Как отправить данные без перезагрузки страницы и использования action в form?

<form class="message__form" action="mail" method="post">
        <!-- Hidden Required Fields -->
      		<input type="hidden" name="project_name" value="Site Name">
      		<input type="hidden" name="admin_email" value="......@gmail.com">
      		<input type="hidden" name="form_subject" value="Form Subject">
    		<!-- END Hidden Required Fields -->
        <p>
          <label class="visually-hidden" for="name">Ваше имя</label>
          <input class="name" id="name" type="text" name="name" value="" placeholder="Ваше имя" required>
        </p>
        <p>
          <label class="visually-hidden" for="phone">Ваше телефон</label>
          <input class="phone" id="phone" type="text" name="phone" value="" placeholder="Ваше телефон" required>
        </p>
        <button class="button-send button" type="submit" name="button">Отправить</button>
      </form>

$("form").submit(function() { //Change
		var th = $(this);
		$.ajax({
			type: "POST",
			url: "./mail.php", //Change
			data: th.serialize()
		}).done(function() {
			alert("Thank you!");
			setTimeout(function() {
				// Done Functions
				th.trigger("reset");
			}, 1000);
		});
		return false;
	});

<?php
$method = $_SERVER['REQUEST_METHOD'];
//Script Foreach
$c = true;
if ( $method === 'POST' ) {
	$project_name = trim($_POST["project_name"]);
	$admin_email  = trim($_POST["admin_email"]);
	$form_subject = trim($_POST["form_subject"]);
	foreach ( $_POST as $key => $value ) {
		if ( $value != "" && $key != "project_name" && $key != "admin_email" && $key != "form_subject" ) {
			$message .= "
			" . ( ($c = !$c) ? '<tr>':'<tr style="background-color: #f8f8f8;">' ) . "
				<td style='padding: 10px; border: #e9e9e9 1px solid;'><b>$key</b></td>
				<td style='padding: 10px; border: #e9e9e9 1px solid;'>$value</td>
			</tr>
			";
		}
	}
} else if ( $method === 'GET' ) {
	$project_name = trim($_GET["project_name"]);
	$admin_email  = trim($_GET["admin_email"]);
	$form_subject = trim($_GET["form_subject"]);
	foreach ( $_GET as $key => $value ) {
		if ( $value != "" && $key != "project_name" && $key != "admin_email" && $key != "form_subject" ) {
			$message .= "
			" . ( ($c = !$c) ? '<tr>':'<tr style="background-color: #f8f8f8;">' ) . "
				<td style='padding: 10px; border: #e9e9e9 1px solid;'><b>$key</b></td>
				<td style='padding: 10px; border: #e9e9e9 1px solid;'>$value</td>
			</tr>
			";
		}
	}
}
$message = "<table style='width: 100%;'>$message</table>";
function adopt($text) {
	return '=?UTF-8?B?'.Base64_encode($text).'?=';
}
$headers = "MIME-Version: 1.0" . PHP_EOL .
"Content-Type: text/html; charset=utf-8" . PHP_EOL .
'From: '.adopt($project_name).' <'.$admin_email.'>' . PHP_EOL .
'Reply-To: '.$admin_email.'' . PHP_EOL;
mail($admin_email, adopt($form_subject), $message, $headers );