(2+1) Boyutlu difüzyon denkleminin eşdeğerlik grupları

Yıl 2017, Cilt: 21 Sayı: 6, 1133 - 1139, 01.12.2017

Saadet Özer

https://doi.org/10.16984/saufenbilder.279668

Öz

Bir diferansiyel denklemler grubu keyfi fonksiyonlar,
parametreler içeriyorsa, elimizde aynı yapıda diferansiyel denklemler ailesi
var demektir. Klasik fiziğin hemen hemen tüm alan denklemleri, içerdiği
parametrelerin farklı yapıları  için,
değişik malzemeleri temsil eder. Eşdeğerlik grupları, verilen bir diferansiyel
denklem ailesini değişmez bırakan dönüşüm grupları olarak tanımlanır. Bu
nedenle diferansiyel denklem ailelerinin eşdeğerlik grupları, aynı aileye ait,
farklı denklemler arası ilişkileri inceleme açısından önemli bir çalışma
alanıdır. Bu çalışmada, lineer olmayan 
difüzyon denklemin eşdeğerlik grupları, Lie grupları uygulaması
çerçevesinde incelenmiş ve sonuçlar tartışılmıştır. 

Anahtar Kelimeler

Eşdeğerlik Grupları, Lie Grupları, difüzyon denklemi

Equivalence groups of (2+1) dimensional diffusion equation

Öz

If a given set of differential equations contain some
arbitrary functions, parameters, we have in fact a family of sets of equations
of the same structure. Almost all field equations of classical physichs have
this property, representing different materials with various paramaters.  Equivalence groups are defined as the group
of transformations which leave a given family of differential equations
invariant. Therefore, equivalence group of family of differential equations is
an important area within the framework of the relations between different
equations of the same family. In this work the equivalence groups  of nonlinear diffusion equation are
investigated as application of Lie groups and their results are discussed. 

Anahtar Kelimeler

Equivalence groups, Lie groups, Diffusion equation

Kaynakça

• [1] Erdoğan. S. Şuhubi, “Dış Form Analizi”, Türkiye Bilimler Akademisi , 2008.
• [2] F. Oliveri, MP Speciale. “Equivalence transformations of quasilinear first order systems and reduction to autonomous and homogeneous form”, Acta Applicandae mathematicae, 122, 1, pp: 447–460, 2012.
• [3] S Özer, E S. Şuhubi, “Equivalence transformations for first order balance equations.” International journal of engineering science, 42, 11, pp: 1305-1324, 2004.
• [4] Lev Vasil’evich Ovsiannikov. “Group analysis of differential equations.” Academic Press, 2014.
• [5] Peter J Olver. “Applications of Lie groups to differential equations”, volume 107. Springer Science & Business Media, 2000.
• [6] Nail’ Kha˘ırullovich Ibragimov. “Elementary Lie group analysis and ordinary differential equations”, volume 197. Wiley New York, 1999.
• [7] Ian Lisle. “Equivalence transformations for classes of differential equations”, PhD dissertation, University of British Columbia, 1992.
• [8] Sophus Lie, “Über integralinvarianten und ihre verwertung für die theorie der differentialgleichungen, leipz. Berichte, 49:369–410, 1897.
• [9] LV Ovsiannikov, “Group relations of the equation of non-linear heat conductivity”, In Dokl. Akad. Nauk SSSR, volume 125, pp: 492–495, 1959.
• [10] Cardoso-Bihlo, Elsa Dos Santos, Alexander Bihlo, and Roman O. Popovych, "Enhanced preliminary group classification of a class of generalized diffusion equations”, Communications in Nonlinear Science and Numerical Simulation, 16, 9 pp: 3622-3638, 2011.
• [11] R. Zhdanov,, V. Lahno. "Group classification of the general second-order evolution equation: semi-simple invariance groups."Journal of Physics A: Mathematical and Theoretical, 40, 19 pp: 5083, 2007.
• [12] A. H. Bokhari, A. Y. Al Dweik, A. H. Kara, F. D. Zaman, “A symmetry analysis of some classes of evolutionary nonlinear (2+ 1)-diffusion equations with variable diffusivity”, Nonlinear Dynamics, 62, 1-2 pp: 127-13, 2010.
• [13] N.M Ivanova, C. Sophocleous, R. Tracina, “Lie group analysis of two dimensional variable-coefficient burgers equation”, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 61, 5pp: 793–809, 2010.
• [14] M.S. Bruzon, M.L. Gandarias, M. Torrisi, R. Tracina, “On some applications of transformation groups to a class of nonlinear dispersive equations, Nonlinear Analysis: Real World Applications, 13, 3, pp: 1139–1151, 2012.
• [15] M Torrisi, R Tracina, “Equivalence transformations and symmetries for a heat conduction model, International journal of non-linear mechanics, 33, 3, pp: 473–487, 1998.
• [16] V. Romano, M. Torrisi, “Application of weak equivalence transformations to a group analysis of a drift-diffusion model. Journal of Physics A: Mathematical and General” , 32, 45, pp: 7953, 1999.
• [17] M Torrisi, R Tracina, “Second-order differential invariants of a family of diffusion equations”, Journal of Physics A: Mathematical and General, 38, 34, pp: 7519, 2005.
• [18] M.L. Gandarias, M. Torrisi, R. Tracina, “On some differential invariants for a family of diffusion equations”, Journal of Physics A: Mathematical and Theoretical, 40, 30, pp: 8803, 2007.
• [19] Nail H Ibragimov, C Sophocleous, “Differential invariants of the one dimensional quasi-linear second-order evolution equation”, Communications in Nonlinear Science and Numerical Simulation, 12, 7, pp: 1133–1145, 2007.
• [20] M. Torrisi, R. Tracina, “Exact solutions of a reaction–diffusion system for proteus mirabilis bacterial colonies”, Nonlinear Analysis: Real World Applications, 12, 3, pp: 1865–1874, 2011.
• [21] C. Tsaousi, R. Tracina, C. Sophocleous, “Differential invariants for third order evolution equations” Communications in Nonlinear Science and Numerical Simulation, 20, 2, pp: 352–359, 2015.
• [22] H. B. Kent, F. B. Estabrook. “Geometric approach to invariance groups and solution of partial differential systems”, Journal of Mathematical Physics, 12, 4, pp: 653-666. 1971.
• [23] Élie Cartan, “Les systemes différentiels extérieurs et leurs applications géométriques” Hermann, Paris, 1971.
• [24] Dominic Edelen, “Applied exterior calculus”, Courier Corporation, GB.1985.
• [25] E. S. Şuhubi. “Explicit determination of isovector fields of equivalence groups for balance equations of arbitrary order part II”, International journal of engineering science, 43, 1, pp:1–15, 2005.