Are Integral Equations Essential for Understanding Physics?

[pantheid]

Hi all, my university is offering a graduate level math course in integral equations for the following semester. I'm not at all familiar with them (I'm assuming they're the opposite of differential equations?), and I'm wondering if you guys think they are at all useful in the field of physics, because I don't recall ever coming across one but it seems like something that could be important.

 

[Physics_UG]

I didn't think there was much there to offer a full course in it. I have never heard it offered as a course before.

 

[micromass]

Can you give us the course contents?

 

[pantheid]

Can you give us the course contents?

Unfortunately, no. The only description is that the course content varies.

 

[micromass]

Unfortunately, no. The only description is that the course content varies.

Then I guess you should talk to the professor who teaches the course.

 

[pantheid]

Then I guess you should talk to the professor who teaches the course.

I like the way you think, but I also just wanted to see if you guys ever use integral equations in your work.

 

[SteamKing]

Integral equations form the mathematical basis for the boundary element method (BEM), which itself grew out of work done in adapting the finite element method (FEM) to the solution of partial differential equations (PDE), like the Laplace, Poisson, and Helmholtz equations. BEMs are useful not only for the solution of problems involving stress and strain, but they have been useful for many years in solving complicated acoustic, aerodynamic, hydrodynamic and electro-magnetic problems. A graduate physicist who plans to work in any of these fields should be familiar with PDEs and integral equations, if for no other reason, to be familiar with how these types of problems are analyzed and solved.

http://en.wikipedia.org/wiki/Integral_equation

http://urbana.mie.uc.edu/yliu/Research/BEM_Introduction.pdf

 

[Physics_UG]

the simple RLC series circuit is modeled by an integro-differential equation.

 

[AlephZero]

Leaving the details (and the "sales and marketing" arguments made by some enthusiasts for one method in preference to another!) I think the basic point is that there are two ways to construct a mathematical model. One is to consider the behavior of the system at each point, which often leads to an ordinary or partial differential equation. The other way is to consider some properties of a finite (or infinite) part of the system, which often leads to an equation involving integrals.

The advantage of the integral equation approach (when it works - for example it often works better for linear problems than nonlinear ones, as SteamKing's list of BEM applications shows) is that the "dimension" of the solution is often reduced by one, i.e. the solution for the whole region is expressed in terms of the behavior on its boundary. That has obvious advantages if the region in infimite. It can also have disadvantages, if trying to express the solution in terms of the boundary is ill-conditioned for physical reasons, independent of the cleverness of the math (for example the transient behavior of a system after the boundary conditions change from one constant state to a different constant state).

I would say integral equations and the numerical methods derived from them have more limited general utility than differential equations, but they are certainly useful for the right type of problems, as well as being interesting mathematics independent of particular applications.

 

[HallsofIvy]

I would not say that "integral equations" are the opposite of "differential equations"! In fact, a standard method of solving integral equations is converting to a corresponding differential equation. And theoretical concepts in differential equations, such as showing that dy/dx=F(x,y), $y(x_0)= y_0$ has a unique solution, can be done by converting to integral equations. (That's because, while the space of all differentiable functions is NOT closed under the operation of differentiation while the space of all integrable functions IS closed under the operation of integration.)