Prerequisites for Studying Proof-based PDE

[Mini-Dow]

Hello QuantNet,

How much knowledge of Ordinary Differential Equations is required for studying theoretical PDE? So far I've taken a course in ODE that is solely computationally based, covering the following:

separable/linear equations, exact differential equations, existence and uniqueness, autonomous equations and stability, second order constant coeff. eqns, 2 by 2 systems, phase plane portraits, matrix exponential, etc.

My question is, does one need to know topics such as Laplace Transform, Fourier Series, Eigenfunctions, etc. in order to succeed in PDE? Any advice is welcome

 

[Daniel Duffy]

Having a good foundation on ODEs (theory, computation) is useful in PDE (theory, computation). In particular, the above transforms allow you to reduce a PDE to a (system of) ODEs etc.

 

[Jacek Podlewski]

I think that some basics of functional analysis and topology (at least in metric spaces) would be very beneficial. I for example hardly saw the whole logic behind Fourier series until I learned the fundamentals of general Hilbert space theory. I don't know how intense and advanced your class will be, but while studying PDE theory you're also likely to come across Sobolev spaces or even touch some distribution theory, which is sort of fun

 

[Daniel Duffy]

I think that some basics of functional analysis and topology (at least in metric spaces) would be very beneficial. I for example hardly saw the whole logic behind Fourier series until I learned the fundamentals of general Hilbert space theory. I don't know how intense and advanced your class will be, but while studying PDE theory you're also likely to come across Sobolev spaces or even touch some distribution theory, which is sort of fun

I think it is overkill at this stage.
Hilbert and Banach formalised a lot of maths and were not really PDE guys.

The golden age of PDE and Mathematical Physics is due to Lagrange, Fourier, Bessel, Poisson etc. And this is the kind of stuff you need in QF as basis. Fluid and heat flow savvy are good backgrounders.

Sobolev spaces are needed in pure FEM; in practice you don't need them, as engineer.

Anyhoo, PDE is a huge area.