Email Form Not Sending Message - HTML

I have this email code from a website template that I want to use for my site. The problem is that when I click on the "Send message" button, the form simply refreshes and nothing is sent. If there is any text data in the fields, it simply disappears. I have tried to look at possible solutions from this HTML Email Form Not Sending Message but to no avail.

Form HTML

<form name="sentMessage" id="contactForm" method='post' novalidate action="public_html/bin/contact_me.php">
   <div class="control-group form-group">
      <div class="controls">
         <label>Full Name</label>
         <input type="text" class="form-control" name="name" id="name" placeholder="Full Name" required data-validation-required-message="Please enter your name.">
         <p class="help-block"></p>
      </div>
   </div>
   <div class="row">
      <div class="col-md-6">
         <div class="form-group">
            <label>Email Address</label>
            <input type="email" class="form-control" name="email" id="email" placeholder="Email Address" required data-validation-required-message="Please enter your email address.">
         </div>
      </div>
      <div class="col-md-6">
         <div class="form-group">
            <label>Phone Number</label>
            <input type="tel" class="form-control" name="phone" id="phone" placeholder="Phone Number" required data-validation-required-message="Please enter your phone number.">
         </div>
      </div>
   </div>
   <div class="form-group">
      <label>Subject</label>
      <input type="text" class="form-control" name="subject" id="subject" placeholder="Subject" required data-validation-required-message="Please enter your phone number.">
   </div>
   <div class="form-group">
      <label>Message</label>
      <textarea rows="10" cols="100" class="form-control" name="message" id="message" placeholder="Please Enter Your Message Here..." required data-validation-required-message="Please enter your message" maxlength="999" style="height:150px;"></textarea>
   </div>
   <div id="success"></div>
   <div>
      <button type="submit" class="btn btn-two">Send message</button>                          
      <p><br/></p>
   </div>
</form>

PHP Code

<?php
// check if fields passed are empty
if(empty($_POST['name'])        ||
   empty($_POST['phone'])       ||
   empty($_POST['email'])       ||
   empty($_POST['subject'])         ||
   empty($_POST['message'])         ||
   !filter_var($_POST['email'],FILTER_VALIDATE_EMAIL))
   {
    echo "No arguments Provided!";
    return false;
   }

$name = $_POST['name'];
$phone = $_POST['phone'];
$email_address = $_POST['email'];
$subject = $_POST['subject'];
$message = $_POST['message'];

// create email body and send it    
$to = 'info@mywebsite.com'; // PUT YOUR EMAIL ADDRESS HERE
$email_subject = "$subject"; // EDIT THE EMAIL SUBJECT LINE HERE
$email_body = "You have received a new message from your website's contact form.\n\n"."Here are the details:\n\nName: $name\n\nPhone: $phone\n\nEmail: $email_address\n\nMessage:\n$message";
$headers = "From: noreply@your-domain.com\n";
$headers .= "Reply-To: $email_address"; 
mail($to,$email_subject,$email_body,$headers);
return true;            
?>

Js

$(function() {

    $("input,textarea").jqBootstrapValidation({
        preventSubmit: true,
        submitError: function($form, event, errors) {
            // something to have when submit produces an error ?
            // Not decided if I need it yet
        },
        submitSuccess: function($form, event) {
            event.preventDefault(); // prevent default submit behaviour
            // get values from FORM
            var name = $("input#name").val();
            var phone = $("input#phone").val();
            var email = $("input#email").val();
            var message = $("textarea#message").val();
            var firstName = name; // For Success/Failure Message
            // Check for white space in name for Success/Fail message
            if (firstName.indexOf(' ') >= 0) {
                firstName = name.split(' ').slice(0, -1).join(' ');
            }
            $.ajax({
                url: "./bin/contact_me.php",
                type: "POST",
                data: {
                    name: name,
                    phone: phone,
                    email: email,
                    message: message
                },
                cache: false,
                success: function() {
                    // Success message
                    $('#success').html("<div class='alert alert-success'>");
                    $('#success > .alert-success').html("<button type='button' class='close' data-dismiss='alert' aria-hidden='true'>&times;")
                        .append("</button>");
                    $('#success > .alert-success')
                        .append("<strong>Your message has been sent. </strong>");
                    $('#success > .alert-success')
                        .append('</div>');

                    //clear all fields
                    $('#contactForm').trigger("reset");
                },
                error: function() {
                    // Fail message
                    $('#success').html("<div class='alert alert-danger'>");
                    $('#success > .alert-danger').html("<button type='button' class='close' data-dismiss='alert' aria-hidden='true'>&times;")
                        .append("</button>");
                    $('#success > .alert-danger').append("<strong>Sorry " + firstName + " it seems that my mail server is not responding...</strong> Could you please email me directly to <a href='mailto:me@example.com?Subject=Message_Me from myprogrammingblog.com;>me@example.com</a> ? Sorry for the inconvenience!");
                    $('#success > .alert-danger').append('</div>');
                    //clear all fields
                    $('#contactForm').trigger("reset");
                },
            })
        },
        filter: function() {
            return $(this).is(":visible");
        },
    });

    $("a[data-toggle=\"tab\"]").click(function(e) {
        e.preventDefault();
        $(this).tab("show");
    });
});


/*When clicking on Full hide fail/success boxes */
$('#name').focus(function() {
    $('#success').html('');
});

How do I solve this?