On an algorithm for solving boundary-value problems for systems of first-order partial differential equations with constant coefficients (Q1912460)

We study linear systems of first-order partial differential equations \[ u_x+ Au_y+ Bu= 0,\tag{1} \] where \(A\) and \(B\) are given constant square matrices and \(u\) is an unknown vector-valued function with values in an \(m\)-dimensional real or complex vector space. We exhibit cases in which the general solution of the system (1) can be expressed in terms of solutions of coupled second-order partial differential equations and the derivatives of these solutions. Then the solution of boundary value problems for the system (1) with a boundary condition of the form \(u= f\) reduces to the successive solution of boundary value problems of the same type for such equations.