On the Tzitzeica Curve Equation
Lewis R. Williams
Fayetteville State University
Faculty Mentor: Nicoleta Bȋlă
Fayetteville State University
Abstract
The Tzitzeica curve equation is a nonlinear ordinary differential equation whose solutions are
called Tzitzeica curves. The aim of this paper is to present the Tzitzeica curve equation along
with new, particular families of Tzitzeica curves.
105
Explorations |Science, Mathematics, and Technology
106
Lewis R. Williams
107
Explorations |Science, Mathematics, and Technology
108
Lewis R. Williams
109
Explorations |Science, Mathematics, and Technology
110
Lewis R. Williams
111
Explorations |Science, Mathematics, and Technology
112
Lewis R. Williams
Figure 1
Figure 2
Figure 3
113
Explorations |Science, Mathematics, and Technology
Figure 4
Figure 5
References
[1] A. F. Agnew, A. Bobe, W. G. Boskoff and B. D. Suceava, Tzitzeica curves
and surfaces, The Mathematica Journal, 12(2010), 1–18.
[2] G. Bluman, J. Cole, Similarity Methods for Differential Equations, Appl.
Math. Sci., Vol. 13, Springer-Verlag New York, Heidelberg, Berlin,
1974.
[3] G. Bluman, S. Kumei, Symmetries and Differential Equations, Appl. Math.
Sci, Vol. 81, Springer-Verlag New York, Heidelberg, Berlin, 1989.
[4] M. Crâșmăreanu, Cylindrical Tzitzeica curves implies forced harmonic
oscillators, Balkan J. Geom. Appl., 7(2002), No. 1, 37–42.
114
Lewis R. Williams
[5] P. E. Hydon, Symmetry Methods for Differential Equations, Cambridge
Texts in Applied Mathematics, Cambridge University Press, 2000.
[6] N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential
Equations: Volume 1, Symmetries, Exact Solutions, and Conservation
Laws, CRC Press, Boca Raton, 1993.
[7] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate
Texts in Mathematics, Vol. 107, Second Edition, Springer-Verlag, New
York, 2000.
[8] P. J. Olver, Classical Invariant Theory, London Mathematical Society,
Student Texts, 44, Cambridge University Press, 1999.
[9] A. Pressley, Elementary Differential Geometry, Springer Undergraduate
Mathematics Series, Springer-Verlag London Limited, 2012.
[10] C. Rogers, W. K. Schief, Bäcklund and Darboux Transformations,
Geometry and Modern Applications in Soliton Theory, Cambridge Texts
in Applied Mathematics, Cambridge University Press, 2002.
[11] V. Rovenski, Geometry of Curves and Surfaces with Maple, Birkhäuser
Boston, 2000.
[12] J. Stewart, Multivarible Calculus: Concepts & Contexts, Brooks Cole,
Cengage Learning, 2010.
[13] G. Tzitzeica, Sur Certaines Courbes Gauches, Ann. de l’Ec. Normale Sup.,
28 (1911), 9–32.
115