Equivalences of Nonlinear Higher Order Fractional Differential Equations With Integral Equations
Bibliographic record
Abstract
ABSTRACT Equivalences of initial value problems (IVPs) of both nonlinear higher order (Riemann–Liouville type) fractional differential equations (FDEs) and Caputo FDEs with the corresponding integral equations are studied in this paper. It is proved that the nonlinearities in the FDEs can be ‐Carathéodory with suitable conditions. The new results generalize the previous results which assumed that the nonlinearities are continuous. For the Caputo FDEs, it is shown in this paper that the continuity assumptions on the nonlinearities used in the literature before are not sufficient for the obtained equivalences. A counterexample is provided to exhibit this. The previous equivalence results with the continuity assumptions alone in the literature have been widely used to study the existence of solutions and numerical solutions of the Caputo FDEs up to now, so according to the new results obtained in this paper, there are no guarantees that the solutions of the integral equations obtained in the literature are the solutions of the Caputo FDEs. New conditions which are stronger than continuity are provided to ensure the equivalences. Sufficient conditions for solutions of the integral equations to be solutions of the Caputo FDEs are obtained. The new equivalence results and the sufficient conditions will be useful for further studying the existence of solutions and numerical solutions of the nonlinear Caputo FDEs via the corresponding integral equations.
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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 | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".