Is it permissible for a wife and children to travel without a mahram, knowing that the flight duration is approximately one hour and fifty minutes, in order to avoid the husband being away from his family for a year?

Most scholars do not permit a woman to travel without a mahram (definition) unless there is a necessity (such as migration). However, some have allowed travel without a mahram when there is no fear of fitna, especially by plane. If your stay achieves a benefit until you can travel to them after six months, then there is no objection to adopting the view that permits their travel without a mahram if it is safe.