Very difficult. I agree with Dutch - stick to chemical treatments (glyphosates). Propogation is through a large network of rhizomes. To eradicate the plant the roots need to be killed. Plan on multiple treatments annually before total achieving total eradication.
I've never utilized this but I have read that the Feds will come and spray Japanese knotweed for no charge in many areas under the Invasive Species Act. Local county extension agencies can be contacted for more details.
What the others said. Also, in my experience, I've had an easier time killing it when I applied the herbicide to the plants when they were young, and kept hitting them as they were coming back up, once they got to around a foot or two tall each time.
This will be my second year trying to eradicate my knotweed. Last year I sprayed roundup at least 5 times, with the last spraying in september at 10%. I've ony had very minor success. In fact the largest patch looks as strong and healthy as it did last year.
there is one way that I have seen work. Mow it down when it sprouts in the spring. When it sprouts again, mow it again. Do this about three or four times. From that point on, use a round-up type herbicide as soon as the leaves on the new sprouts open up. It may take a year or two, but this seems to work.
I've been told by folks that is you can fence off the area and let some hogs into it, they will kill it. I saw this once and it seemed to be working. I would be worried about tramp metal, nails, etc. it is around old buildings being ingested by the pigs.
I attended a seminar and this is what was suggested.
1. Cut it down around mid summer when most of the root energy has been used for growth.
2. Allow it to resprout requiring it to draw more energy from the roots.
3. Cover it with something like black plastic for about 2 weeks. This starves the roots that are now in a weakened state.
4. Remove the plastic and whatever resprouts when it is 1-2' use glyphosate on it.
This puts the roots in a very bad state and the glyphosate being a systemic root killer has a much better chance of killing them.