I have changed the domain name of my website but the same hosting company. Google has cached the same website under the old name. Once google spider looks at the new domain name does it see it as duplicate content.
No duplicate trouble unless the old domain name is also resolving. It the old domain name is still active you should place a 301 permanent redirect on the old domain. If the rest of the URL after the domain stays the same for each page, then redirect every URL to it's new version, instead of sending all URLs to the new Home Page.
If you don't need the old domains history, just use the Google removal tool. Once you get an email from Google that the content on the old domain has been removed, upload the content to the new one and you will be 100% safe.