Dissertation topic in one field and degree in another field? Feeing kind of stuck here. I started out as a math PhD student then switched to engineering. I thought by doing so I would be able to learn new skills and expertise in that particular field. But I quickly realized my advisor had a side interest in a math-ish field that is tangentially related to his main area (maybe that’s THE reason I was hired in the first place) but obviously his knowledge is limited in this new field. 

Somehow I managed to publish 3 papers in engineering/applied math journals (by fluke? I thinkso) but since everything I know about this field is self-taught I don’t feel confident in my knowledge. I don’t know enough math to become a mathematician nor can I claim expertise in this engineering field I am supposedly in and time is running out. 

My biggest fear is actually that the committee members are going to think I have been doodling around doing some unrelated math stuff and reject my dissertation (does this happen? My advisor doesn’t seem worried about this possibility at all). What are my options here really? I suppose if I manage to graduate I will just get a regular job outside of academia, but I do want my degree as I have been here for too long.
You write that you do not feel confident in your knowledge. However, experts in your field have put your knowledge to the test, and deemed it worthy of publication in a journal. Not once, not twice, but three times. Trust those experts. If they think your knowledge is good enough for three journal publications, then the committee members are highly likely to think similarly positively about your dissertation.

A solid background in mathematics is fertile ground for an exceptional career in any engineering discipline. If you stop worrying about the limitations of your knowledge, and instead accept that everyone's knowledge is by definition limited, you have a bright future ahead of you.