In responding to this question, here is a statement concerning the issue issued by the Council of Senior Scholars in the Kingdom of Saudi Arabia dated 12/4/1398 AH which says:
Firstly: Whoever lives in a land where the people can distinguish night from day by the rising of the dawn and the setting of the sun, but their day is very long in summer and very short in winter, is obliged to offer the five daily prayers at the times that are known in sharia, because of the general meaning of the verses in which Allah says: “Duly establish the Prayer (Salat) from midday till the darkness of the night, and recite the Qur’an in the early dawn. Verily, the recitation of the Qur’an in the early dawn is ever witnessed (attended by the angels in charge of mankind of the day and the night)” (al-Isra’ 17:78)
“Verily, Salat (the prayer) is enjoined on the believers at fixed hours” (al-Nisa’ 4:103)
And it is proven from ‘Abd-Allah ibn ‘Amr ibn al-‘Aas (may Allah be pleased with him) that the Messenger of Allah (peace and blessings be upon him) said: “The time for Zuhr is when the sun passes its zenith and the shadow of a man is equal in length to him, so long as ‘Asr has not come. The time of ‘Asr is so long as the sun has not turned yellow. The time for Maghrib is so long as the twilight has not yet disappeared. The time for ‘Isha’ is until halfway through the night. The time for Fajr is from dawn, so long as the sun has not yet risen. When the sun rises, then refrain from praying, for it rises between the horns of the Shaytan.”
There are other ahadith which define in word and deed the times for the five daily prayers, but no differentiation is made between long days and short days, or long nights and short nights, so long as the times of prayer can be determined by the signs explained by the Messenger of Allah (peace and blessings be upon him).
With regard to the times for fasting the month of Ramadan, those Muslims who are accountable must refrain from eating, drinking and everything else that breaks the fast on every day of the month from dawn till sunset in their country, so long as the day can be distinguished from the night in their country, and the total of day and night is twenty-four hours. It is permissible for them to eat, drink, have intercourse and so on only during their night, even if it is short.
The sharia of Islam is universal and applies to all people in all lands. Allah says: “and eat and drink until the white thread (light) of dawn appears to you distinct from the black thread (darkness of night), then complete your Sawm (fast) till the nightfall” (al-Baqarah 2:187)
If a person is unable to fast the whole day because it is too long, or because he knows from signs or by experience or on the advice of a trustworthy, skilled doctor, or he is almost certain that that fasting will cause him to die or to become severely ill, or will make his sickness worse or will slow down his recovery from sickness, then he should break his fast, and make up the days that he did not fast during any other month when he can make them up. Allah says: “So whoever of you sights (the crescent on the first night of) the month (of Ramadan i.e. is present at his home), he must observe Sawm (fasts) that month, and whoever is ill or on a journey, the same number [of days which one did not observe Sawm (fasts) must be made up] from other days” (al-Baqarah 2:185)
“Allah burdens not a person beyond his scope” (al-Baqarah 2:286)
“and He has not laid upon you in religion any hardship” (al-Hajj 22:78)
Secondly: The one who lives in a land where the sun does not set during the summer and the sun does not rise during the winter, or he lives in a land where the day lasts for sixth months and the night lasts for six months for example, should offer the five daily prayers during each twenty-four hour period, and he should try to work out their times, based on the closest land to him where the times of the five prayers are distinct from one another, because of what it says in the hadith about the Isra’ and Mi’raaj, that Allah enjoined upon this Ummah fifty prayers every day and night, then the Prophet (peace and blessings be upon him) kept asking his Lord to reduce it until He said: “O Muhammad, they are five prayers each day and night.” (Muslim).
And it was narrated that Talhah ibn ‘Ubayd-Allah (may Allah be pleased with him) said: A man from among the people of Najd, with disheveled hair, came to the Messenger of Allah (peace and blessings be upon him), and we could hear the sound of his voice but we could not understand what he was saying, until he drew close to the Messenger of Allah (peace and blessings of Allah be upon him), and he was asking about Islam. The Messenger of Allah (peace and blessings be upon him) said: “Five prayers each day and night.” He said, Do I have to do any (prayers) other than that? He said, “No, unless you do them voluntarily” (al-Bukhari and Muslim).
It is proven that the Prophet (peace and blessings be upon him) told his companions about the Dajjal, and they said: How long will he stay on earth? He said: “Forty days, a day like a year, a day like a month, a day like a week, and the rest of the days like your days.” We said: O Messenger of Allah, on that day which is like a year, will the prayers of one day be sufficient for us? He said: “No. Work out the time (for prayer).” (Muslim). He did not regard that day that will be like a year as being a single day in which five prayers would be sufficient, rather he enjoined praying five prayers every twenty-four hours, and he commanded them to space them out during the time based on the times on an ordinary day in their country.
So the Muslims in the country asked about here have to define the times for prayer based on the closest country to them in which night is distinct from day, and thus the times for the five daily prayers may be known by their sharia signs (in the closest country) in every twenty-four hour period.
Similarly they have to fast the month of Ramadan, and they should work out the beginning and end of the month of Ramadan, and the times to start and end the fast every day, and the time of dawn and sunset every day, in the closest country to them where night is distinct from day and the total of night and day is twenty-four hours, according to the hadith of the Prophet (peace and blessings be upon him) about the Dajjal that we have quoted above, in which he told his companions how to work out the times of prayer. There is no difference in this case between fasting and prayer. And Allah is the Source of strength. May Allah send blessings and peace upon our Prophet Muhammad and his family and companions.