All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, it should be mentioned that praying silently in silent prayers [like Thuhr and ‘Asr] or loudly in prayers whose recitation is audible [like Fajr, Maghrib and ‘Isha] is among the Sunan (supererogatory acts of worship following the example of the Prophet ) which do not make the prayer void if one does not observe it, so it is desirable to perform the prayers during the day silently and perform the prayers during the night loudly, but if the praying person does the opposite, his prayer is valid.
If a person misses a loud prayer, like Fajr, he should make it up loudly whether he makes it up at night or in the day. Also, if one misses a silent prayer, he should make it up silently as well. This is the view of the Maaliki and Hanafi Schools and one of the opinions of the Shaafi’i School of jurisprudence.
The Fiqh Encyclopedia reads: “The Maaliki and Hanafi Schools of jurisprudence and Shaafi’i School in their opinion which is the opposite of their most correct opinion, and Abu Thawr and Ibn Al-Munthir, are of the view that what should be considered in regard to the way or recitation in prayer is the time of the missed prayers, so that the making up of the prayer should be according to the way it is performed at its prescribed time, and there is no difference of opinion among these scholars between a person who prays individually or the one who leads the prayer [Imaam]…”
Shaykh Ibn ‘Uthaymeen had chosen this view as he said: “A person should make up for the missed prayer as it is; because making up for the prayer should be according to the manner of performing it at its prescribed time; this is the known rule. Therefore, if one makes up a night prayer by day, he should recite it loudly, and if he were to make up a prayer that is performed during the day at night, then he should recite silently. The evidence for this is the following:
1- The saying of the Prophet, sallallaahu ‘alayhi wa: “Whoever slept through a prayer or forgot it, should offer it whenever he remembers it.” Since the command here refers to the very prayer, then it also refers to the manner of performing that prayer, and the manner of the prayer is to recite out loud if the prayer is basically performed at night, and to recite silently if the prayer is basically performed in the day.
2- The Hadeeth narrated by Abu Qataadah when they overslept and missed the Fajr prayer with the Prophet, sallallaahu ‘alayhi wa sallam, and he said: “And he [i.e. the Prophet] prayed Fajr in the same manner he used to pray it every day.”
3- “Making up for the prayer should be exactly in the same manner it is performed at its prescribed fixed time.” [End quote]
On the other hand, the Shaafi'i School of jurisprudence is of the view that what should be taken into account is the time of making up the prayer and not the time it used to be performed at its prescribed fixed time; so if one makes up a loud prayer in the day, he should pray it silently, and if one makes up a silent prayer by night, he should pray it loudly. However, the Hanbali School of jurisprudence is of the view that if a person makes up for a silent prayer, he should not recite loudly whether he makes it up at night or in the day, and if he makes up a loud prayer in the day, he should not recite it loudly according to the correct view of the School.
However, the first view could be the closest to the truth.
Allaah Knows best.