Numerical Treatment of the Coupled Fredholm Integro-Differential Equations by Compact Finite Difference Method
Bibliographic record
Abstract
The work introduces a novel numerical method for solving the Fredholm Integro-Differential Equations (FIDEs) and system of Fredholm Integro-Differential Equations (SFIDEs) by employing the fourth-order compact finite difference methods in conjunction with Simpson's quadrature rule.The accuracy of the proposed scheme is rigorously evaluated using 𝑙 2 and 𝑙 ∞ norms, while the computational efficiency is measured by assessing the CPU-time values, demonstrating a notable reduction in computational cost compared to standard finite difference schemes.The significance of this approach lies in its ability to maintain high levels of accuracy, addressing a common challenge in traditional methods.The methods presented exhibit fourth accuracy in space, as evidenced by numerical experiments.The mentioned work signifies a notable progress in tackling problems related to FIDEs and SFIDEs.It introduces a robust and efficient numerical methodology that proves particularly effective in situations where obtaining exact solutions poses challenges.This advancement is crucial as it addresses a common difficulty faced in the solution of FIDEs and SFIDEs problems, offering a reliable numerical approach that can handle complex scenarios and contribute to more accurate and practical solutions in various fields of study.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| 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 | 0.000 |
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".