I am doing my MSc thesis which can be easily characterized by the phrase " building a system of equations". It is a bit of new territory for me because in my graduate course work (mostly in resource and environmental economics), we didn't really discuss modelling systems that in depth. So I would put my job more into the realm of applied mathematics.
So I have created a system of equations (12) on paper that I intend to place into a spatially explicit optimal control framework. Part of the challenge is going to be learning how to code this monster, but that is another problem for another day. Furthermore, I am fairly confident I can't really solve this problem by hand because it involves both space and time.
What I would like to know is how I can kind of conceptually safeguard against my model just being nonsensical. Are there any kinds of mathematical modelling principles that I can abide by to save me some headaches before I invest a lot of time into coding the model only to find out that what I have in theory isn't really reasonable?
Edit: I tried to articulate my problem the best I could, as simple as I could. If what I am asking isn't clear, please let me know and I will correct it.
This is quite a subtle matter to my understanding. Also you might specify your question a bit better to get more solid answers. But this vague question allows for creative answers so here you go: (Most of my background are in applied math so I will speak from my perspective.)
If you are in the realm of Differential Equations, the rules are many and although this is where my research interests lay in, I can't claim to be an expert at such. As a rule of thumb for beginners, (a) check that you have enough initial, boundary conditions; (b) check that you don't have a non-trivial kernel in your operators also (c) check that your equations are well-posed (At least not famously ill-posed lol). Refer to Evans, Lawrence if you have the time for it.
If you not in the realm of differential equations, you basically need to check your solution space is at least non-trivial. How many variables do you have? How many constraints are you putting in? If you are over specifying the constraints how are you going to deal with it? If you are under specifying the constrains how to select your candidate from the resulting pool?
You should also ask: Do you really understand your question and equations? Many nice (and difficult! and effective!) models are but a few if not one or two equations (less the boundary/initial conditions) but they are elegant and beautiful. It is of philosophical and mathematical interests to have such models instead. Math, although it's tempting, might I say, is not about sticking as many equations as possible in a problem. Too many useless information makes the model less trustworthy and have a higher chance introducing some issues that you didn't mean to.
At last a modeling text, although aimed for undergrads, I still enjoyed a lot: Strang, Gilbert Intro to Applied Math. It could be treated with rigor, or could be read over coffee.