The organization determined the first day of the holy month using astronomical calculations, according to the Iranian Cultural Center in Jakarta.
Based on the calculations, Monday, May 13, will be the first day of Shawwal, marking Eid al-FItr.
It also said the Eid al-Adha will fall on Tuesday, July 20.
Muhammadiyah is a major Islamic non-governmental organization in Indonesia.
Ramadan is the ninth month of the Islamic calendar, observed by Muslims worldwide as a month of fasting, prayer, reflection and community.
Some religious scholars rely on astronomical calculations to determine the beginning of lunar months while most of them believe visual sighting of the moon should be used in this case.