Indian Passport Name and Green Card name mismatch- Travel issues?

[pg365]

Hi,

I entered US on K1 visa (fiancee) and changed my last name after marriage. I received my green card but now the last name on green card and on Indian Passport is different. I am planning to travel to India and already booked the tickets using the name thats on the green card. Can I travel and use the ticket that has my new last name and use the Indian passport that has my old name. I have all the documentation regarding the name change.

Please guide through this situation. Whats best for me to do

Change the name on Indian passport or change the name on ticket.