A nonlocal boundary value problem for a third-order differential equation

The nonlocal boundary value problem
\begin{gather*} w_{xxt}+d(x,t)w_t+\eta(x,t)w_{xt}+a(x,t)w_x+b(x,t)w=f(x,t),\\ w(0,t)=\lambda w(l,t),\quad w_x(0,t)=g_0(t),\quad w(x,0)=\varphi(x),\\ 0<x<l,\quad 0<t<T. \end{gather*}
is studied. The authors present new conditions for solvability, construct a family of approximate solutions, and establish convergence rate of approximate solutions to an exact solution.