Lucas collocation method to determination spherical curves in euclidean 3-space

Year 2018, Volume: 3 Issue: 3, 44 - 58, 31.12.2018

Muhammed Cetin Huseyin Kocayigit Mehmet Sezer

Abstract

In this study, we give a necassary and sufficient condition for an arbitrary-speed regular space curve to lie on a sphere centered at origin. Also, we obtain the position vector of any regular arbitrary-speed space curve lying on a sphere centered at origin satisfies a third-order linear differential equation whose coefficients is related to speed function, curvature and torsion. Then, a collocation method based on Lucas polynomials is developed for the approximate solutions of this differential equation. Moreover, by means of the Lucas collacation method, we approximately obtain the parametric equation of the spherical curve by using this differential equation. Furthermore, an example is given to demonstrate the efficiency of the method and the results are compared with figures and tables.

Keywords

Spherical curves, Frenet frame, Lucas polynomial and series, collocation points

References

• [1] Wong, Y.C., A Global Formulation of the Condition for a Curve to Lie in a Sphere, Monatsh Math., 67 : 363-365, 1963.
• [2] Breuer, S and Gottlieb D., Explicit Characterization of Spherical Curves, Proceedings of the American Mathematical Society, 27(1): 126-127, 1971.
• [3] Wong, Y.C., On an Explicit Characterization of Spherical Curves, Proceedings of the American Mathematical Society, 34(1): 239-242, 1972.
• [4] ¨Ozdamar, E. and Hacısaliho˘glu, H.H., Characterizations of Spherical Curves in Euclidean n-Space, Fen Fak¨ultesi Tebli˘gler Dergisi, 23 : 109-125, 1974.
• [5] Mehlum, E and Wimp, J., Spherical Curves and Quadratic Relationships for Special Functions, J. Austral. Math. Soc. Ser. B, 27 : 111-124, 1985.
• [6] Karamete, A. and Sezer, M., A Taylor collocation method for the solution of linear integro-differential equations, 79-9, (2002), 987-1000.
• [7] Sezer, M., Karamete, A. and G¨ulsu, M., Taylor polynomial solutions of systems of linear differential equations with variable coefficients, International Journal of Computer Mathematics, 82-6, (2005), 755-764.
• [8] Y¨uzbas¸ı, S¸ . and Sezer, M., An exponential matrix method for solving systems of linear differential equations, Mathematical Methods in the Applied Sciences, 36, (2013), 336-348.
• [9] C¸ etin,M., Sezer,M., G¨uler, C., “Lucas Polynomial Approach for System of High-Order Linear Differential Equations and Residual Error Estimation”, Mathematical Problems in Engineering, Volume 2015, Article ID 625984, 14 pages, (2015).
• [10] C¸ etin, M., Sezer, M., Kocayi˘git, H., An Efficient Method based on Lucas Polynomials for Solving High-Order Linear Boundary Value Problems, Gazi University Journal of Science, 28(3), 483-496, (2015).
• [11] Do Carmo, M.P., Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976.
• [12] Millman, R.S. and Parker, G.D., Elements of Differential Geometry, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1977. [13] O’Neill, B., Elementary Differential Geometry, Academic Press Inc., 1966.
• [14] Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd ed.), CRC Press, Washington, 1998.
• [15] K¨uhnel, W., Differential Geometry Curves-Surfaces-Manifolds, American Mathematical Society, 2006.
• [16] F. A. Oliveira, Collocation and residual correction, Numer. Math., 36 (1980), 27-31.
• [17]˙I. C¸ elik, Approximate calculation of eigenvalues with the method of weighted residuals-collocation method, Applied Mathematics and Computation, 160(2) (2005), 401-410.
• [18]˙I. C¸ elik, Collocation method and residual correction using Chebyshev series, Applied Mathematics and Computation, 174(2) (2006), 910-920.
• [19] S. Shahmorad, Numerical solution of the general form linear Fredholm-Volterra integro-differential equations by the Tau method with an error estimation, Applied Mathematics and Computation, 167(2) (2005), 1418-1429.
• [20] Filipponi, P., Horadam, A.F., Second Derivative Sequences of Fibonacci and Lucas Polynomials, The Fibonacci Quarterly, 1994, 31(3), 194-204.
• [21] Koshy, T., Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, John Wiley &Sons, Inc., 2001.