Praise be to Allaah.
For one who is praying on his own, the adhaan is Sunnah, but it is prescribed for him to do it in the case of a prayer that is being made up, and he should perform it in full. So if it is adhaan for Fajr prayer, he should say, al-salaatu khayrun min al-nawm (prayer is better than sleep) twice, because it is part of the adhaan that is prescribed (for that prayer). The fact that giving the adhaan is Sunnah for one who is praying alone is the report narrated by al-Bukhaari (609) from Abd al-Rahmaan ibn Abd-Allah ibn Abd al-Rahmaan ibn Abi Sasaah al-Ansaari from his father, who told him that Abu Saeed al-Khudri said to him: I see that you love sheep and the wilderness. When you are with your sheep or in your wilderness and you give the call to prayer, then raise your voice with the call, for no jinn or man or anything within the voice range of the muezzin hears the call, but it will bear witness for him on the Day of Resurrection. Abu Sa'eed said: I heard this from the Messenger of Allaah (peace and blessings of Allaah be upon him).
Shaykh Ibn Uthaymeen (may Allah have mercy on him) was asked: What is the ruling on giving the adhaan and iqaamah for one who is praying on his own?
He replied: Giving the adhaan and iqaamah is Sunnah for one who is praying on his own, and they are not obligatory, because he does not have anyone with him to call with the adhaan. But because the adhaan is remembrance and veneration of Allah, and is a call to himself to prayer and prosperity, (it is Sunnah) and the iqaamah, likewise, is also Sunnah. The fact that it is mustahabb to give the adhaan is indicated by what is said in the hadeeth of Uqbah ibn Aamir (may Allah be pleased with him) who said: I heard the Messenger of Allah (blessings and peace of Allah be upon him) say: Your Lord wonders at a shepherd at the top of a mountain who gives a call to prayer, and Allah says: Look at this slave of Mine giving the adhaan and iqaamah for prayer and fearing Me. I have forgiven My slave and granted him admittance to Paradise.' End quote from Fataawa al-Shaykh Ibn Uthaymeen, 12/161
It says in Fataawa al-Lajnah al-Daa'imah (6/61): When giving the adhaan for Fajr prayer, the muezzin says in the adhaan, al-salaatu khayrun min al-nawm (prayer is better than sleep). If I am on my own and not in a group, should I say al-salaatu khayrun min al-nawm' in the adhaan or not?
Answer: Yes, you should say it, because it makes no difference with regard to the adhaan whether one is giving the adhaan on one's own or with other people around, and because this is part of the phrases of the adhaan that are prescribed for the adhaan of Fajr. End quote.
It is Sunnah to recite out loud in Fajr prayer and in the first two rakahs of Maghrib and Isha', both for the imam and the one who is praying on his own.
So if you pray Fajr on your own, you should recite out loud.
See also the answer to question no. 6130.
And Allah knows best.