Elementary Primer on Partial Differential Equations (PDEs)
Bibliographic record
Abstract
Chapters 10–12 described continuum models of engineered membranes that involved partial differential equations (PDEs). This appendix gives a brief introduction to the classification of PDEs, linear PDEs, nondimensionalization of PDEs, and numerical methods for solving PDEs. Uniqueness and existence results are not discussed here since they involve advanced results in functional analysis that are outside the scope of this book. For a comprehensive advanced treatment of PDEs at a graduate mathematics level see [115]. Linear, Semilinear, and Nonlinear Partial Differential Equations PDEs are typically classified into linear, semilinear, and nonlinear. Linear PDEs are important as several methods exist to obtain closed-form solutions of these PDEs (under simple boundary conditions), including separation of variables, superposition, Fourier series, Laplace transform, and Fourier transform. The solution to some semilinear PDEs can be obtained using the symmetry method or method of characteristics. When a linear or semilinear PDE has more sophisticated boundary conditions (as is the case with engineered membranes), the PDE needs to be solved numerically. Similarly, nonlinear PDEs typically do not have an exact solution and must also be solved numerically. Consider a generic PDE of two scalar variables (space x and time t ), where u ( x, t ) is a function of the variables x and t . A linear PDE has the form where a ( x, t ) , b ( x, t ) , c ( x, t ), and d ( x, t ) are generic functions of x and t but not u . A PDE is semilinear if the coefficient function a ( x, t ) of the highest partial derivative is dependent on x and t but not u . Therefore, (A.1) is a semilinear PDE if the coefficient functions can be expressed as a ( x, t ) , b ( x, t, u ) , c ( x, t, u ), and d ( x, t, u ). If the PDE is neither linear nor semilinear, it is nonlinear. Linear, semilinear, and nonlinear PDEs are all used to model the dynamics of engineered membranes. For example: Linear. Poisson's equation ((10.3) on page 180), the Nernst–Planck equation ((10.9) on page 182), the Poisson–Fermi equation ((11.23) on page 233), and the Fokker– Planck equation ((14.28) on page 318).
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.094 | 0.048 |
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".