Exact and Numerical Treatment of a Special Kind of the Pantograph Model via Laplace Technique
Bibliographic record
Abstract
The Pantograph is a device of practical application in electric trains, by which the current is collected. The mathematical problem of this device is generally given by the delay differential equation φ’(t) = αy(t) + βφ(γt), where α and β are real constants and γ is a proportional delay parameter. In the literature, a special attention has been given to the particular case γ = −1. The objective of this paper is to extend the application of the Laplace transform (LT) combined with the Adomian decomposition method (ADM) to analyze the above model at such particular case of γ. The solution will be determined in exact form which agrees with the corresponding results in the literature. Various properties of the obtained exact solution are discussed in detail. Moreover, it will be declared that for sufficiently small values of α compared to β there exists an accurate approximate solution. The accuracy of the approximate solution is numerically validated. In addition, some numerical results are conducted for the behavior of the present solution at selected values of α and β.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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".