(2+1) Boyutlu difüzyon denkleminin eşdeğerlik grupları

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

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.

Keywords:

Equivalence groups, Lie groups, Diffusion equation,

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[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.