Edit:
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?
Hello,
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.
The courses and book you mentions above is for basic engineering, physics problems. I think in your case, it's more about the applications than theorem. But, here are some books that you should check out and it combines a lot of pure math:
Single and Multi Variable Calculus: Calculus, 7ed by James Stewared
Fourier Transform and Analysis: A First Course in Wavelets with Fourier Analysis by Albert Boggess and Francis J. Narcowich (You should have basic knowledge about linear algebra and some Differential Equation in order to understand Fourier Transform)
Linear Algebra: Linear Algebra and its Application by Gilber Strang. More Advance we have, Linear Algebra Done Right by Axler
Differential Equation: Ordinary Differential Equation by Morris Tenenbau and Harry Pollard. More advance: Introduction to Ordinary Differential Equation by Agarwal, or even deeper: Ordinary Differential Equations by Edward L. Ince
Partial Differential Equation: Partial Differential Equation for Scientists and Engineer by Stanley. More advance: Partial Differential Equation by Strauss (he talks about Fourier Transform in this book too), or Partial Differential Equation by Lawrence C. Evans (I love this book because it's very detail)
About reading papers,let says a paper in Mathematics about PDE; it requires that your knowledge must be very deep in PDE in order to understand that paper, not just reading some basic books.