Should I worry about closed form solutions in research?

Question

I am an MSc student studying operations research (specifically bioinvasions) and it seems that a lot of the mathematics in this area focuses quite heavily on PDE's, Optimal Control Theory, and Algorithms/Numerical methods in order to solve the bioeconomic models. Most of this math I don't understand at the moment, but I think that I would have the capacity to learn them once I got into the theory. Now, I haven't taken a course on ODE's or PDE's in quite a while and from what I remember in the classes, we focused quite a lot on closed form solutions and solving various classifications of equations by hand.

What I am concerned about is that I have lost the ability to be able to fluently generate these closed form solutions by hand and what impact this would have on my own research. Is it detrimental that I might not be able to do partial fractions, for example. So, I suppose I am curious how often higher level math in applied research is dependent on the ability to solve equations by hand in closed form or if I should put more of my attention towards algorithms and computational (I mean computational in a coding sense) solutions?

Answer 1

The techniques that you studied in an introductory course in differential equations are all easily automated- tools like Mathematica and Maple can do these computations much more reliably than you could ever do them. You should make use of such tools when appropriate, but you should also make an effort to learn more about the underlying analysis of differential equations and about various numerical methods for solving equations.

Answer 2

I think it is useful to be able to identify situations where you are likely to obtain a closed form solution, even if you are not fluent in the process of obtaining the solution itself. I will not try to estimate the probability of coming across a pde/ode for which you can get the exact solution in closed form.

Answer 3

I am an applied mathematician and I have been applying PDE's, Optimal Control Theory, Finite Element Analysis, and Numerical Analysis to engineering problems and financial analysis for the past 25 years.

IMHO, you need to be able to solve by hand the simpler ODEs and PDEs to understand what is going on in more complicated problems. Specifically, I think you need to be able to solve initial value problems of the following forms:

1. $y'(t) = f(y(t))$ where $t, y(t)$, and $f(y(t),t)$ are real and $f$ is a polynomial,
2. $a y''(t) + b y'(t) + c y(t) = f(t)$ where $a,b,c,t$ and $y(t)$ are reals and $f(t)$ is a polynomial, trigonometric, or exponential function,
3. $y'(t) = A y(t)$ where $A$ is a real valued matrix and $y(t)$ is a real vector, and $t$ is real, and
4. $a u_{xx} + b u_{yy} = c$ where $a, b, c, x, y$, and $u(x,y)$ are real and the value of $u$ is given on the boundary of a rectangular region.

Being able to solve these problems by hand will enable you to understand solutions provided by numerical methods.

I also found that understanding projections in $\mathbb R^n$ (and Hilbert Spaces) was useful for learning optimal control theory. (A real matrix $M$ is called a projection matrix if $M=M^T$ (symmetric) and $M^2=M$ (idempotent).)

Answer 4

I had a mid-career crisis about this 2 years ago, when I realised that I was expending brains and time to come up with closed-form solutions to save on computer runs. With compute becoming cheaper and cheaper, this didn’t seem to be a winning formula.

But see Borwein & Crandall (2013). And for bioeconomics, is there a premium for algorithms that can run quickly on limited compute?