Please help, numerical methods related to curvature flow

[cousinbalin]

Hello everyone, I'm hoping someone can help me here. The actual question I have may be a little bit dumb (and you may not even need to know how to solve this problem in order to help me). The image attached below shows the problem that I'm currently trying to solve involving a numerical methods solution for a curvature flow model. If you have any trouble reading the terms, you can read see a higher-resolution version here. I think that I have a good general idea on how to proceed and have already written a fair portion of the MATLAB code.

At the present, the main thing that's holding me up is the x_1αα and x_2αα terms. I feel stupid because I feel like it should be obvious, but I don't know what these are (how to define/parameterize them). The only thing I can think of is to take it to mean x_1αα = x₁(α,t=α), i.e. x₁ at point α and t=α. But that doesn't seem to make any sense to me since α is in radians (ranging from 0 to 2π). Thoughts?

(click here for higher resolution version)


Meanwhile, if anyone has any helpful tips or suggestions in general regarding the implementation in MATLAB or otherwise it wouldn't hurt as a verification that I'm proceeding correctly.

Thanks!

 

[Deleted member 4993]

Hello everyone, I'm hoping someone can help me here. The actual question I have may be a little bit dumb (and you may not even need to know how to solve this problem in order to help me). The image attached below shows the problem that I'm currently trying to solve involving a numerical methods solution for a curvature flow model. If you have any trouble reading the terms, you can read see a higher-resolution version here. I think that I have a good general idea on how to proceed and have already written a fair portion of the MATLAB code.

At the present, the main thing that's holding me up is the x_1αα and x_2αα terms. I feel stupid because I feel like it should be obvious, but I don't know what these are (how to define/parameterize them). The only thing I can think of is to take it to mean x_1αα = x₁(α,t=α), i.e. x₁ at point α and t=α. But that doesn't seem to make any sense to me since α is in radians (ranging from 0 to 2π). Thoughts?

View attachment 2492 (click here for higher resolution version)


Meanwhile, if anyone has any helpful tips or suggestions in general regarding the implementation in MATLAB or otherwise it wouldn't hurt as a verification that I'm proceeding correctly.

Thanks!

As far as I can tell:

x₁ is a function of \(\displaystyle \alpha\)

then:

dx₁/dα = x_1α

and

d²x₁/dα² = x_1αα