What is Dawud’s fasting?
“Dawud’s Fasting” means fasting by every other day; one day you perform and the other day you do not. It is called as such because the Prophet Dawud (as) fasted this way. This fasting was named by Prophet Muhammad himself and he stated its importance as, “The best fasting is the one Dawud performed; He fasted one day and gave up fasting for a day.” (Bukhari, Sawm, 56; Muslim, Siyam, 181) Also, the Prophet Muhammad (saw) said, “The most beloved fast to Allah is the one Dawud performed.” (Bukhari, Tahajjud, 7)
Source: Presidency Of Religious Affairs The Turkey, High Board of Religious Affairs FATWAS