Taylor-Matrix Collocation Method to Solution of Differential Equations Characterizing Spherical Curves in Euclidean 4-Space

Abstract

In this study we consider a third order linear differential equation with variable coefficients characterizing spherical curves according to Frenet frame in Euclidean 4-Space . This equation whose coefficients are related to special function, curvature and torsion, is satisfied by the position vector of any regular unit velocity spherical curve. These type equations are generally impossible to solve analytically and so, for approximate solution we present a numerical method based on Taylor polynomials and collocations points by using initial conditions. Our method reduces the solution of problem to the solution of a system of algebraic equations and the approximate solution is obtained in terms of Taylor polynomials.

Keywords

• Curves in Euclidean Space,Spherical curves,Taylor matrix method,Frenet frame,Linear differential equations; Matrix and collocation method

References

1. 1. Euler, L. 1778. De curvis trangularibis, Acta Academica Petropol; 1780: 3-30.
2. 2. Fujivara, M. 1914. On space curves of constant breadth, Thoku Mathematical Journal; 5: 179-184.
3. 3. Blaschke, W, Leipziger Berichte; 1917, 67, pp 290.
4. 4. Wong, Y-C. 1963. A global formulation of the condition for a curve to Lie in a sphere, Monatshefte für Mathematik, 67(4), 363-365.
5. 5. Reuleaux, F, The Kinematics of Machinery; Trans. By Kennedy A.B.W. Dover Publishers: New York, 1963.
6. 6. Gluck. H. 1966. Higher curvatures of curves in Euclidean space, The American Mathematical Montly, 73, 699-704.
7. 7. Bruer, S, Gottlieb, D. 1971. Explicit characterization of spherical curves, Proceedings of the American Mathematical Society, 27(1), 126-127.
8. 8. Dannon, V. 1981. Integral characterizations and the theory of curves, Proceedings of the American Mathematical Society, 81(4), 600-602.

9. 9. Sezer, M, Integral properties and applications of a Frenet-like differential equation system, II. National Mathematical Symposium, Ege University, İzmir, 1989, 1, pp 435-444.
10. 10. Akdoğan, Z, Mağden, A. 2001. Some characterization of curves of constant breadth in En space, Turkish Journal of Mathematics; 25, 433-444.
11. 11. Sezer, M. 1996. A method for the approximate solution of the second order linear differential equations in terms of Taylor Polynomials, International Journal of Mathematical Education in Science and Technology, 27(6), 821- 834.
12. 12. Hacısalihoğlu, H.H, Differential Geometry; Ankara Univ. Press: Ankara, Turkey, 1993.
13. 13. O’Neill, B, Elemantary Differential Geometry; Academic Press Inc: 1966.
14. 14. Karger, A, Novak, J, Space Kinematics and Lie Groups; Gordon And Breach Science Publishers: 1985. 15. Millman, R.S, Parker, G.D, Elements of Differential Geometry; Prentice-Hall, Inc., Englewood Cliffs Press: New Jersey, 1977.