Is the fast valid for someone who forgot the intention for the beginning of Muharram and did not remember it until the adhan of Fajr?

It is permissible to fast a supererogatory fast with an intention made before the dawn Athan (call to prayer) or during the day, as long as one has not consumed anything that breaks the fast. This is based on what was reported from Aisha, may Allah be pleased with her, that the Prophet, peace and blessings be upon him, said, "Indeed, I am fasting," after he asked her if they had anything for him to eat. This indicates the permissibility of intending a supererogatory fast during the day before zawal (midday). As for an obligatory fast, the intention for it must be made during the night.