Numerical Solutions of Weakly Singular Nonlinear Volterra Integral Equations With Smooth and Nonsmooth Solutions Over Large Time Intervals
Bibliographic record
Abstract
ABSTRACT This paper proposes a spectral collocation method for solving weakly singular nonlinear Volterra integral equations over large time intervals, addressing both smooth and nonsmooth solutions. For smooth solutions, classical scaled Laguerre polynomials (SLPs) are employed. For nonsmooth solutions, a new set of generalized Laguerre functions (GLFs) is introduced. To efficiently approximate weakly singular integrals, the interval is transformed into a half‐line using a suitable mapping. The Gauss points of the GLFs are then used as the collocation points to approximate the resulting integral. Also, the convergence of the proposed method is analyzed in the weighted ‐norm. Finally, the accuracy and reliability of the new approach are demonstrated through several examples.
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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 | 0.005 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| 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".