fractional derivative

Abstract

We find conditions for the unique solvability of the problem uxy(x, y) = f(x, y, u(x, y), (D0ru)(x, y)), u(x, 0) = u(0, y) = 0, x ∈ [0, a], y ∈ [0, b], where (D0ru)(x, y) is the mixed Riemann-Liouville derivative of order r = (r1, r2), 0 < r1, r2 < 1, in the class of functions that have the continuous derivatives uxy(x, y) and (D0ru)(x, y). We propose a numerical method for solving this problem and prove the convergence of the method.