Can You Make Intention to Fast after Fajr?
Question

Should the intention to fast Ramadan be made at night or during the day? If someone tells you at the time of duha (forenoon) that today is Ramadan, should you make it up or not?
Praise be to Allah.Can you make an intention to fast Radaman after Fajr?
It is essential to make the intention to fast the month of Ramadan at night before Fajr. It is not sufficient to start fasting it that day without the intention. Whoever finds out at the time of Duha that today is Ramadan and makes the intention of fasting has to refrain from eating until sunset, and he also has to make up that day. 
Ibn ‘Umar narrated from Hafsah (may Allah be pleased with her) that the Prophet (peace and blessings of Allah be upon him) said: “Whoever does not have the intention of fasting before Fajr, there is no fast for him.” (Narrated by Imam Ahmad, the authors of al-Sunan, Ibn Khuzaymah and Ibn Hibban; they classed it as sahih and marfu’. )
Can I make an intention for voluntary fasting on the same day?
This is with regard to obligatory fasts. With regard to nafl fasts , it is permissible to make the intention to fast on the day, if you have not eaten or drunk or had intercourse after Fajr, because it was proven in the hadith of `Aishah (may Allah be pleased with her) that the Prophet (peace and blessings of Allah be upon him) entered upon her one day at Duha time and said, “Do you have anything (any food)?” She said, “No.” He said, “Then I am fasting.”  (Narrated by Muslim) 
And Allah knows best.