The title might seem vague but please bear with me:
I always liked pure mathematics, I believe I understood the purpose of applied mathematics as well, but just until I am finishing up now my thesis. The thesis involves PDEs and numerical schemes. I spent almost now 1.5 year with the subject. I didn't read a lot of papers, but the ones that I consulted give me the impression that they are 'everywhere', and it doesn't seem like there is a purpose (a unique or many) to what is going on, I mean what are applied mathematicians doing anyway?
Are they taking problems from real life / i.e. physics and trying to solve them using the mathematical machinery? Or are they taking some mathematical machinery and looking for fields to apply it? Is there a historical or contextual explanation to the research going on?
More importantly, how do you decide if an idea is quite original and worth publishing in the field ? I can argue that any 'application' of known results proven by other Mathematicians, is just an application and not an original solution?
I assume to believe that, for example, some biologists are doing some research on live creatures and they stumble upon a problem which will take them years to crack just because they didn't have enough background in ODEs or mathematical modeling, so when they solve it they will publish the results, but it is still known to mathematicians that this ODE problem has known solutions. Of course, their publication will contain other results not just the problem solved but the consequence of these results. This is very understandable to me and I have no issue with it.
On the other hand, what I do not understand is how does an applied mathematician 'pick' his problems? say someone who doesn't come from any of 'real life' backgrounds, like physics, chemistry, biology, geology... One who is just doing applied mathematics.