I have Japanese knotweed growing into my garden from an adjoining neighbour's garden.
Does anyone please have any advice of how to get rid of this awful plant?
I'm in the UK.
Japanese knotweed (Polygonum cuspidatum) and hogweed (Heracleum spp.) are two completely different things. Knotweed kind of looks like bamboo, it has a hollow stem and takes over like crazy. Hogweed is an Apiaceae and has flowers that look a bit like Queen Anne's Lace.
If you've got knotweed, it's awful stuff to get rid of. This fact sheet from the NH DOT describes some of the practices they're supposed to be using to avoid spreading it around. (Too bad they still mow it and spread it around...)
This page describes a technique using tarps to smother it. If you try this, let us know how it works -- I live in fear of the japanese knotweed around us invading our yard...
Since you can't smother it, you can try one of two other approaches. See this page by the Alien Plant Working Group of the U.S. National Park Service:
Do not let any part of the plant get away from you! Bag it and toss it in the dumpster.
I generally try to avoid resorting to chemical warfare. For Japanese knotweed I might make an exception. If you want to go this route, make sure you read the label and follow directions:
You should talk to your neighbor to see if you can work together to get rid of it. If you don't, it will probably migrate back into your yard later anyway.
It will keep coming in as long as your neighbor has it.
Patience and persistence will win, or at least get you to a standoff in the bad-neighbor case. Roundup and digging work. For full-grown plants, the root bole can be quite hard to get out - use a D-handle gardening fork, or even a pickax.
If you don't cut the plant first, Roundup often just stunts the plant, making it produce a lot of small foliage. Dig it out then.
Even if you don't have the wherewithal to dig it all out, just keeping it cut down will slow it down a lot - don't just let it go because you don't have time to get rid of all of it.
This is what I know of as hogweed in the UK: http://en.wikipedia.org/wiki/Hogweed
When people complained of knotweed I assumed they meant something else (but I could be wrong).
Some hogweeds are native, but the Giant one was imported as an ornamental and escaped. From what I remember, it has shallow roots and pulls out fairly easy? I do remember my primary school back in the 80s getting paranoid about it and telling us not to touch it! in reality the problem is only the sap. Some people are allergic to the sap, and it can also result in skin burns/reaction in the sun. So if this is the stuff, be sure to use gloves.