EXPLORING THE APPLICATIONS OF FRACTIONAL CALCULUS IN MODELING REAL-WORLD PHENOMENA

Abstract

The paper looks into the application of the high order calculus in one of the real-world phenomena and its efficiency compared to the classical integer-order calculus. The results indicate that the fractional models make the prediction process more accurate by 21 percent making it between 68 and 89 percent and errors are less by a factor of over 50 percent, 32 to 14 percent. It is also demonstrated in the experiment that with the application of fractional calculus, it becomes possible to stabilize the system to a greater extent of 18 percent and the accuracy of long-term predictions is also enhanced by 62-84 percent. Fractional models have a very high memory effect representation (85%) when compared to classical models (45%), and indicate that they are able to model the complex dynamics of a system.

The breadth of the field of fractional calculus is illustrated by domain analysis showing 30, 27, and 25 percent performance improvements in biological systems, physics and engineering respectively. However, the study records an increase in computation time by 35 percent as among the limitations. Physics (62 percent) and engineering (58 percent) have the highest adoption rates.

Overall, the data helps to confirm the hypothesis that the notion of fractional calculus is more specific, coherent, and comprehensive in the modeling of complex systems. The study observes the need of improved computing methods and broader applications in order to achieve the full potential of the application of fractional-order models in science and engineering.