Affiliation:
1. Bitlis Eren Un. Dep. of Math.,
2. Bitlis Eren Un. Dep. of Math. (Master Std.) ,
3. Celal Bayar Un. Dep. of Math .,
Abstract
Abstract
A Laplace operator and harmonic curve have very important uses in various engineering science such as quantum mechanics, wave propagation, diffusion equation for heat, and fluid flow. Additionally, the differential equation characterizations of the harmonic curves play an important role in estimating the geometric properties of these curves. Hence, this paper proposes to compute some new differential equation characterizations of the harmonic curves in Euclidean 3-space by using an alternative frame named the N-Bishop frame. Firstly, we investigated some new differential equation characterizations of the space curves due to the N-Bishop frame. Secondly, we firstly introduced some new space curves which have the harmonic and harmonic 1-type vectors due to alternative frame N-Bishop frame. Finally, we compute new differential equation characterizations using the N-Bishop Darboux and normal Darboux vectors. Thus, using these differential equation characterizations we have proved in which conditions the curve indicates a helix.
Reference30 articles.
1. [1] Ali R., Mofarreh F., Alluhaibi N., Ali A., Ahmad I., On differential equations characterizing Legendrian submanifolds of Sasakian space forms, Mathematics, 8,2, 150, 2020, 1-10.
2. [2] Arslan K., Kocayigit H., Onder M., Characterizations of space curves with 1-type Darboux instantaneous rotation vector, Commun. Korean Math. Soc., 31,2, 2016, 379-388.
3. [3] Balki O.P., Kocayiğit H., Differential representation of the Lorentzian spherical timelike curves by using Bishop frame, Thermal Science, 23,6, 2019, 2037-2043.
4. [4] Bektaş M., Külahcı M., Differential equations characterizing spacelike curves in the 3- dimensional lightlike cone, Palestine Journal of Mathematics, 6,1, 2017.
5. [5] Bishop R.L., There is more than one way to frame a curve, The American Mathematical Monthly, 82, 3, 1975, 246-251.