change of surname on teir1 dependant visa

Hello,
I am currently on tier 1 dependent in the UK and Indian passport holders. I have my birth surname on my passport and visa. But now I want to change my surname to my husbands surname.
I can easily get a new Indian passport with my new married name, however Indian Commission does not transfer any visas to the new Indian passport.

The question is, how do I go about changing the name in my Tier 1 dependant visa? I can't find any information on UKBA site. Can I just show both my new passport and old passport at the border even though I have different names in them? Or how can I transfer the visa to my new passport and change the name? Or Can I just change it when my next extension?

Thank you,
Tiree.