I gone through many resources and documentation , still i faced only the errors in removing and adding the new events to fullcalendar .
Also i tried by using refetch events to refresh , still not worked for me .
var auto_refresh=setInterval(function()
{ $("#calendar").fullCalendar('refetchEvents');
},10000);
Also i tried this , it takes the data correctly but events couldnt add to the calendar
var auto_refresh=setInterval(function()
{
var url_event='take_events.php?id='+ id +'$role='+ role;
$("#calendar").fullCalendar('removeEvents');
$("#calendar").fullCalendar('addEventSource', url_event);
},10000);
Here i solved by adding true to the addeventsource
Please note that this will work for only 3. versions of fullcalendar.