On the Geometric and Physical Properties of Conformable Derivative

Year 2024, Volume: 12 Issue: 2, 60 - 70, 14.04.2024

Aykut Has , Beyhan Yılmaz , Dumitru Baleanu

https://doi.org/10.36753/mathenot.1384280

Cited By: 21

https://izlik.org/JA39RB96PL

Abstract

In this article, we explore the advantages geometric and physical implications of the conformable derivative. One of the key benefits of the conformable derivative is its ability to approximate the tangent at points where the classical tangent is not readily available. By employing conformable derivatives, alternative tangents can be created to overcome this limitation. Thanks to these alternative (conformable) tangents, physical interpretation can be made with alternative velocity vectors. Furthermore, the conformable derivative proves to be valuable in situations where the tangent plane cannot be defined. It enables the creation of alternative tangent planes, offering a solution in cases where the traditional approach falls short. Geometrically speaking, the conformable derivative carries significant meaning. It provides insights into the local behavior of a function and its relationship with nearby points. By understanding the conformable derivative, we gain a deeper understanding of how a function evolves and changes within its domain. A several examples are presented in the article to better understand the article and visualize the concepts discussed. These examples are accompanied by visual representations generated using the Mathematica program, aiding in a clearer understanding of the proposed ideas. By combining theoretical explanations, practical examples, and visualizations, this article aims to provide a comprehensive exploration of the advantages and geometric and physical implications of the conformable derivative.

Keywords

Conformable derivative , Curvatures , Frenet frame , Surface

References

• [1] Khalil, R., Horani, M., Yousef,A., Sababheh, M.: A new definition of fractional derivative. Journal of Computational and Applied Mathematics. 264, 65-70 (2014).
• [2] Abdeljawad, T.: On conformable fractional calculus. Journal of Computational and Applied Mathematics. 279, 57-66 (2015).
• [3] Thabet, H., Kendre, S., Baleanu, D., Peters, J.: Exact analytical solutions for nonlinear systems of conformable partial differential equations via an analytical approach. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics. 84 (1), 109-120 (2022).
• [4] Ibrahim, R. W., Baleanu, D., Jahangiri, J. M.: Conformable differential operators for meromorphically multivalent functions. Concrete Operators. 8 (1), 150-157 (2021).
• [5] Au, V. V., Baleanu, D., Zhou, Y., Can, N. H.: On a problem for the nonlinear diffusion equation with conformable time derivative. Applicable Analysis. 101 (17), 6255-6279 (2022).
• [6] Al-Jamel, A., Masaeed, M. A., Rabei, E. M., Baleanu, D.: The effect of deformation of special relativity by conformable derivative. Revista Mexicana de Fisica, 68 (5), 050705 (2022).
• [7] Asjad, M. I., Ullah, N., Rehman, H. U., Baleanu, D: Optical solitons for conformable space-time fractional nonlinear model. Journal of Advances in Mathematics and Computer Science. 27 (1), 28-41 (2022).
• [8] Masaeed, M. A., Rabei, E. M., Al-Jamel, A. A., Baleanu, D.: Extension of perturbation theory to quantum systems with conformable derivative. Modern Physics Letters A. 36 (32), 2150228 (2021).
• [9] Yajima, T., Yamasaki, K.: Geometry of surfaces with Caputo fractional derivatives and applications to incompressible two-dimensional flows. Journal of Physics A: Mathematical and Theoretical. 45, 065201 (2012).
• [10] Yajima, T., Oiwa, S., Yamasaki, K.: Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas. Fractional Calculus and Applied Analysis. 21 (6), 1493-1505 (2018).
• [11] Lazopoulos, K. A., Lazopoulos, A. K.: Fractional differential geometry of curves and surfaces. Progress in Fractional Differentiation and Applications. 2 (3), 169-186 (2016).
• [12] Aydın, M. E., Mihai, A., Yoku¸s, A.: Applications of fractional calculus in equiaffine geometry: plane curves with fractional order. Mathematical Methods in the Applied Sciences. 44 (17), 13659-13669 (2021).
• [13] Gözütok, U., Çoban, H. A., Sağıroğlu, Y.: Frenet frame with respect to conformable derivative. Filomat 33 (6), 1541-1550 (2019).
• [14] Has, A., Yılmaz, B.: Special fractional curve pairs with fractional calculus. International Electronic Journal of Geometry. 15 (1), 132-144 (2022).
• [15] Has, A., Yılmaz, B., Akkurt, A., Yıldırım, H.: Conformable special curves in Euclidean 3-Space. Filomat. 36 (14), 4687-4698 (2022).
• [16] Has, A., Yılmaz, B.: Effect of fractional analysis on magnetic curves. Revista Mexicana de Fisica. 68 (4), 1–15 (2022).
• [17] Yılmaz, B., Has, A.: Obtaining fractional electromagnetic curves in optical fiber using fractional alternative moving frame. Optik - International Journal for Light and Electron Optics. 260 (8), 169067 (2022).
• [18] Yılmaz, B: A new type electromagnetic curves in optical fiber and rotation of the polarization plane using fractional calculus. Optik - International Journal for Light and Electron Optics. 247 (30), 168026 (2021).
• [19] Aydın, M. E., Bekta¸s, M., Öğrenmiş, A. O., Yokus, A.: Differential geometry of curves in Euclidean 3-space with fractional order. International Electronic Journal of Geometry. 14 (1), 132-144 (2021).
• [20] Aydın, M. E., Kaya, S.: Fractional equiaffine curvatures of curves in 3-dimensional affine space. International Journal of Maps in Mathematics. 6 (1), 67-82 (2023).
• [21] Ögrenmiş, M.: Geometry of curves with fractional derivatives in Lorentz plane. Journal of New Theory. 38, 88-98 (2022).
• [22] Gözütok, N. Y., Gözütok, U.: Multivariable conformable fractional calculus. Filomat. 32 (2), 45-53 (2018).
• [23] Has, A., Yılmaz, B.: Measurement and calculation on conformable surfaces. Mediterranean Journal of Mathematics. 20 (5), 274 (2023).
• [24] Has, A., Yılmaz, B.: $C_\alpha-$curves and their $C_\alpha-$frame in fractional differential geometry. In Press (2023).