Subsequent car starting after initial start for the day

Car starts fine the first time. After driving 15 minutes or so and tuning car off for 15 to 20 minutes, car would not start. It tries to start but acts like it is flodded or not getting gas, After is sets for 20 to 30 minutes it starts up fine. This happened two days in a row. Took it to a mechanic and was told the computer did not show any "faults" and the mechanic tried to duplicate the problem for a week and couldn't. Got the car back but don't trust it. It hasn't failed in three days.