Generalized RKM Method for Solving Sixth-Order Fractional Ordinary Differential Equations
Bibliographic record
Abstract
Different types of DEs have a wide range of applications in both engineering and science.Typically, when modelling a physical quantity's variation, Recently, it has been found that models based on the theory of fractional-order derivatives and integrals provide an exceptionally good description of a wide range of scientific phenomena.The FDE is a differential equation that contains some derivatives of non-integer powers order.FDEs have become increasingly significant in the theoretical and applied parts of a wide variety of scientific and technical disciplines in recent years.The high-order ODE can be reduced to systems of first order ODEs, which can then be solved.Directly attacking the issue with numerical methods, however, is much more efficient in terms of accuracy, number of function evaluations, and processing time.In this article, the RKM method for solving ordinary differential equations has been introduced.This numerical approach has been generalized to be suitable for solving a class of fractional differential equations (FDEs).However, the developed RKM approach with three-and four-stages for solving sixth-order FODEs is developed.Moreover, this technique was used to solve various test problems, these examples to which the developed method was applied were for various functions with different values of and then, the solutions of the developed numerical method were compared with the exact solution, the numerical results proved the efficiency and accuracy of the modified technique.
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How this classification was reachedexpand
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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".