Say there is thermo-conductive string which can be described by a plane curve f(x, y)=0. Heat cannot diffuse to ambient but only along the curve. How to describe this heat equation on the curve?
I google it and it seems the Laplacian is now called Laplace-Beltrami operator. There is also a thesis on this topic but it does not list the plane curve case explicitly (see esp. sections 2.5 and 5.1): https://www.diva-portal.org/smash/get/diva2:652933/fulltext01.pdf.
Can you not just solve the heat equation in 1D for an interval with the usual methods and then map it to the curve in 2D? I don’t see why the curvature or surroundings of the string matter (per your description) or did you leave out some additional complications?
So you suggest the curvature of the curve has nothing to do with the heat diffusion so that I can simply project the flat 1D solution to any smooth plan curve? If so, it will be no new physics.
Then if heat can diffuse into ambient, I suppose there will be some interference if two points are far apart along a curve but their normal distance is very close?
Doesn’t the Laplace–Beltrami operator on a curve simplify to the ordinary 1d Laplace operator if you parameterize the curve by arc length?
But things may be more complicated if you are not talking about a mathematical curve, but a physical wire (of nonzero diameter) that you want to approximate by a 1d equation along the curve. As you bend a physical wire, I suspect that it changes the corresponding 1d approximate heat-flow equation depending on the curvature. But this is not about differential geometry, it is about the difference in Laplace solutions for a cylinder vs. a torus, as well as distortions in the material properties by strain (and there are other effects in nanoscale wires) — I’m guessing that there are corrections of order (d/R)^2, where d is the wire diameter and R is the radius of curvature (squared because it shouldn’t depend on the sign of R?).
Yes, this is my main worry. But I cannot think through what is important. Section 5.1 of Ref[1] seems solving a similar problem (steady state instead of time-dependent). But he did not use the flat 1D approach. I don’t quite follow how he exactly did it.
That reference still seems to be dealing with mathematical curves of infinitesimal thickness, just doing it by embedding it in higher dimensions?
To get the lowest-order geometrical curvature correction for a finite-thickness wire numerically, you could implement numerical solutions of heat flow around a torus as a function of radius (this could be done with 2d axisymmetric calculations for 3d wires, or 1d axisymmetric calculations for 2d wires, exploiting linearity to expand non-axisymmetric flows in Fourier series to reduce them to a set of axi-symmetric calculations). For 2d annular domains you can probably solve everything analytically (e.g. by conformal mapping or Bessel expansions). By comparing the difference in heat-flow rate (e.g. the rate at which a sinusoidal heat variation damps out) to a straight wire, you should hopefully be able to pull out a lowest-order correction depending only on d/R for thermal variations much slower than d. If you search “heat conduction torus” there are a bunch of hits on e.g. google scholar, so hopefully someone has already worked this out.
But to work out the effect of strain you probably need empirical data for your material — if you google “strain-dependent thermal conductivity” you get lots and lots of papers. (If you’re lucky maybe this will be negligible for typical materials and d/R values? You’d have to plug in some numbers to check.)
Diffusion through the ambient medium is a different ballgame if this is non-negligible, but you might be able compute this with integral-equation-type methods (i.e. a nonlocal interaction based on the Green’s function of the homogeneous background medium), though of course this becomes more complicated if it is time-dependent (hence retarded Green’s functions).
(Caveat: I mostly work with wave equations.)
If this is not just a mathematical exercise but an approximation for a some situation in the physical world - for me it is difficult to imagine such a thing without substantial thermal exchange with the outer world.
Can you tell us a bit more about the problem?
Essentially, I want to compute the density distribution of a certain molecule (block copolymers which can undergo phase separation thus form patterns in the space) confined in a long curved tube. To save computation, I thought it should be possible to compute in one dimension if the density distribution is homogeneous along the radius direction. To avoid mass exchange with the outer world, you just let the wall of tube be some hard wall.
To describe the probability of finding a particular particle s in location \mathbf{x}, we solve the following modified diffusion equation (MDE),
\frac{\partial q(\mathbf{x}, s)}{\partial s} = \nabla^2q + w(\mathbf{x})q
Then the density \phi(\mathbf{x}) can be obtained via an integration,
\phi(\mathbf{x}) = \int ds\; q(\mathbf{x}, s)q^{\dagger}(\mathbf{x}, s)
where q^\dagger is a complementary function to q which can be obtained via solving a similar MDE as q.
Follow Steven’s idea, I should indeed solve my problem in a tube rather than on a string. Or, maybe the 1D version should be equivalent to the flat 1D as suggested by @abraemer.
For further details, you can consult paper 13&17 listed here (full-text available there) where I have developed ETDRK4 solvers to solve the MDEs in the Euclid space.
I love mathematicians - they have such great and simple solutions 
If it is exactly one molecule enclosed in a channel, then the interaction with the channel walls is probably much stronger than the influence of the channel curvature anyway. If you neglect the first, you can neglect the second, too.
If we consider an abstract mathematical problem of diffusion in a channel of given (circular?) cross-section, then my physical feeling tells me, as long as the curvature radius of the channel is large as compared to the channel radius, the problem can be considered as 1D one, as abraemer suggested. As stevengj suggested, you can model it by a torus to get the feeling of the channel curvature influence.
If you tell us more about the actual problem without using euphemisms like “heat diffusion” 
, you might get more useful input.
After reading this article, I finally understand what you mean here. I should use arc length parameterization instead of the conventional choice of parameters, such as \theta for circle and ellipse. With arc length parameterization, I can use conventional Laplacian, i.e.
\nabla^2 \equiv \frac{\partial^2 }{\partial s^2} \equiv f(\theta)\frac{\partial^2 }{\partial \theta^2} + g(\theta)\frac{\partial }{\partial \theta^2}
where s is the arc length, \theta is the Polar coordinate, and f(\theta) and g(\theta) depend on the plane curve.
For my problem, I can choose parametrization freely, so I will stick to the arc length parameterization. And if I want to plot the results on the actual curve, I then convert the arc length to the Cartesian coordinates.
Sadly, it looks like a trivial problem now in our field: there is no any new insight and does not justify for a publication 