"Do you solve engineering problems with use of programming methods, or do you solve engineering problems with the use of programming methods?"
Which one is correct? Are both wrong? If so, what is a grammatical and idiomatic way of expressing this?
I wouldn't use either - they're both too stilted and use too many words to convey the meaning. When's the last time you heard someone say "with the use of"? I'm not sure I ever have.
A simpler and cleaner grammatical and idiomatic way of saying the same thing that I would use is either "I solved problem X using method Y" or "I solved X by using Y".
I suspect you solve problems (generically) by appropriate use of available tools. When you solve a specific problem, you do so by a particular use of those tools. When your attempt to solve a problem fails, it's because the use you made of those tools wasn't suitable.
Notice that I give examples with no article, with the indefinite article "a" and with the definite article "the"; all three can be apt. The answer depends on whether you have a specific use in mind (the one you did this morning) or a general use; see the "Generic/Non-countable" section of: http://www.icaltefl.com/zero-article-in-english-grammar
I would say
Do you solve engineering problems with the use of programming methods ?
is the correct way to phrase the sentence.
I don't have much knowledge on grammar, so sorry I can't help :(
E: But as far as I understand, most of the time a preposition precedes the noun. But, I too would like someone to explain this clearly :3
"..use of iterative numerical analysis methods"
, I just wanted to highlight my real problem so I "cropped" the sentence :)"solve problems with programming methods"
, or"solve problems by programming methods"
, or".. via programming methods"
?