A non well formed numeric value encountered

Because you are passing a string as the second argument to the date function, which should be an integer.

string date ( string $format [, int $timestamp = time() ] )

Try strtotime which will Parse about any English textual datetime description into a Unix timestamp (integer):

date("d", strtotime($_GET['start_date']));

This error occurs when you perform calculations with variables that use letters combined with numbers (alphanumeric), for example 24kb, 886ab ...

I had the error in the following function

function get_config_bytes($val) {
    $val = trim($val);
    $last = strtolower($val[strlen($val)-1]);       
    switch($last) {
        case 'g':
            $val *= 1024;
        case 'm':
            $val *= 1024;
        case 'k':
            $val *= 1024;
    }
    return $this->fix_integer_overflow($val);
}

The application uploads images but it didn't work, it showed the following warning:

Solution: The intval() function extracts the integer value of a variable with alphanumeric data and creates a new variable with the same value but converted to an integer with the intval() function. Here is the code:

function get_config_bytes($val) {
    $val = trim($val);
    $last = strtolower($val[strlen($val)-1]);
    $intval = intval(trim($val));
    switch($last) {
        case 'g':
            $intval *= 1024;
        case 'm':
            $intval *= 1024;
        case 'k':
            $intval *= 1024;
    }
    return $this->fix_integer_overflow($intval);
}

The function fix_integer_overflow

// Fix for overflowing signed 32 bit integers,
// works for sizes up to 2^32-1 bytes (4 GiB - 1):
protected function fix_integer_overflow($size) {
    if ($size < 0) {
        $size += 2.0 * (PHP_INT_MAX + 1);
    }
    return $size;
}

$_GET['start_date'] is not numeric is my bet, but an date format not supported by strtotime. You will need to re-format the date to a workable format for strtotime or use combination of explode/mktime.

I could add you an example if you'd be kind enough to post the format you currently receive.