In this study, the effect of fractional derivatives, whose application area is increasing day by day, on curves is investigated. As it is known, there are not many studies on a geometric interpretation of fractional calculus. When examining the effect of fractional analysis on a curve, the Caputo fractional analysis that fits the algebraic structure of differential geometry is used. This is because the Caputo fractional derivative of the constant function is zero. This is an important advantage and allows a variety of fractional physical problems to be based on a geometric basis. This effect is examined with the help of examples consistent with the theory and visualized for different values of the Caputo fractional analysis. The difference of this study from others is the use of Caputo fractional derivatives and integrals in calculations. Fractional calculus has applications in many fields such as physics, engineering, mathematical biology, fluid mechanics, signal processing, etc. Fractional derivatives and integrals have become extremely important as they give more numerical results than classical solutions in solving various problems in many fields. In addition, many problems that cannot be answered by classical analysis have been solved by Caputo fractional analysis. In this context, the curvatures of a curve are calculated by Caputo fractional analysis and obtained differently from the classical result. It is aimed to characterize the curve more accurately with the numerically more accurate calculation of the curvatures.
Fractional derivative, Caputo fractional analysis, special curves, curvatures, Frenet frame
HAS, AYKUT and YILMAZ, BEYHAN
"Effect of fractional analysis on some special curves,"
Turkish Journal of Mathematics: Vol. 47:
5, Article 9.
Available at: https://journals.tubitak.gov.tr/math/vol47/iss5/9