A Numerical Approach for Solving the System of Differential Equations Related to the Spherical Curves in Euclidean 3-Space

Yıl 2019, Cilt: 11 Sayı: 2, 84 - 96, 31.12.2019

Seda Çayan , Hüseyin Kocayiğit , Mehmet Sezer

Öz

In 1971, integral form of spherical curve in 3-dimensional Euclidean space  was given in [3]. The explicit characterization of the spherical curves in n-dimensional Euclidean space was  given in [12]. Morever, the position vector of spherical curves in Euclidean 3-space was determined in [10]. In the present  work, a)  it is given the system of differential equations of the spherical curves in 3-dimensional Euclidean space; b) it is shown that the numerical solutions of this system of differential equations are obtained in the truncated Taylor series form by using Taylor matris collocation method; c) an example together with error analysis are given to demonstrate the validity and applicability of present  method.

Anahtar Kelimeler

Spherical curves, Taylor matrix collocation method, residual error analysis

Kaynakça

• Ayd{\i}n, T.A., Sezer, M., {\em Taylor-matrix collocation method to solution of differential equations characterizing spherical curves in Euclidean 4-Space}, Celal Bayar University Journal of Science, \textbf{15}(2019), 1--7, DOI:10.18466/cbayarfbe.416121.
• Ayd{\i}n, T.A., Sezer, M., {\em Hermite polynomial approach to determine spherical curves in Euclidean 3-space}, New Trends in Mathematical Sciences, \textbf{6}(2018), 189--199.
• Breuer, S., Gottileb, D., {\em Explicit characterization of spherical curves}, Proceedings of the American Mathematical Society, \textbf{27}(1971), 126--127.
• Bulut, V., \c{C}al{\i}\c{s}kan, A., {\em Spherical images of special Smarandache curves in $E^3$}, International J. Math. Combin., \textbf{3}(2015), 43--54.
• \c{C}etin, M., Kocayi\u{g}it, H., Sezer, M., {\em Lucas collocation method to determination spherical curves in euclidean 3-space}, Communication in Mathematical Modeling and Applications, \textbf{3}(2018), 44--58.
• \c{C}etin, M., Kocayi\u{g}it, H., Sezer, M., {\em On the solution of differential equation system characterizing curve pair of constant Breadth by the Lucas collocation approximation}, New Trends in Mathematical Sciences, \textbf{4}(2016), 168--183.
• Deshmukh, S., Chen, B., Alghanemi, A., {\em Natural mates of Frenet curves in Euclidean 3-space}, Turkish Journal of Mathematics, \textbf{42}(2018), 2826--2840, DOI:10.3906/mat-1712-34.
• Eisenhart, L.P., A Treatise on The Differential Geometry of Curves and Surfaces, Dover Publications, Inc., Mineola-New York, 2004.
• Gray, M., Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, CRC Press, 1997.
• Hac{\i}saliho\u{g}lu, H. Hilmi., Diferansiyel Geometri, Hac{\i}saliho\u{g}lu Yay{\i}y{\i}nc{\i}l{\i}k, Ankara, 1998.
• Keskin, O., Yayl{\i}, Y., {\em An application of N-Bishop frame to spherical images for direction curves}, International Journal of Geometric Methods in Modern Physics, \textbf{14}(2017), 1750162.
• Kocayi\u{g}it, H., Yaz, N., \c{C}amc{\i}, \c{C}., Hac{\i}saliho\u{g}lu, H. Hilmi.,{\em On the explicit characterization of spherical curves in n-dimensional Euclidean space}, J. Inv. Ill-Posed Problems, \textbf{11}(2003), 245--254.
• O'Neill, B., Elementary Differential Geometry, Second Edition, Academic Press, 2006.
• Okullu, P.B., Kocayi\u{g}it, H., Ayd{\i}n, T.A., {\em An explicit characterization of spherical curves according to Bishop Frame and an approximately solution}, Thermal Science, (2019), DOI:10.2298/TSCI181101049B.
• \"{O}zdamar, E., Hac{\i}saliho\u{g}lu, H. Hilmi., {\em Characterizations of spherical curves in Euclidean n-Space}, Ankara \"{U}niversitesi, Fen Fak\"{u}ltesi Tebli\u{g}ler Dergisi, \textbf{23}(1974), 109--125.
• Sezer, M., {\em Differential equations and integral characterizations for $E^4$ spherical curves}, Do\u{g}a Tr. J. Math, \textbf{13}(1989), 1125--131.
• Sezer, M., Karamete, A., G\"{u}lsu, M., {\em Taylor polynomial solutions of system of linear differential equations with variable coefficients}, International Journal of Computer Mathematics, \textbf{82}(2005), 755--764.
• \c{S}ahiner, B., Sezer, M., {\em Determining constant breadth curve mate of a curve on a surface via Taylor collocation method}, New Trends in Mathematical Sciences, \textbf{6}(2018), 103--115.
• Wong, Y., C., {\em On an explicit characterization of spherical curves}, Proceedings of the American Mathematical Society, \textbf{34}(1972), 239--242.
• Wong, Y.C., {\em A global formulation of the condition for a curve to lie in a sphere}, Monatshefte f\"{u}r Mathematik, \textbf{67}(1963), 363--365.
• Yang, Y., Yun, Y., {\em Moving frame and integrable system of the discrete centroaffine curves in $R^3$}, arXiv preprint, arXiv:1601.06530, (2016).