On the Generalized Fractional Convection–Diffusion Equation with an Initial Condition in Rn
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Bibliographic record
Abstract
Time-fractional convection–diffusion equations are significant for their ability to model complex transport phenomena that deviate from classical behavior, with numerous applications in anomalous diffusion, memory effects, and nonlocality. This paper derives, for the first time, a unique series solution to a multiple time-fractional convection–diffusion equation with a non-homogenous source term, based on an inverse operator, a newly-constructed space, and the multivariate Mittag–Leffler function. Several illustrative examples are provided to show the power and simplicity of our main theorems in solving certain fractional convection–diffusions equations. Additionally, we compare these results with solutions obtained using the AI model DeepSeek-R1, highlighting the effectiveness and validity of our proposed methods and main theorems.
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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 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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