Quartic spline collocation methods for second-order two-point boundary value ODE problems
Bibliographic record
Abstract
Collocation methods based on quartic splines are presented for second-order two-point boundary value problems. In addition to the boundary conditions specified by the problem, extra boundary conditions are introduced in order to uniquely determine the quartic spline approximation. The standard quartic spline collocation method gives fourth order bounds. Two optimal methods, namely the extrapolated (one-step) and the deferred-correction (two-step) methods, are formulated based on appropriate extra boundary conditions and an appropriate perturbation of the operators of the differential equation, boundary conditions and extra boundary conditions. The convergence analysis and error bounds are developed using a Green's functions approach. The analysis shows that the maximum discrete error is of sixth order, and the maximum global error is of fifth order for the optimal methods. The properties of the matrices arising from the optimal methods for a certain class of problems are studied. Non-optimal collocation methods based on different extra boundary conditions are also investigated. Problems with layers are also considered, and a grid refinement technique is presented. The theoretical behavior of the collocation methods is verified by numerical 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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".