A New Operational Matrices Based-Technique for Fractional Integro Reaction-Diffusion Equation Involving Spatiotemporal Variable-Order Derivative
Bibliographic record
Abstract
In this article, a novel generalized model of the nonlinear fractional integro-reaction-diffusion equation (NFIRDE) with spatiotemporal variable-order (SVO) is introduced, where the variable order derivatives are equipped with the Atangana–Baleanu–Caputo (ABC) sense. This model represents a great generalization of a significant type of NFIRDE and their applications. Moreover, A novel, efficient and fully spectral shifted Legendre tau technique is developed to solve the proposed model. Despite the difficulty of applying this mechanism to solve this type of equations, due to the presence of nonlinear terms and the SVO functions that appear in the traditional differential and integral operational matrices. We deduce some new operational matrices that play the fundamental role in facilitating the implementation of the tau method. These operational matrices represent the SVO ABC-derivative, the integro term within the model, as well as the vector multiplications with the space-time Kronecker product. As a result, the proposed model is restructured into a system of nonlinear algebraic equations, which simplifying the solving process. We illustrate our method’s effectiveness and validity with numerical examples with both smooth and non-smooth solutions. Our findings show that the proposed tau method delivers accurate results and exhibits non-local properties.
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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 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| 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 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".