Functional differential equation

[Matterwave]

Homework Statement

Solve:

$$\frac{\delta F[f]}{\delta f(x)}=b(x)f(x)^2F[f]$$

For b(x) a fixed smooth function.

Homework Equations

$$\left.\frac{dF[f+\tau h]}{d\tau}\right|_{\tau=0}\equiv \int\frac{\delta F[f]}{\delta f(x)}h(x)dx$$

The Attempt at a Solution

This isn't a homework problem since I'm self-studying this, but I don't know how to solve this equation. Based off of the analogy from regular ordinary differential equations, I tried the solution:

$$F[f(x)]=\int e^{b(x)f(x)^3/3}dx$$

So, constructing the function derivative, I start with:

$$F[f+\tau h]=\int e^{b(x)(f(x)^3+3f(x)^2\tau h(x)+3f(x)\tau^2 h(x)^2+\tau^3 h(x)^3)/3}dx$$

Taking the derivative, and then taking ##\tau\rightarrow 0## I get:

$$\left.\frac{dF[f+\tau h]}{d\tau}\right|_{\tau=0}=\int \left(b(x)f(x)^2e^{b(x)f(x)^3/3}\right)h(x)dx$$

Therefore:

$$\frac{\delta F[f]}{\delta f(x)}=b(x)f(x)^2e^{b(x)f(x)^3/3}$$

This almost looks right except there's no integral over the exponential term, so that term is not F[f]...I feel like this is close, but certainly not right.

I don't know any other method of trying to solve this problem. Thanks.

 

[Matterwave]

Oh wait, I think I found the solution, I think I just put the integral in the wrong place, but can someone verify this for me? I think the actual solution is:

$$F[f]=e^{\int \frac{b(x)f(x)^3}{3}}dx $$

So that:

$$F[f+\tau h]=e^{\frac{1}{3}\int b(x)(f(x)^3+3f(x)^2\tau h(x)+3f(x)\tau^2 h(x)^2+\tau^3 h(x)^3)}dx$$

And then:

$$\left.\frac{dF[f+\tau h]}{d\tau}\right|_{\tau=0}=\left(e^{\int \frac{b(t)f(t)^3}{3} dt}\int b(x)f(x)^2h(x)\right)dx$$

Giving finally:

$$\frac{\delta F[f]}{\delta f(x)}=b(x)f(x)^2e^{\frac{1}{3}\int b(t)f(t)^3 dt}$$

Which looks right to me.

Does this look right to you guys? I think I might be missing a constant of integration so that the final answer should be more like:

$$F[f]=Ce^{\int \frac{b(x)f(x)^3}{3}}dx $$

Or something? Is there anything special about the constant of integration in this case, or is it just a constant number like the the case of regular ODE's?

 

[haruspex]

If F[] is a solution, can you show that C.F[] is also a solution?

 

[Matterwave]

Yes, I think CF[] is also a solution by just plug-and-chug. That constant will always remain on the outside of all integrals and expressions, so it'll just stay. What I was wondering is, in the case of an ODE, C is a constant number. In the case of a functional ODE can C be something more than a constant? I was thinking maybe it could be a fixed function C(x), but it doesn't seem like that would work since F[f] should be a number, and not a function.

 

[haruspex]

Yes, I think CF[] is also a solution by just plug-and-chug. That constant will always remain on the outside of all integrals and expressions, so it'll just stay. What I was wondering is, in the case of an ODE, C is a constant number. In the case of a functional ODE can C be something more than a constant? I was thinking maybe it could be a fixed function C(x), but it doesn't seem like that would work since F[f] should be a number, and not a function.

Good question. It does see to me that $$\frac{\delta (C(x)F[f])}{\delta f(x)}=C(x)\frac{\delta F[f]}{\delta f(x)}$$

 

[Matterwave]

Good question. It does see to me that $$\frac{\delta (C(x)F[f])}{\delta f(x)}=C(x)\frac{\delta F[f]}{\delta f(x)}$$

Yea I thought that too, but it seems:

$$F[f]=C(x)e^{\int \frac{b(t)f(t)^3}{3} dt}$$

Would return a function, given a function as an input, instead of a number. So it wouldn't be a proper functional. Perhaps it would be a "function valued functional", so to speak...

 

[haruspex]

Yea I thought that too, but it seems:

$$F[f]=C(x)e^{\int \frac{b(t)f(t)^3}{3} dt}$$

Would return a function, given a function as an input, instead of a number. So it wouldn't be a proper functional. Perhaps it would be a "function valued functional", so to speak...

Then maybe I misunderstand the notation. What's the range of the integral? Can you give me a web link for the topic?

 

[Matterwave]

I'm actually reading this problem from a book, it's problem 3-5 in Klauder, J.R. A Modern Approach to Functional Integration. Klauder only gives the problem in the differential form (first equation of my first post), the integration limits are implicit depending on the support of the functions under consideration (I assume ##(-\infty,\infty)## for simplicity, no explicit form for the support of the functions is given). The notion of a functional has been so far a mapping of functions into real numbers. So that ##F[f]## should return a number given a function f, which doesn't depend on the argument of f, but only on the whole function f itself.

 

[haruspex]

I'm actually reading this problem from a book, it's problem 3-5 in Klauder, J.R. A Modern Approach to Functional Integration. Klauder only gives the problem in the differential form (first equation of my first post), the integration limits are implicit depending on the support of the functions under consideration (I assume ##(-\infty,\infty)## for simplicity, no explicit form for the support of the functions is given). The notion of a functional has been so far a mapping of functions into real numbers. So that ##F[f]## should return a number given a function f, which doesn't depend on the argument of f, but only on the whole function f itself.

I managed to find a web item on functional integration, and I now get the idea, roughly. But it leaves me struggling to understand the differential notation in the OP. The LHS looks like it means the change in F[f] for a small change in the value of f at a specific location x. But it's hard to think that would be other than zero in any reasonable F[]. Does Klauder give any clues?

 

[Matterwave]

He gives only the definition in equation 2 of OP. That is the definition for that notation. I think it could equally be as useful to use the notation:

$$\frac{\delta F[f]}{\delta f}(x)$$

Basically you can see that this functional derivative, whatever it is, should be a function of (x) since we are going to integrate over dx on the right hand side of equation 2.