On admissible curves and their evolution equations in pseudo-Galilean space
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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.
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ISRP Style
H. S. Abdel-Aziz, H. M. Serry, F. M. El-Adawy, A. A. Khalil, On admissible curves and their evolution equations in pseudo-Galilean space, Journal of Mathematics and Computer Science, 25 (2022), no. 4, 370--380
AMA Style
Abdel-Aziz H. S., Serry H. M., El-Adawy F. M., Khalil A. A., On admissible curves and their evolution equations in pseudo-Galilean space. J Math Comput SCI-JM. (2022); 25(4):370--380
Chicago/Turabian Style
Abdel-Aziz, H. S., Serry, H. M., El-Adawy, F. M., Khalil, A. A.. "On admissible curves and their evolution equations in pseudo-Galilean space." Journal of Mathematics and Computer Science, 25, no. 4 (2022): 370--380
Keywords
- Admissible curves
- evolution equations
- Frenet frame
- pseudo-Galilean space
- spacelike and timelike curves
MSC
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
-
[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
-
[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
-
[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
-
[5]
R. Balakrishnan, R. Blumenfeld, Transformation of general curve evolution to a modified Belavin-Polyakov equation, J. Math. Phys., 38 (1997), 5878--5888
-
[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
-
[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
-
[8]
M. Dede, C. Ekici, On parallel ruled surfaces in Galilean space, Kragujevac J. Math.,, 40 (2016), 47--59
-
[9]
M. Desbrun, M.-P. Cani, Active implicit surface for animation, In: Proc. Graphics Interface--Canadian Inf. Process. Soc., 1998 (1998), 143--450
-
[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
-
[11]
B. Divjak, Geometrija pseudogalilejevih prostora, Ph.D. Thesis, University of Zagreb, Zagreb (1997)
-
[12]
B. Divjak, Curves in pseudo-Galilean geometry, Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 41 (1998), 119--130
-
[13]
B. Divjak, Z. Milin-Šipuš, Special curves on ruled surfaces in Galilean and pseudo-Galilean spaces, Acta Math. Hungar.,, 98 (2003), 203-215
-
[14]
B. Divjak, Z. Milin-Šipuš, Minding isometries of ruled surfaces in pseudo-Galilean space, J. Geom., 77 (2003), 35--47
-
[15]
C. Ekici, M. Dede, On the Darboux vector of ruled surfaces in pseudo-Galilean space, Math. Comput. Appl., 16 (2011), 830--838
-
[16]
A. S. Fokas, J. Lenells, A new approach to integrable evolution equations on the circle, Proc. A., 477 (2021), 28 pages
-
[17]
M. Gage, R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69--96
-
[18]
E. F. D. Goufo, I. T. Toudjeu, Analysis of recent fractional evolution equations and applications, Chaos Solitons Fractals, 126 (2019), 337--350
-
[19]
M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285--314
-
[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
-
[21]
H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 51 (1972), 477--485
-
[22]
M. Hisakado, M. Wadati, Moving discrete curve and geometric phase, Phys. Lett. A, 214 (1996), 252--258
-
[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
-
[24]
G. L. Lamb Jr, Solitons on moving space curves, J. Mathematical Phys., 18 (1977), 1654--1661
-
[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
-
[26]
K. Nakayama, Motion of curves in hyperboloid in the Minkowski space, J. Phys. Soc. Japan, 67 (1998), 3031--3037
-
[27]
K. Nakayama, H. Segur, M. Wadati, Integrability and the motion of curves, Phys. Rev. Lett., 69 (1992), 2603--2606
-
[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
-
[29]
A. I. Prilepko, A. B. Kostin, I. V. Tikhonov, Inverse problems for evolution equations, De Gruyter, 2020 (2020), 379--389
-
[30]
A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC Press, Boca Raton (1998)
-
[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)
-
[32]
D. J. Unger, Developable surfaces in elastoplastic fracture mechanics, Int. J. Fract., 50 (1991), 33--38
-
[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
-
[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
-
[35]
O. G. Yıldız, M. Tosun, A note on evolution of curves in the Minkowski spaces, Adv. Appl. Clifford Algebr., 27 (2017), 2873--2884