Ok, user Chinny84 made comment that truly helps narrow the focus of my question. Basically, I'm asking for a self-study course of Mathematical Methods.
Thanks to his recommendation I identified a book that is very close to the answer:
Mathematical Methods in the Physical Sciences by Mary L. Boas
Supposedly, the book covers most of the Math a person needs to learn in order to understand the simulations I presented.
Does this book suit my needs?
I have always been interested on the idea to study Math at an Undergraduate level to be able to implement simulations like the followings:
Finite Difference Simulation of the Wave Equation
Heat Equation in Two Dimensions
Besides my precollege experience in Math, I have only been able to do very simple simulations due to my lack of knowledge of Advanced Math. Every time I ask about the inner workings of the simulations I get redirected to papers that look like foreign language to me and, unfortunately, I depend on easy-to-read tutorials to understand them, and while I have found very helpful explanations I'm more interested in reaching the solution myself.
I checked some of the questions related to book recommendations for self-study and while they are very helpful I'm interested in learning by example (There is a book called: "Accelerated C++ Practical Programming by Example" that fits very well my study habits)
Thanks in advance.