When do these methods for PDEs apply?

[logarithmic]

I'm doing a course in PDEs, where the lecturer hasn't really explain when all these methods for solving PDEs are suitable.

How do you decide which method to use to solve PDEs? Can someone explain for which class of PDEs do the following methods work:
- similarity solutions
- separation of variables
- method of characteristics
- Laplace transform
- Fourier transform
- Green's function.

 

[HallsofIvy]

When they work!

For a better answer you need to look into the separation of partial differential equations as "parabolic", "elliptic", or "hyperbolic" equations. (And with variable coefficients you can have equations that are "hyperbolic" for some values of the variables and "parabolic" or "elliptic" for others.)

 

[jasonRF]

Halls of Ivy is of course right, and you should not hesitate to ask your professor for his/her take on it. If they are teaching that class they likely have some real insight into the topic (at least you would hope so).

Here is a stab at a couple of these from someone on the practical side (I am not a mathematician who knows deep theory of pdes. I actually just know very elementary stuff) ...

Separation of variables: As a bare minimum, the pde has to be separable, of course, and the boundaries need to be along constant coordinate surfaces. Thus, if you have a 2-D Laplace equation to solve in the region bounded on the inside by a circle and the outside by a square, this technique will not work! EDIT: Also, since you will almost always need a sum of eigenfunctions to satisfy the boundary conditions, the pde needs to be linear.

Characteristics: I have personally only found these useful when dealing with hyperbolic equations (ie wave equations). This can be a very nice approach for some nonlinear wave problems as well.

Fourier transforms: This can somewhat depend upon what you count as a Fourier transform (complex argument? real argument only? distributions? bounded domain? etc.), but usually this means infinite or semi-infinite in some dimension. Limitation is that this approach will only find solutions that have a Fourier transform! Likewise for Laplace transforms (bilateral? unilateral? etc.) EDIT: forgot to mention that if the pde isn't linear, then the transformed equation is in general not so nice. If the coefficients are not constant in the variable you are transforming, the technique can still be useful but you end up with derivatives after the transform (this is usually only helpful if the resulting pde is of lower order than the original).

Green's functions: The basic idea is that to solve a nonhomogenous pde with what can be a complicated source term, you can instead find the "simple" solution to a relatively simple source (delta function), approximate the complicated source term as a sum of these simple sources, and then sum up the simple solutions just like you did the sources. This of course requires linearity. And the sum is really an integral. Sometimes you actually just care about a delta function source in order to understand some fundamental source (like a simple dipole antenna located above the ground, for example), and you have no intention of using the Green's function in an integral to solve for a more complicated source.

Green's functions can also be useful for turning a differential equation into an integral equation. This is often done in order to apply numerical methods for solving integral equations, such as the method of moments that is often used in electromagnetics. If your prof has any good insight please post it. Thanks!

Good luck,

Jason