A Unified Framework Using Orthogonal Hybrid Functions for Solving Linear and Nonlinear Fractional Differential Systems
Why this work is in the frame
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Bibliographic record
Abstract
This paper presents a novel and computationally efficient numerical method for solving systems of fractional-order differential equations using orthogonal hybrid functions (HFs). The proposed HFs are constructed by combining piecewise constant orthogonal sample-and-hold functions with piecewise linear orthogonal right-handed triangular functions, resulting in a flexible and accurate approximation basis. A central innovation of the method is the derivation of generalized one-shot operational matrices that approximate the Riemann–Liouville fractional integral, enabling direct integration of differential operators of arbitrary order. These matrices act as unified integrators for both integer and non-integer orders, enhancing the method’s applicability and scalability. A rigorous convergence analysis is provided, establishing theoretical guarantees for the accuracy of the numerical solution. The effectiveness and robustness of the approach are demonstrated through several benchmark problems, including fractional-order models related to smoking dynamics, lung cancer progression, and Hepatitis B infection. Comparative results highlight the method’s superior performance in terms of accuracy, numerical stability, and computational efficiency when applied to complex, nonlinear, and high-dimensional fractional-order systems.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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