On the convergence of solving a nonlinear Volterra‐type integral equation for surface divergence based on surface thermal information
Bibliographic record
Abstract
We analyze a nonlinear integral equation for calculating free‐surface divergence that was proposed by Szeri (2017, https://doi.org/10.1002/2016JC012312 ). When given the temperature and heat flux at a free surface, the surface divergence can be calculated through a nonlinear singular Volterra‐type integral equation. The two given functions in the integral equation satisfy auxiliary conditions through a higher dimensional partial differential equation. We prove the existence and uniqueness of the solution of the integral equation. We also prove the local linear convergence of the corresponding Picard iteration method for solving the integral equation when the surface heat flux is a real‐analytic function of time. The rate of convergence is derived explicitly, which depends on the function of surface heat flux. Numerical examples are provided to validate the convergence performance.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.008 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".