Calculator Inputs
Plotly Graph
Example Data Table
The table previews the first rows from the active simulation dataset.
| Step | Time | Exact Radius | Euler Radius | Exact Curvature | Euler Curvature | Exact Speed | Exact Area | Exact Volume |
|---|---|---|---|---|---|---|---|---|
| 0 | 0.0000 | 5.000000 | 5.000000 | 0.380731 | 0.380731 | -0.380731 | 31.415927 | 78.539816 |
| 1 | 0.0500 | 4.980942 | 4.980963 | 0.381604 | 0.381603 | -0.381604 | 31.296179 | 77.942222 |
| 2 | 0.1000 | 4.961839 | 4.961883 | 0.382485 | 0.382483 | -0.382485 | 31.176157 | 77.345542 |
| 3 | 0.1500 | 4.942693 | 4.942759 | 0.383373 | 0.383370 | -0.383373 | 31.055856 | 76.749782 |
| 4 | 0.2000 | 4.923502 | 4.923591 | 0.384269 | 0.384265 | -0.384269 | 30.935275 | 76.154945 |
| 5 | 0.2500 | 4.904266 | 4.904377 | 0.385173 | 0.385168 | -0.385173 | 30.814412 | 75.561036 |
| 6 | 0.3000 | 4.884985 | 4.885119 | 0.386084 | 0.386078 | -0.386084 | 30.693263 | 74.968060 |
| 7 | 0.3500 | 4.865657 | 4.865815 | 0.387004 | 0.386996 | -0.387004 | 30.571827 | 74.376019 |
Formula Used
This page uses a radial ball model for fractional mean curvature flow.
H_s(R) = ω · R^(−s)
dR/dt = −H_s(R) = −ω · R^(−s)
R(t) = max(R0^(s+1) − (s+1)·ω·t, 0)^(1/(s+1))
T_ext = R0^(s+1) / ((s+1)·ω)
V_n(R) = [π^(n/2) / Γ(n/2 + 1)] · R^n
A_(n−1)(R) = n · [π^(n/2) / Γ(n/2 + 1)] · R^(n−1)
Exact values come from the closed form radius law. Euler values come from forward time stepping with the chosen number of steps.
How to Use This Calculator
- Enter the ambient dimension n for the radial set.
- Choose fractional order s between zero and one.
- Set the starting radius R0 for the interface.
- Enter curvature constant ω for your chosen normalization.
- Pick the analysis time t you want to inspect.
- Choose Euler steps to compare exact and discrete paths.
- Press Calculate Flow to generate results and graphs.
- Use CSV or PDF buttons to export the simulation table.
FAQs
1. What does this calculator measure?
It estimates how a radially symmetric interface shrinks under a fractional curvature law. It reports radius, curvature, speed, boundary measure, enclosed volume, extinction time, and a discrete Euler comparison.
2. Is this a full solver for arbitrary shapes?
No. This page is an educational radial model. It is useful for spheres, circles, and symmetric benchmark cases, but it does not solve the full nonlocal interface equation for general shapes.
3. Why is the fractional order restricted between zero and one?
The common fractional mean curvature setting uses a parameter s in the open interval from zero to one. Values near the endpoints can also become numerically unstable in simple teaching models.
4. What does the constant ω represent?
ω is a normalization constant for the curvature law in the radial model. Different texts use different conventions, so this input lets you match the scaling used in your notes or research setup.
5. Why do exact and Euler radii differ?
The exact radius uses the closed form evolution. The Euler radius uses finite time steps. When the step count is small, the discrete path accumulates approximation error. More steps usually improve agreement.
6. What happens at extinction time?
The radial solution collapses to radius zero at the computed extinction time. After that point, this calculator holds radius, boundary measure, and enclosed volume at zero.
7. Can I use this in coursework or reports?
Yes. The page includes a data table, chart output, and CSV or PDF export tools. You can cite it as a numerical benchmark, but mention that it uses a radial approximation.
8. Which units should I use?
Use one consistent unit system. Radius, time, area, and volume outputs follow the scale implied by your chosen radius unit, time unit, dimension, and curvature constant.