Solving the fourth-order nonlinear boundary value problem by a boundary shape function method
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
A method to construct the boundary shape function (BSF) and then two novel methods are developed to obtain the solutions of fourth-order singularly perturbed beam equation and nonlinear boundary value problem (BVP). In the first-type algorithm, the free function is a series of complete basis functions, while the corresponding BSFs are new bases. The trial functions with fractional powers exponential are suitable for the singularly perturbed beam equation under fixed-end and simply supported boundary conditions. With the aid of the BSF, we can improve the asymptotic and uniform approximations to exactly satisfy the prescribed boundary conditions. In the second-type algorithm, the solution of a nonlinear BVP is viewed as a boundary shape function, while the free function is regarded as a new variable. With this means, the fourth-order nonlinear BVP is exactly converted to an initial value problem with a new variable, the terminal value of which is unknown, when the initial conditions are given. The computed order of convergence and an error estimation are given. Numerical illustrations, including the singularly perturbed examples, show that the present methods, based on the new idea of the BSF, are highly effective, accurate, and fast convergent.
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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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