A reliable algorithm for the approximate solution of the nonlinear Lane‐Emden type equations arising in astrophysics
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
In this paper, we present a reliable algorithm to obtain the approximate solution of the nonlinear Lane‐Emden type equations arising in astrophysics. The suggested algorithm is based upon the operational matrix of integration for Jacobi polynomials and the collocation method. Convergence analysis and numerical stability of the suggested method are provided. Numerical results for several interesting nonlinear cases of the Lane‐Emden type equations such as the standard Lane‐Emden equation, the white‐dwarf equation, and the isothermal gas spheres equation, as well as Richardson's theory of thermionic current are discussed. These numerical results are shown in the form of tables and figures for the particular cases of Jacobi polynomials such as the Legendre polynomial (P1), the Chebyshev polynomials of the second kind (P2), the Chebyshev polynomials of the third kind (P3), the Chebyshev polynomials of the fourth kind (P4), and the Gegenbauer (or ultraspherical) polynomials (P5). Numerical results are also compared with those that were derived earlier by applying some well‐known and recently developed numerical methods and it is observed that our numerical results are more accurate. The maximum absolute errors and the root mean square errors are calculated by using P1, P2, P3, P4, and P5 for comparison purposes.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.010 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 it