On the Application of Complex Delta Function Leading to New Fractional Calculus Formulae Involving the Generalized Hypergeometric Function and Kinetic Equation
Bibliographic record
Abstract
The sun is a vital component of our natural environment, and kinetic equations are important mathematical models that show how quickly a star’s chemical composition changes. Taking inspiration from these facts, we develop and solve a novel fractional kinetic equation by calculating the Laplace transform of hypergeometric functions in the complex coefficient parameter. This was a challenging task because the function cannot be integrated concerning the coefficient parameters using classical methods due to the infinite number of singular points of the gamma function involved in it. We achieved it using the distributional representation of the generalized hypergeometric function. Moreover, on the one hand, the role of the delta function is vital to represent the electromotive forces, and on the other, the solution of differential equations of engineering and mathematical physics led to a class of hypergeometric functions. This article is the confluence of both. Therefore, innovative characteristics concerning the Fox-Wright and several related important functions are applied for the simplification of the obtained outcomes. A popular class of fractional transforms involving generalized hypergeometric functions are evaluated using the delta function, and as a distribution, numerous additional features of this function are described.
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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 | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".