Is it permissible to fast make-up days (for Ramadan) on the second or third days of Eid?

Eid al-Fitr is only one day, which is the first day of Shawwal. The notion that it is three days is a common custom among people, but it does not carry any religious ruling. The Prophet, peace and blessings be upon him, forbade fasting on the day of Eid al-Fitr and the Day of Sacrifice (Eid al-Adha). As for the second and third days of Shawwal, it is not forbidden to fast them; therefore, it is permissible to fast them as a make-up for missed fasts or as voluntary fasting.