In the areas that are situated on extreme latitudes, you mentioned that there are four methods of calculating the times for the Eisha and Fajr prayers. How would these times be used for fasting in Ramadan? It seems that any of the methodologies enumerated could result in 20+ hour fasts, which does not seem reasonable for an entire month of fasting. It may become very difficult for one to fast for 20 hours and more.
Firstly, with regards to fasting, the texts of the Qur’an and Sunnah clearly indicate that the time for its commencement is at true dawn (al-fajr al-sadiq) and it ends at sunset (ghurub). There is also the consensus of the Ummah (ijma) on this.
Secondly, as mentioned in an earlier post, one of the methods of calculating the times of Eisha and Fajr prayers respectively is the method of Aqrab al-Ayyam, which means to calculate the times according to the last day when Dawn did actually set in. (See for details a post on this website in the Salat section).
Now, for example, on the last day when true dawn (al-fajr al-sadiq) set in, it was around 1:20 A.M. This time will remain the beginning time for the fasts throughout the period where the sun fails to descend fully below the western horizon.
Indeed, the fasts may be twenty hours long, but this is something one will have to adhere to. It should be remembered that the duration of the fast on the last day when true dawn did actually appear, was also twenty hours. Now, when true dawn actually does appear (meaning we experience actual fajr al Sadiq), then (obviously) we must begin our fasts from that time, thus there is no alternative but to follow this ruling during the days when we don’t experience the actual appearing of Fajr al-Sadiq, for the duration of fasts in both times is similar.
One should always keep in mind that certain rulings may definitely be difficult to practice upon, but the rewards by Allah in the hereafter are immense.
However, if old and weak people are unable to bear the long fasts, and they fear becoming extremely ill, it would be permitted for them not to keep the fasts in Ramadhan and thereafter make them up (qadha) when the duration of the fasts becomes less.
And Allah Knows Best
[Mufti] Muhammad ibn Adam
Leicester , UK