Can she travel to complete her studies without a mahram because of necessity?
We ask Allah to make you steadfast in adhering to His religion and to protect you from trials both evident and unseen, and to guide your parents to Islam as He guided you.
But if there is a case of necessity that forces her to travel without a mahram, then cases of necessity make permissible that which is ordinarily forbidden. Undoubtedly protecting one’s religious commitment is one of the five necessities; in fact it is the most important and most essential of them, for the sake of which one may sacrifice one’s life and everything one possesses.
The prohibition on a woman travelling without a mahram is a prohibition of the means that may lead to evil, because it may lead to fitnah and immorality. Matters that are forbidden as a means of preventing evil may be permitted in cases of necessity and when there is a definite interest to be served.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
The basic principle is that everything that may be a means that leads to fitnah is not permissible; the means that leads to immorality must be prevented, unless there is an interest that may be served.
End quote from Majmoo‘ al-Fataawa, 15/419
Ibn al-Qayyim (may Allah have mercy on him) said:
It is haraam to be alone with a non-mahram woman, to travel with her, or to look at her unnecessarily, as a preventative measure and so as to block the means that may lead to evil.
End quote from Ighaathat al-Lahfaan, 1/362; Zaad al-Ma‘aad, 4/78; I‘laam al-Muwaqqi‘een, 2/159
Therefore, if a woman needs to travel in order to preserve her religious commitment, and she fears fitnah for herself if she does not travel, and she does not have any mahram to accompany her on her journey, it is permissible for her to undertake the journey, so as to ward off the greater evil that may result if she does not travel.
The scholars have stated that it is permissible for a woman to travel without a mahram in cases of necessity, especially if the necessity is to preserve her religious commitment and for fear of fitnah.
Ibn Qudaamah (may Allah have mercy on him) said in al-Mughni (3/230):
If a captive woman travels after escaping from the clutches of the kuffaar, that is a necessary journey, and no analogy can be drawn between it and a journey undertaken by choice, hence she may set out on her own. End quote.
Al-Haafiz Ibn Hajar (may Allah have mercy on him) said:
Al-Baghawi said: The scholars did not differ concerning (the ruling) that it is not permissible for a woman to travel for any purpose other than that which is obligatory (i.e., Hajj), unless she is with her husband or a mahram, except in the case of a kaafir woman who becomes Muslim in dar al-harb (non-Muslim territory that is hostile to Islam), or a female captive who escapes. Others added: or a woman who becomes separated from her travelling companions and is found by a trustworthy man; it is permissible for him to accompany her until he brings her back to her group. End quote.
Fath al-Baari, 4/76
An-Nawawi (may Allah have mercy on him) said:
Al-Qaadi said: The scholars are unanimously agreed that a woman does not have the right to set out on a journey, except in the case of Hajj and ‘Umrah, unless she is accompanied by a mahram, except in the case of hijrah from dar al-harb (migration from non-Muslim territory that is hostile to Islam), in which case they are unanimously agreed that she is obliged to migrate from there to Muslim territory, even if she has no mahram with her, because her staying in the kaafir land is haraam, if she cannot practice Islam openly and she fears for her religious commitment and her life.
End quote from Sharh an-Nawawi ‘ala Muslim, 9/104
Based on that, there is nothing wrong with you travelling or staying to complete your studies without a mahram, even if you are staying in a kaafir country, so long as that country is safer for you than your own country, and you do not fear that you may become confused about your religion or drift away from it there.
Nevertheless, your husband should try to travel to join you and stay with you, even if that costs him extra money. And if you are able to help him achieve that with some official papers or money, then do so.
And Allah knows best.