Studying the mathematics of dynamic phenomena seems to branch out in an overwhelming number of directions. There are univariate and multivariate varieties, continuous and discrete varieties, deterministic and stochastic varieties, and perhaps more distinctions, from which each combination seems to lead to its own field of theory. As someone self studying math who has the experience of a first course in ODEs, is most of the way through what would be a first course in PDEs, and has occasionally glanced into the field of difference equations, with an eye toward stochastic and financial applications, I am wondering if my further studies might benefit from learning some Time Scale Calculus or Dynamic Equations on Time Scales.
The most basic ideas of the subject don't seem too hard to grasp, but a lot of literature seems orientated toward a "purer" proof-ridden sort of math, borderline graduate-level or researcher-grade material. My interest is more in the camp of computation/conceptual understanding. Is it reasonable to study dynamic phenomena as Dynamic Equations on Time Scales in order to eliminate the need to learn continuous and discrete subjects individually, or is it important to understand the underlying math continuously and discretely before generalizing it? If I learn a few key topics in Time Scale Calculus, can I use the subject to understand what solution methods for continuous problems will port over to discrete problems and vice-versa, i.e., can I solve problems as a "function of time scale" and then plug in whatever time scale I need at the end to arrive at correct results?
It is essential to understand calculus in continuous cases. Time scale is a generalization, but you cannot understand it unless you learn continuous calculus.
2.If I learn a few key topics in Time Scale Calculus, can I use the subject to understand what solution methods for continuous problems will port over to discrete problems and vice-versa, i.e., can I solve problems as a "function of time scale" and then plug in whatever time scale I need at the end to arrive at correct results?
The answer depends on a problem sometimes; it may become complicated; for example, consider the definition of an exponential function. For $T=\mathbb{R}$, if you try to calculate exponential in the time scale set up, it will be complicated.