On admissible curves and their evolution equations in pseudo-Galilean space

Volume 25, Issue 4, pp 370--380 http://dx.doi.org/10.22436/jmcs.025.04.07

Publication Date: August 12, 2021 Submission Date: June 18, 2021 Revision Date: July 08, 2021 Accteptance Date: July 11, 2021

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 929 Downloads
• 1905 Views

Authors

H. S. Abdel-Aziz - Department of Mathematics, Sohag University, Sohag 82524, Egypt. H. M. Serry - Department of Mathematics, Suez Canal University, Ismailia, Egypt. F. M. El-Adawy - Department of Mathematics, Suez Canal University, Ismailia, Egypt. A. A. Khalil - Department of Mathematics, Sohag University, Sohag 82524, Egypt.

Abstract

The evolution equations of some forms of admissible curves in the pseudo-Galilean Space \(G_{3}^1\) are investigated in this paper. In more detail, we use two separate methods to obtain coupled nonlinear partial differential equations of time evolution in terms of their curvatures. The first method studies the evolution equations for admissible curves via the frame field, while the second studies the evolution equations via the velocity vector. Then, the position vectors of the evolving curves are formulated. Also, we conduct comparative research of the evolution equations for curves in different spaces. We furthermore present some models as an application of the evolution equations of the curvature and torsion for admissible curves, confirming our theoretical results.

Share and Cite

Keywords

• Admissible curves
• evolution equations
• Frenet frame
• pseudo-Galilean space
• spacelike and timelike curves

MSC

•  53A35

References

• [1] N. H. Abdel-All, M. A. Abdel-Razek, H. S. Abdel-Aziz, A. A.Khalil, Geometry of evolving plane curves problem via lie group analysis, Stud. Math. Sci., 2 (2011), 51--62
  • View Article


• [2] N. H. Abdel-All, R. A. Hussien, T. Youssef, Evolution of curves via the velocities of the moving frame, J. Math. Comput. Sci., 2 (2012), 1170--1185
  • View Article


• [3] N. H. Abdel-All, S. G. Mohamed, M. T. Al-Dossary, Evolution of generalized space curve as a function of its local geometry, Appl. Math., 5 (2014), 2381--2392
  • View Article


• [4] K. Alkan, S. C. Anco, Integrable systems from inelastic curve flows in 2and 3dimensional Minkowski space, J. Nonlinear Math. Phys., 23 (2016), 256--299
  • View Article


• [5] R. Balakrishnan, R. Blumenfeld, Transformation of general curve evolution to a modified Belavin-Polyakov equation, J. Math. Phys., 38 (1997), 5878--5888
  • View Article


• [6] D. Baldwin, Ü.Göktaş, W. Hereman, L. Hong, R. S. Martino, J. C. Miller, Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs, J. Symbolic Comput., 37 (2004), 669--705
  • View Article


• [7] S. Cengiz, E. B. Koc Ozturk, U. Ozturk, Motions of curves in the pseudo-Galilean space $G^1_3$, Math. Probl. Eng., 2015 (2015), 6 pages
  • View Article


• [8] M. Dede, C. Ekici, On parallel ruled surfaces in Galilean space, Kragujevac J. Math.,, 40 (2016), 47--59
  • View Article


• [9] M. Desbrun, M.-P. Cani, Active implicit surface for animation, In: Proc. Graphics Interface--Canadian Inf. Process. Soc., 1998 (1998), 143--450
  • View Article


• [10] Q. Ding, W. Wang, Y. Wang, A motion of spacelike curves in the Minkowski 3-space and the KdV equation, Phys. Lett. A, 374 (2010), 3201--3205
  • View Article


• [11] B. Divjak, Geometrija pseudogalilejevih prostora, Ph.D. Thesis, University of Zagreb, Zagreb (1997)
  • View Article


• [12] B. Divjak, Curves in pseudo-Galilean geometry, Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 41 (1998), 119--130
  • View Article


• [13] B. Divjak, Z. Milin-Šipuš, Special curves on ruled surfaces in Galilean and pseudo-Galilean spaces, Acta Math. Hungar.,, 98 (2003), 203-215
  • View Article


• [14] B. Divjak, Z. Milin-Šipuš, Minding isometries of ruled surfaces in pseudo-Galilean space, J. Geom., 77 (2003), 35--47
  • View Article


• [15] C. Ekici, M. Dede, On the Darboux vector of ruled surfaces in pseudo-Galilean space, Math. Comput. Appl., 16 (2011), 830--838
  • View Article


• [16] A. S. Fokas, J. Lenells, A new approach to integrable evolution equations on the circle, Proc. A., 477 (2021), 28 pages
  • View Article


• [17] M. Gage, R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69--96
  • View Article


• [18] E. F. D. Goufo, I. T. Toudjeu, Analysis of recent fractional evolution equations and applications, Chaos Solitons Fractals, 126 (2019), 337--350
  • View Article


• [19] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285--314
  • View Article


• [20] N. Gürbüz, Three classes of non-lightlike curve evolution according to Darboux frame and geometric phase, Int. J. Geom. Methods Mod. Phys., 15 (2018), 16 pages
  • View Article


• [21] H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 51 (1972), 477--485
  • View Article


• [22] M. Hisakado, M. Wadati, Moving discrete curve and geometric phase, Phys. Lett. A, 214 (1996), 252--258
  • View Article


• [23] Z. Kucukarslan-Yuzbasi, E. Cavlak-Aslan, M. Inc, D. Baleanu, On exact solutions for new coupled nonlinear models getting evolution of curves in Galilean space, Therm. Sci., 23 (2019), 227--233
  • View Article


• [24] G. L. Lamb Jr, Solitons on moving space curves, J. Mathematical Phys., 18 (1977), 1654--1661
  • View Article


• [25] H. Q. Lu, J.S. Todhunter, T. W. Sze, Congruence conditions for nonplanar developable surfaces and their application to surface recognition, CVGIP: Image Understanding, 58 (1993), 265--285
  • View Article


• [26] K. Nakayama, Motion of curves in hyperboloid in the Minkowski space, J. Phys. Soc. Japan, 67 (1998), 3031--3037
  • View Article


• [27] K. Nakayama, H. Segur, M. Wadati, Integrability and the motion of curves, Phys. Rev. Lett., 69 (1992), 2603--2606
  • View Article


• [28] H. Oztekin, H. G. Bozok, POSITION VECTORS OF ADMISSIBLE CURVES IN 3-DIMENSIONAL PSEUDOGALILEAN SPACE $G^1_3$, Int. Electron. J. Geom., 8 (2015), 21--32
  • View Article


• [29] A. I. Prilepko, A. B. Kostin, I. V. Tikhonov, Inverse problems for evolution equations, De Gruyter, 2020 (2020), 379--389
  • View Article


• [30] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC Press, Boca Raton (1998)
  • View Article


• [31] J. A. Sethian, Level set methods and fast marching methods: Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, Cambridge university press, Cambridge (1999)
  • View Article


• [32] D. J. Unger, Developable surfaces in elastoplastic fracture mechanics, Int. J. Fract., 50 (1991), 33--38
  • View Article


• [33] I. Waini, A. Ishak, I. Pop, Unsteady flow and heat transfer past a stretching/shrinking sheet in a hybrid nanofluid, Int. J. Heat Mass Transf., 136 (2019), 288--397
  • View Article


• [34] F. Yang, Sun, Y.-R. Li, H. X.-X. Li, C.-Y. Huang, The quasi-boundary value method for identifying the initial value of heat equation on a columnar symmetric domain, Numer. Algorithms, 82 (2019), 623--639
  • View Article


• [35] O. G. Yıldız, M. Tosun, A note on evolution of curves in the Minkowski spaces, Adv. Appl. Clifford Algebr., 27 (2017), 2873--2884
  • View Article