Inhomogeneous Whittaker Equation with Initial and Boundary Conditions
Bibliographic record
Abstract
In this study, a semi-analytical solution to the inhomogeneous Whittaker equation is developed for both initial and boundary value problems. A new class of special integral functions Ziκ,μf(x), along with their derivatives, is introduced to facilitate the construction of the solution. The analytical properties of Ziκ,μf(x) are rigorously investigated, and explicit closed-form expressions for Ziκ,μf(x) and its derivatives are derived in terms of Whittaker functions Mκ,μ(z) and Wκ,μ(z), confluent hypergeometric functions, and other special functions including Bessel functions, modified Bessel functions, and the incomplete gamma functions, along with their respective derivatives. These expressions are obtained for specific parameter values using symbolic computation in Maple. The results contribute to the broader analytical framework for solving inhomogeneous linear differential equations with applications in engineering, mathematical physics, and biological modeling.
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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".