Praise be to Allah.
Firstly: a vow (nadhr) to do an act of worship must be fulfilled
If someone swears by Allah that if He grants him such and such, he will fast on Mondays and Thursdays for the rest of his life, then this is a vow to do an act of worship and must be fulfilled, because the Prophet (blessings and peace of Allah be upon him) said: “Whoever vows to do an act of obedience to Allah, let him do that act of obedience, and whoever vows to do an act of disobedience to Him, let him not do that act of disobedience.” Narrated by al-Bukhaari (6202).
Secondly: it is not valid to observe any fast during Ramadan except the Ramadan fast
It is not valid to observe any fast during Ramadan except the Ramadan fast, according to scholarly consensus. So one should not fast in fulfilment of a vow, or to make up a missed fast, or observe a supererogatory (naafil) fast. Even if a man was travelling and did not fast, he does not have the right to observe any fast in Ramadan except the Ramadan fast.
Similarly, it is not valid to fast on the day of either Eid or the days of at-tashreeq (the 11th, 12th and 13th of Dhu’l-Hijjah).
Al-Bukhaari (6212) narrated that Ziyaad ibn Jubayr said: I was with Ibn ‘Umar, and a man asked him: I vowed to fast every Tuesday or Wednesday so long as I live, and this day coincides with the Day of Sacrifice. He said: Allah has enjoined fulfilling vows, but we are prohibited to fast on the Day of Sacrifice. He repeated his question, and he said the same thing again, without adding anything to it.
Al-Bukhaari (1998) narrated that ‘Aa’ishah and Saalim ibn ‘Umar (may Allah be pleased with them) said: No concession was granted allowing fasting on the days of at-tashreeq, except for one who could not find a sacrifice (hadiy).
Al-Haafiz Ibn Hajar said: There is scholarly consensus that it is not permissible to fast on the day of (Eid) al-Fitr or the day of (Eid) al-Adha, whether that is a voluntary fast or a fast in fulfilment of a vow. End quote.
An-Nawawi (may Allah have mercy on him) said in al-Majmoo‘ (6/315): ash-Shaafa‘i and the companions (may Allah have mercy on them) said: Ramadan is only for the fast of Ramadan, and it is not valid to observe any other fast during the month. So if one who is at home or one who is travelling or one who is sick intends to offer a fast as expiation, or in fulfilment of a vow, or to make up a missed fast, or as a voluntary fast, or he makes his intention to fast general [without intending specifically that it be for Ramadan], his intention is not valid, and his fast is not valid, either for the purpose that he intended or for Ramadan. End quote.
Ibn Qudaamah (may Allah have mercy on him) said in al-Mughni: The traveller does not have the right to fast during Ramadan for any other purpose, such as in fulfilment of a vow or to make up a missed fast, because not fasting is permitted as a concession and so as to make things easier for him, so if he does not want to make things easier for himself, he must do that which is originally enjoined [i.e., fasting for Ramadan].
So if he intends to fast for a reason other than Ramadan, his fast is not valid, whether as the Ramadan fast or as whatever he intended. This is the correct view according to our madhhab and it is the view of most of the scholars. End quote.
Thirdly: what should a person do, if he vowed to fast on Mondays and Thursdays, then Ramadan begins?
If someone vowed to fast on Mondays and Thursdays for the rest of his life, then Ramadan comes, he should fast on those two days with the intention of fasting Ramadan, and he does not have to make up those days or offer any expiation. This is the view of ash-Shaafa‘i, and is also narrated from Ahmad.
The reason for that is that these days are not included in the vow, because Ramadan will inevitably include Mondays and Thursdays. This is the explanation given by ash-Shaafa‘i.
Or his fast is valid for both days, as al-Khuraqi said, because it is not possible to fast in fulfilment of the vow on that day, so he does not have to do it.
It says in al-Majmoo‘ (8/482): If he vowed to fast on every Monday, he does not have to make up the Mondays of Ramadan, because he knows that there will inevitably be Mondays in Ramadan, so these days are not included in his vow, and he does not have to make them up.
With regard to those that coincide with the days of Eid, there are two scholarly views:
The first view is that he does not have to make them up. This is the view of al-Muzani, by analogy with the days that happened to be in Ramadan.
The second view is that he must make them up, because he made a vow to fast on days that would not necessarily coincide with the days of Eid, so if they do coincide with the days of Eid, he must make them up. End quote.
It says in Mughni al-Muhtaaj (6/239): If he vowed to fast on Mondays for the rest of his life, he does not have to make up the Mondays of Ramadan, as the month inevitably includes Mondays, of which there are four for certain, because the vow does not include them, as it is already obligatory to fast on those days. End quote.
Al-Mirdaawi said in al-Insaaf (11/137): al-Khuraqi said: It is valid for him to fast for both Ramadan and in fulfilment of his vow. This is mentioned in one report from Imam Ahmad (may Allah have mercy on him), as was narrated by al-Mirwadhi, and Ibn ‘Aqeel stated that it is the correct view in his Tadhkirah. … In al-Fusool he said: He does not have to fast on another day [to make up the fast in fulfilment of his vow], not because his fast in Ramadan suffices for both; rather it is because it is not possible [to fast in fulfilment of his vow in Ramadan]. End quote.
The Permanent Committee issued a fatwa to this effect. It says in their Fataawa (23/268): Question: I have an aunt, my father’s sister, who vowed to fast every Monday and Thursday each week, for as long as she is able to do that, and she is not paralysed or incapacitated. Now, the blessed month of Ramadan is approaching – may Allah grant many blessed returns thereof to us and to you and to the Arab and Muslim nation. What is the ruling on the vow that she made? Please note that she began these fasts after last Ramadan, in 1401 AH. Is every Monday and Thursday of Ramadan counted as something that she must make up, or what is the ruling regarding that? I hope that you can clarify the matter for me, so that I can explain the matter to her in detail, because she asked me to find out, and Allah will help a person so long as he helps his brother. May Allah bless you.
Answer: it is not valid for her to fast on Mondays and Thursdays during Ramadan in fulfilment of her vow, and she does not have to make them up or offer expiation for that.
‘Abdullah ibn Qa‘ood, ‘Abdullah ibn Ghadyaan, ‘Abd ar-Razzaaq ‘Afeefi, ‘Abd al-‘Azeez ibn ‘Abdillah ibn Baaz. End quote.
The view of the Hanbalis is that he must make up the days and offer expiation.
It says in Sharh Muntaha al-Iraadaat (3/477): If he vowed to fast on the day So-and-so arrived, then that person’s arrival coincided with Ramadan, or coincided with the day of Eid, or coincided with the menses of a woman who made such a vow, then it must be made up and expiation must be offered, because it is a vow to which he committed himself and did not fulfil it, like any other vow. End quote.
Fourthly: making a vow to fast one day, which then coincides with a day on which it is prohibited to fast
If that day coincides with Eid al-Fitr, or Eid al-Adha, or the days of at-tashreeq, it is haraam to fast on those days. According to the Hanbalis, it must be made up and expiation offered, for missing the right time.
The Shaafa‘is have two opinions regarding making up that day, the more correct of which is that it does not have to be made up.
It says in Mughni al-Muhtaaj (6/240): The same applies to Eid and the days of at-tashreeq. If any of them falls on a Monday, that fast should not be made up according to the more correct view, by analogy with Mondays in Ramadan.
According to the other view, it should be made up, because Monday would not necessarily coincide with those days [when a person vowed to fast]. End quote.
No expiation is to be offered according to the Shaafa‘is, and this was also narrated in one report from Ahmad. See: al-Insaaf (11/134).
Fifthly: vowing to fast on a day which coincides with the day of ‘Arafah
If that coincides with the day of ‘Arafah, then you should fast with the intention of fulfilling the vow, and there is the hope that you will also have the reward of fasting on the day of ‘Arafah, in sha Allah.
See also the answer to question no. 128256 .
And Allah knows best.