Applications of Standard Methods for Solving the Electric Train Mathematical Model With Proportional Delay
Bibliographic record
Abstract
In electric trains, the current is collected via a certain device, called the Pantograph. The governing mathematical model of such physical problem is well-known as the Pantograph delay differential equation (PDDE): y’(t)=ay(t)+by(ct), where c is a proportional delay parameter. In the literature, a special case of the PDDE was analyzed when c=−1. The objective of this paper is to determine the general solution of the PDDE for arbitrary c. In addition, it will be shown that the obtained general solution reduces to exact one at c=−1. Such exact solution is expressed in terms of several types of functions, e.g., hyberbolic, Mittag-Leffler, and trigonometric functions. Moreover, it is declared that the exact trigonometric solution is periodic with periodicity 2π/√b2−a2 which agrees with the corresponding results in the literature. Furthermore, the solution of PDDE is provided at almost all possible cases of the involved parameters a, b, and c. Finally, the solution Ambartsumian delay equation (ADE), which has an application in the theory of surface brightness in the Milky Way, will be recovered as a special case of our 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 | 0.001 | 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) | 0.004 | 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".