Overview
This is a small spectral visualization project comparing Fourier mode decay in normal diffusion and time-fractional subdiffusion on the one-dimensional interval Omega = (0, 1).
The project uses a simple sine eigenbasis to make the difference between exponential diffusion decay and slower subdiffusive decay visible. The Mittag-Leffler function is approximated with a lightweight piecewise implementation for qualitative visualization.
What I Studied / Implemented
- Sine eigenbasis on Omega = (0, 1)
- Normal diffusion decay of the form exp(-lambda_n t)
- Subdiffusion decay through a Mittag-Leffler-type factor
- High-frequency decay signatures related to inverse problems with unknown terminal time
- Numerical plots for comparing mode behavior across frequencies and times
- Noise sensitivity of high-frequency terminal coefficients
Methods or Workflow
The implementation computes eigenvalues and mode amplitudes for a simple one-dimensional spectral setting. It then compares how each mode decays under normal diffusion and under a time-fractional subdiffusion model.
The project is designed to build intuition from a small numerical example before moving to more complicated inverse-problem settings.
Outputs
- Five figures covering mode decay, high-frequency signatures, terminal profiles by alpha, terminal profiles by terminal time, and noise sensitivity
- CSV outputs:
mode_decay_table.csv,high_frequency_signature_table.csv, andprofile_grid.csv - A small, inspectable Python codebase
- Repository README documentation
What I Learned
The visualization makes high-frequency decay behavior easier to see. It also clarifies why terminal-time information and decay rates matter when thinking about inverse problems involving diffusion-type models.
Repository
Limitations
This is a toy spectral visualization, not a full inverse solver or a full reproduction of a paper. The Mittag-Leffler values are approximated for qualitative visualization. The project does not implement Levenberg-Marquardt methods, finite element methods, 2D problems, or a complete parameter-recovery pipeline.