Maybe possible, maybe not.

You'll need to hack the database back into the server (google for hack attach Paul Randal)

Then, if that's successful, you'll need to see if you can take a tail log backup. Requires that the database was in full recovery with an unbroken log chain at the time of the disaster.

Then you'll probably have to restore the database again, leaving it in the RECOVERING state, then apply all log backups that were taken between when the full backup you used was taken and the time of the disaster. If any are missing or if there was any break in the log chain you won't be able to restore.