I am a mediocre grad student. Yet my supervisor approved a extension of my PhD for another whole year (including funds).

When I left college I came directly to work with him in a highly competitive institution. He offered my few projects, and since, 4 year ago I was an arrogant kid that thought that can solve everything, I chose the toughest one. My field is mathematical physics, working in spherical and rotating fluids.
Well, now I have FINALLY 2 papers in preparation, after zero publication. 1 extra years seems to be ok, I felt secure that everything will run smoothly after years of depression.

When I started to work with him, I was asked to derive a paper of him. At that time the paper looked like "chinese" to me. I derived it almost entirely, and based all my research in some results of that paper.

Checking at my notes from almost 4 years ago. I found out that there was a exclamation mark near one of the equations of my supervisor's paper. I couldn't get one of his results. Keeping track of the implications of that, and now that I am a little more flunet in "chinese" I found out that this mistakes,, leads to HUGE BAD CONSEQUENCES.

For the people who knows about partial differential equations, this is the tipical 3-d ugly equation that, given some approximation and assumptions becomes separable, i.e, easy to solve, with reliable eigenvectors and eigenvalues.

Well, one of the assumptions doesn't lead to separation of variables, therefore the equation is not easy to solve, and the results presented in that paper (at the time, there were very few other papers addressing the problem, with different approaches to solve the equations) are put into a question mark.

I've been working with these equations during my whole PhD believing in the result.

I can learn how to solve 3-d ugly equations, but that's a new thesis project by itself, so far I never done it, it takes a lot of computer programing efforts, and the results may not be publisheable.

I'm under a tremendous anxiety and pressure. I feel like I wasted my youth and time. I have no excuses and I don't know what to say to my advisor. I suppose to derive the paper entirely in the past and I didn't do it perfectly. Which bring the consequences I'm getting now.

I'm sure I'm not gonna get more funds, I also cost quite much as an international student. I've been suffering of depression, but since I got my papers ready to be published, I found a second boost to my spirit. heck I was working quite efficiently the last 4 months and I was willing to work 14-16 hrs a day in order to get my PhD in a year.

Now I feel totally lost, I'm sure about this results, very nasty thoughts are coming to my mind. I've been fighting to get this PhD so much, and suddenly it seems I wont have papers to apply for jobs, and I wont be able even to finish. I am really panicking. I know what I'm talking about, this is a huge problem, and I'm sure about the mistake.

What should I do?
Thanks