Answer
Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
Fasting Ashura without intention from the night before is valid as long as nothing that breaks the fast was consumed after dawn. However, the reward for the full day is not granted; instead, the person receives reward only from the time of intention. This is based on the hadith, “Actions are but by intentions,” and the scholarly explanation that applies to voluntary fasts such as Ashura.
With regard to your question about forming the intention on the same day, we would like to state the following:
The Validity of Intention Formed After Dawn
There is sound evidence from the Prophet (peace and blessings be upon him) that the intention to observe a voluntary fasting that one forms n the morning of the day itself is valid, as long as the person has not ingested anything that breaks the fasting after dawn.
Aishah narrated that the Prophet entered upon his family one day and said: “Do you have anything (i.e., any food)?” They said, “No.” He said, “Then I am fasting.” (Muslim)
Difference Between Voluntary and Obligatory Fasting Intentions
The word idhan (translated here as “then,”) indicates that it is permissible to form an intention to observe a voluntary fast on the day itself. This is unlike the obligatory fasts, which are not valid unless one forms the intention the night before. The Prophet said; “Whoever does not form the intention to fast before dawn, has not fasted.” (Abu Dawud and At-Tirmidhi)
This hadith refers to obligatory fasts. So, your fast is valid.
With regard to the question about whether you will receive the reward for fasting for the whole day or from the time of the intention only, the late Saudi Muslim scholar, Sheikh Ibn Uthaymeen, said:
Will You Be Rewarded for the Whole Day?
There are two scholarly opinions on this matter. The first is that he will get the reward from the beginning of the day, because the correct Islamic fasting is only from the beginning of the day.
The second view is that he will get a reward from the time of his intention. If he formed the intention at midday, then he will be rewarded for half a day.
This is the correct view because the Prophet said, “The reward of deeds depends upon the intentions and every person will get the reward according to what he has intended.” (Al-Bukhari)
This man only formed his intention during the day, so he will be rewarded from the time he formed the intention.
Based on the correct opinion, if a fast is connected to a particular day, such as fasting on Mondays and Thursdays, and fasting on the 13th, 14th and 15th of each Hijri month or fasting three days each month, and the person forms his intention during the day, he will not be rewarded for the entire day. (Al-Sharh al-Mumti`, 6/373)
Ruling on Fasting Ashura Without Intention at Dawn
The same applies to the case of one who did not intend to fast Ashura until after dawn had come. In this case, he will not receive the reward for fasting Ashura, which is the expiation of one year, because his case is not like that of the person who fasted the whole day. His fast is valid from the moment he first formed his intention to fast.
Allah will reward him in the general sense of the reward for fasting during the month of Muharram, which is the best kind of fasting after Ramadan. (Al-Sharh al-Mumti`, 6/471)
Allah Almighty knows best.
Source: www.islamqa.info