Ode's to PDE's: Essential Knowledge for Advanced Math/Physics

[Visceral]

So I just finished this course. I feel like I haven't learned as much as I need to for PDE's. How much overlap is there from ODE's to PDE's? I would assume a lot.

Basically, I feel like I know a few methods for solving first and second order equations, and also how to do it in a matrix, but that's about it. There are a lot of things in our book that we skipped altogether, such as series solutions to differential equations, introduction to PDE's, Sturmville/Eigenvalue problems, etc...

What is essential to take from ODE's as one travels to PDE's and more advanced math/physics courses?

 

[clope023]

So I just finished this course. I feel like I haven't learned as much as I need to for PDE's. How much overlap is there from ODE's to PDE's? I would assume a lot.

Basically, I feel like I know a few methods for solving first and second order equations, and also how to do it in a matrix, but that's about it. There are a lot of things in our book that we skipped altogether, such as series solutions to differential equations, introduction to PDE's, Sturmville/Eigenvalue problems, etc...

What is essential to take from ODE's as one travels to PDE's and more advanced math/physics courses?

Odd that you skipped series solutions, you'll need the method of frobenius to derive the Bessel function, which is used as a solution method to the wave equation. Sturm-Liouville and eigenvalues too, though you'll probably see some of that in linear algebra. PDE's can be a very self-sustained course and the derivation of the equations themselves isn't that bad if you understand partial derivatives and a little physics. The most ode I used in pde was separation of variables and the characteristic equation tbh, not much else, lots of proofs if your professor goes heavy into Fourier analysis though.

 

[ahsanxr]

This sounds exactly like what happened to me in my ODE course this semester. Our instructor spent so much time on reviewing the linear algebra needed for Systems of DE's that we skipped series solutions and PDE's and Storm-Louville altogether and barely touched Fourier Series/Boundary Value Problems.

 

[snipez90]

Basic knowledge of how to solve ODEs is most essential. Several of the analytical techniques for solving PDEs aim at reducing particular PDEs to ODEs.

Series solutions and eigenvalue problems are going to involve the same basic ideas and methods in either ODE or PDE theory, but obviously you'll need to be a lot more comfortable with vector calculus for PDE (or more importantly, you'll need to get past notation, which is generally a nightmare in PDE theory, but seems worse for series solutions). It might be a good idea to skim these topics now (within the context of ODEs) to brush up on the relevant analysis concepts and theorems (notably some power series methods from complex analysis and perhaps some basic Fourier and functional analysis for eigenvalue problems) before heading into PDE theory.

 

[Visceral]

Basic knowledge of how to solve ODEs is most essential. Several of the analytical techniques for solving PDEs aim at reducing particular PDEs to ODEs.

Series solutions and eigenvalue problems are going to involve the same basic ideas and methods in either ODE or PDE theory, but obviously you'll need to be a lot more comfortable with vector calculus for PDE (or more importantly, you'll need to get past notation, which is generally a nightmare in PDE theory, but seems worse for series solutions). It might be a good idea to skim these topics now (within the context of ODEs) to brush up on the relevant analysis concepts and theorems (notably some power series methods from complex analysis and perhaps some basic Fourier and functional analysis for eigenvalue problems) before heading into PDE theory.

So basically, I should go over essentially what my prof skipped in that course? I've already taken linear algebra, too...so I don't know if that will change anyone's opinions on what I should brush up on.

You are saying PDE "theory"... I think the courses I am taking are not so much involved in theory. Here is the course description for the PDE class I am taking next semester, MA 501.

http://www2.acs.ncsu.edu/reg_records/crs_cat/MA.html

From what I hear, its essentially the same as MA 401, but "grad level", so slightly harder I think.

 

[Chunkysalsa]

My class didnt even do systems of equations. Just First/second/higher order methods and the laplace transform. The laplace transforms is useful but like half of my EE textbooks have a chapter on it lol.

 

[snipez90]

You are saying PDE "theory"... I think the courses I am taking are not so much involved in theory. Here is the course description for the PDE class I am taking next semester, MA 501.

http://www2.acs.ncsu.edu/reg_records/crs_cat/MA.html

From what I hear, its essentially the same as MA 401, but "grad level", so slightly harder I think.

Okay I see. I don't think you necessarily have to do preparation beforehand then. When I referred to series solutions, I meant general power series methods for differential equations that typically cannot be solved using the basic ODE techniques. On the other hand, Fourier series solutions are typically treated within the context of separation of variables. It looks like your course will cover both topics though.

 

[Visceral]

Okay I see. I don't think you necessarily have to do preparation beforehand then. When I referred to series solutions, I meant general power series methods for differential equations that typically cannot be solved using the basic ODE techniques. On the other hand, Fourier series solutions are typically treated within the context of separation of variables. It looks like your course will cover both topics though.

From what I have heard from others who have taken the course, we basically go through all or almost all of Partial Differential Equations with Fourier Series and Boundary Value problems by Nakhle Asmar.