Classical solution of the boundary-value problem for hyperbolic equation in half-region

Using method of characteristics the analytical solution of the first problem for the hyperbolic equation $(\partial_t-a^{(1)}\partial_x+b^{(1)})(\partial_t-a^{(2)}\partial_x+b^{(2)})u=f(t,x),$ is under construction. Conditions of the coordination of initial and boundary condition are deduced.