Recovering Time-Dependent Coefficients in a Two-Dimensional Parabolic Equation Using Nonlocal Overspecified Conditions via ADE Finite Difference Schemes
Bibliographic record
Abstract
This study delves into the nonlocal inverse boundary-value problem for a second-order, two-dimensional parabolic equation within a rectangular domain.The primary focus is to identify the unknown coefficient and propose a resolution to the problem.The second-order, two-dimensional convection equation is addressed through the direct application of the alternating direction explicit (ADE) finite difference scheme.An adaptation of the ADE scheme is formulated to accommodate mixed boundary conditions, utilizing suitable expressions at the boundaries.Furthermore, unconditional stability is scrutinized through a series of examples.Each ADE scheme typically comprises two substeps, known as upward and downward sweeps, during which values computed at the new time level are incorporated into the discretization template.The inverse problem is restructured into a nonlinear regularized least-square optimization problem, with a defined boundary for the unknown factor, and is effectively resolved using the MATLAB subroutine lsqnonlin from the optimization toolbox.Given the typically ill-posed nature of the problem under investigation, where minor errors in the input data can significantly affect the output, Tikhonov's regularization technique is employed to produce stable and regularized results.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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 source (direct Gemma or distilled Codex), 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".