What Advanced Topics Follow a Basic Course in Partial Differential Equations?

[Remixex]

OK so i finished my first course of Differential equations at Uni and i have some curious questions
The last equations we solved were PDEs solved with Variation of parameters and having to homogenize the boundary conditions
They were all Sturm-Liouville problems as they called them, we assumed that the function itself u(x,t) could be written via multiplication of 2 functions X(x) and T(t) and then one solved for each.
Stuff such as Heat diffusion, the Wave equation, The Laplace equation in 2D were solvable with these methods
The question is, where does it go from here in therms of DEs? Is there more to it or the rest are only numerical methods to solve more complicated DEs?
Just a question that spawned in my mind, I'm only in second year right now so i still have a long way to go before i get the physics degree I'm after

 

[jackmell]

The question is, where does it go from here in therms of DEs?

Non-linear ones. Keep in mind that we live in a massively non-linear world from the atomic scale to astronomical. The easy linear ones you first study don't do justice in fully describing natural phenomena. Take the pendulum. You study the easy linear one first and can only push the pendulum slightly before the DE fails to describe it's motion. But if you study the non-linear DE you can push the pendulum so much that it rotates completely around and around and the non-linear DE will describe it's motion. The secrets of the Universe lie in non-linear differential equations. Many go through life puzzled about the world around them, not understanding why things are the way they are. If you study non-linear DEs long enough, you'll understand and that in my opinion will give you peace of mind: I may not like what's happening, but at least I know why it's happening. :)

 

[jasonRF]

OK so i finished my first course of Differential equations at Uni and i have some curious questions
The last equations we solved were PDEs solved with Variation of parameters and having to homogenize the boundary conditions
They were all Sturm-Liouville problems as they called them, we assumed that the function itself u(x,t) could be written via multiplication of 2 functions X(x) and T(t) and then one solved for each.
Stuff such as Heat diffusion, the Wave equation, The Laplace equation in 2D were solvable with these methods
The question is, where does it go from here in therms of DEs? Is there more to it or the rest are only numerical methods to solve more complicated DEs?

I agree that nonlinear problems are very important and interesting. They show up in many many areas - physics, biology, etc.

Even for linear equations you will learn additional techniques: the method of characteristics comes to mind as a general technique for wave-like problems, and integral transform techniques are a personal favorite.

However, I think that the more interesting aspects of linear equations are analytical methods to find approximate solutions. One example you have probably seen in intro physics is ray optics; it is a high frequency approximate solution to the wave equation. It is valid when the wavelength is much smaller than the dimensions and radius of curvature of objects. The next level approximation for EM waves is physical optics, etc. Some approximation techniques yield more physical insight that exact solutions, which is why I think they are so useful.

Other examples: variational principles often provide useful methods (you will likely see the Ritz method in quantum mechanics), and in perturbation theory you use solutions to a simpler equation that is "close" to your equation of interest to find an approximate solution (again, you will see in quantum mechanics). Also, those "exact" integral transform techniques I mentioned above often yield integrals that we cannot do analytically - but if a parameter of hte problem is very small or very large (ie observing the scattered wave from an object many wavelengths away) then various asymptotic techniques can apply and yield very useful results that provide insight.

Just a question that spawned in my mind, I'm only in second year right now so i still have a long way to go before i get the physics degree I'm after

Knowing Sturm-Liouville problems will certainly make your upper division EM and Quantum classes easier to understand.

jason

 

[SteamKing]

Separation of variables is one technique which can be used to solve PDEs. Other methods include the use of complex variables and mapping the problem onto certain simple regions, and numerical methods, including finite-difference, finite element, and boundary element methods.

Most real-world PDEs are solved numerically using either finite elements, boundary elements, or a mixture of the two. Problems in elasticity, heat flow, E-M, and fluid flow have all been treated quite well numerically, and, as computers have increased in power, the tools used to solve these problems are becoming available to more people.

 

[jack476]

The question is, where does it go from here in therms of DEs? Is there more to it or the rest are only numerical methods to solve more complicated DEs?
Just a question that spawned in my mind, I'm only in second year right now so i still have a long way to go before i get the physics degree I'm after

A few different ways:
-Analysis, real or complex, which deals with the formal properties of functions and operations on real or complex numbers
-Differential geometry, which I personally found to be the most natural continuation of advanced (multivariable and vector) calculus and DEs.
-Numerical methods (most DEs can be analyzed only with numerical or approximate techniques).
-"Extraordinary" differential equations, based on fractional calculus, which are DEs with order that may be a real or complex number rather than just an integer (ie a "0.7th order DE" or one where the order of the equation at any point is itself a function of the variable of the DE). My personal favorite subject, because it's just so weird. This is more of an area of current research than something you'd be likely to encounter in college.
-Dynamics and chaos theory
-Finite differences and difference or sum equations (differential equations in discrete variables)

PDEs are just the beginning, it only gets more terrifying fun from here on out.