Evidence of adequate funds for U.S. university
From my experience, during the application process, you only need to sign the form that you are responsible and have the necessary funds to cover the tuition and living fees. You will need to have this evidence (scholarships, own funds, RA/TA assistantships) only after the admission and mostly for the F-1 visa application.

I would definitely contact the department you are applying to for additional information. If they actually require the firm evidence, you are limited to the following options (as far as I know):


Applying for a loan in your home country. – In this case, the letter stating that you are approved for XX,XXX USD will be enough. Depending on the regulations within the financial institution, you might only have to cover some fees in case you don't need the loan.
Applying for an education loan in the US financial institution (tricky, but possible). 
Borrowing money from friends and family for a short period. Can be the cheapest option; however, it is not a firm evidence of sufficient funds. One might actually require the statements for the three-five months period and it will become obvious that this amount of money is not really natural and probably an instantaneous pump.
In some cases, you might state that you own some property (house) in your country that you will sell in case of admission. It might satisfy the admission committee (though again, asking for financial documents before considering an application is IMHO too much).
Finding a guarantor (preferably in the USA, but maybe in your home country) that states he is responsible for your tuition and living costs. You will have to submit the documents for his financial situation then.


To sum it up, I would definitely confirm with the department the requirements as they seem to be a little harsh. They might also offer some other options, I doubt it's their first case.