Okay, I totally get what you mean now.
Might wanna change the title slightly to
smooth edges of animated texture.
Using an animated texture, you want to make it fade as it goes closer to the edges.
Kinda like this one.
And as far as I know, I haven't been able to do it, so can't help you here.
But don't take my word for it.
If you ever find out, tell me
You could try to PM those guys that make really good VFX here on Hive if they know of any workarounds, but I don't.
Unless, of course, you're not refering to animated texture.
In that case, just edit the texture in your favourite image processor.