1. bookVolume 46 (2021): Issue 3 (September 2021)
Journal Details
License
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Journal
First Published
24 Oct 2012
Publication timeframe
4 times per year
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English
access type Open Access

Some New Characterizations of The Harmonic and Harmonic 1-Type Curves in Euclidean 3-Space

Published Online: 17 Sep 2021
Page range: 235 - 254
Received: 09 Apr 2020
Accepted: 01 Mar 2021
Journal Details
License
Format
Journal
First Published
24 Oct 2012
Publication timeframe
4 times per year
Languages
English
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.

Keywords

[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. Search in Google Scholar

[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. Search in Google Scholar

[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. Search in Google Scholar

[4] Bektaş M., Külahcı M., Differential equations characterizing spacelike curves in the 3- dimensional lightlike cone, Palestine Journal of Mathematics, 6,1, 2017. Search in Google Scholar

[5] Bishop R.L., There is more than one way to frame a curve, The American Mathematical Monthly, 82, 3, 1975, 246-251. Search in Google Scholar

[6] Bükçü B., Karacan M.K., Special Bishop motion and Bishop Darboux rotation axis of the space curve, Journal of Dynamical Systems and Geometric Theories, 6, 1, 2008, 27-34. Search in Google Scholar

[7] Bükçü B., Karacan M.K., On the slant helices according to Bishop frame of the timelike curve in Lorentzian space, Tamkang Journal of Math., 39,3, 2008, 255-262. Search in Google Scholar

[8] Bükçü B., Karacan M.K., The slant helices according to Bishop frame, International Journal of Computational and Mathematical Sciences, 3, 2, 2009,63-66. Search in Google Scholar

[9] Chen B.Y., On the total curvature of immersed manifolds, VI: Submanifolds of finite type and their applications, Bull. Ins. Math. Acad. Sinica, 11, 1983, 309-328. Search in Google Scholar

[10] Chen B.Y., Total mean curvature and submanifolds of finite type, World Scientific, 1984. Search in Google Scholar

[11] Chen B.Y., Ishikawa S., Biharmonic surface in pseudo-Euclidean space, Mem. Fac. Sci. Kyushu Univ., A45, 1991, 323-347. Search in Google Scholar

[12] Çakır O, Şenyurt S., Differential equations for a space curve according to the unit Darboux vector, Turk. J. Math. Comput. Sci., 9, 2018, 91-97. Search in Google Scholar

[13] https://en.wikipedia.org/wiki/Laplace_operator Search in Google Scholar

[14] Inoguchi J., Biharmonic curves in Minkowski 3-space. International J. Math. Math. Sci. 21, 2003, 1365-1368. Search in Google Scholar

[15] Keskin O., Yaylı Y., An application of N-Bishop frame to spherical images for direction curves, International Journal of Geometric Methods in Modern Physics, 14,11, 2017, 1750162. Search in Google Scholar

[16] Kocayigit H., Hacisalihoglu H.H., 1-type and biharmonic Frenet curves in Lorentzian 3- space, Iran. J. Sci. Technol. Trans. A Sci., 33, 2009, 159-168. Search in Google Scholar

[17] Kocayigit H., Hacisalioğlu H.H., 1-type curves and biharmonic curves in Euclidean 3- space, International Electronic Journal of Geometry, 4, 1, 2011, 97-101. Search in Google Scholar

[18] Kocayigit H, Hacisalihoglu HH,. Biharmonic curves in contact geometry, Communications Faculty of Sciences University of Ankara-series A1 Mathematics and Statistics, 61, 2, 2012, 35-43. Search in Google Scholar

[19] Kocayigit H., Ozdemir A., Çetin M., Asartepe S.O., Characterizations of timelike curves according to the Bishop Darboux vector in Minkowski 3-space, International Mathematical Forum, 19, 2013. Search in Google Scholar

[20] Kocayigit H., Kazaz M., Arı Z., Some characterizations of space curves according to Bishop frame in Euclidean 3-space, Journal of Abstract and Computational Mathematics, 1,1, 2016, 47-57. Search in Google Scholar

[21] Krutitskii P.A., The modified jump problem for the Laplace equation and singularities at the tips, Journal of Computational and Applied Mathematics, 183,1, 2005, 232-240. Search in Google Scholar

[22] Onder, M., Kocayigit H., Canda E., Differential equations characterizing timelike and spacelike curves of constant breadth in Minkowski 3-space E_1^3, Journal of the Korean Mathematical Society, 48, 4, 2011, 849-866. Search in Google Scholar

[23] Ou Y.L., p-Harmonic morphisms, biharmonic morphisms, and nonharmonic biharmonic maps, J. Geom. Phys., 56, 2006, 358-374. Search in Google Scholar

[24] Reuter M., Wolter F.E., Peinecke N., Laplace–Beltrami spectra as ‘Shape-DNA’of surfaces and solids, Computer-Aided Design, 38, 4, 2006, 342-366. Search in Google Scholar

[25] Scofield P.D., Curves of constant precession, The American Mathematical Monthly, 102, 6, 1995, 531-537. Search in Google Scholar

[26] Stump D.M., Whatson P.J., Fraser W.B., Mathematical modelling of interwound DNA supercoils, J.of Biomechanics, 33,2000,407-413. Search in Google Scholar

[27] Uzunoğlu B., Gök İ., Yaylı Y., A New approach on curves of constant precession, Applied Mathematics and Computation, 275, 2016, 317-323. Search in Google Scholar

[28] Verma D.A, Laplace transformation approach to simultaneous linear differential equations, New York Science Journal, 12, 7, 2019. Search in Google Scholar

[29] Yılmaz S., Turgut M., A new version of Bishop frame and an application to spherical images, Journal of Mathematical Analysis and Applications, 371, 2, 2010, 764-776. Search in Google Scholar

[30] Yılmaz S., Özyılmaz E., Turgut M., New spherical indicatrices and their characterizations, An. St. Univ. Ovidius Constanta, 18, 2, 2010, 337-354. Search in Google Scholar

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