CONSERVATION LAWS FOR THE (1 + 2)-DIMENSIONAL WAVE EQUATION IN BIOLOGICAL ENVIRONMENT

doi:10.1016/S0252-9602(13)60078-8

Abstract

Abstract:

The derivation of conservation laws for the wave equation on sphere, cone and flat space is considered. The partial Noether approach is applied for wave equation on curved surfaces in terms of the coefficients of the first fundamental form (FFF) and the partial Noether operator´s determining equations are derived. These determining equations are then used to construct the partial Noether operators and conserved vectors for the wave equation on different surfaces. The conserved vectors for the wave equation on the sphere, cone and flat space are simplified using the Lie point symmetry generators of the equation and conserved vectors with the help of the symmetry conservation laws relation.

Key words: partial Noether operators, first fundamental form (FFF), conservation laws

CLC Number:

35R01

Adil JHANGEER. CONSERVATION LAWS FOR THE (1 + 2)-DIMENSIONAL WAVE EQUATION IN BIOLOGICAL ENVIRONMENT[J].Acta mathematica scientia,Series B, 2013, 33(5): 1255-1268.

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References

[1] Bokhari A H, Al-Dweik A Y, Mahomed F M, Zaman F D. Conservation laws of a non linear (n+ 1) wave equation. Nonlinear Anal: Real World Appl, 2010, 11(4): 2862–2870

[2] Azad H, Mustafa M T. Symmetry analysis of wave equation on sphere. J Math Anal Appl, 2007, 333: 1180–1188

[3] Bessel-Hagen E. ¨Uber die Erhaltumgsatzeder Elektrodynamik. Math Ann, 1921, 84: 258–276

[4] Blauman G, Kumei S. Symmetries and Differential Equation. New York: Springer-Verlag, 1989

[5] Cheviakov A F. GeM software package for computaion of symmetries and conservation laws of differential equation. Comp Phys Commun, 2007, 176(1): 48–61

[6] Frese R N, P`amies J C, Olsen J D, Bahatyrove S, Van der Weij-de Wit C D, Aartsma T J, Otto C, Hunter C N, Frenkel D, Van Grondelle R. Protein shape and crowding drive domain formation and curvature in biological membranes. Biophs J, 2008, 94: 640–647

[7] G¨oktas ¨U, Hereman W. Symbolic computation of conserved densties for system of nonlinear evolution equation. J Symb Comput, 1997, 24: 591–621

[8] Hereman W, Adams P J, Eklund H L, Hickman M S, Herbst B M. Direct methods and symbolic software for conservation laws of nonlinear equations//Yan Z, ed. Advances of Nonlinear Waves and Symbolic Computation. New York: Nova Science, 2009: 19–79

[9] Hereman W, Colagrosso M, Sayers R, Ringler A, Deconinck B, Nivala M. Continous and discrete homotopy
operators and the computation of conservation laws//Wang D, Zheng Z, ed. Differential Equations with Symbolic Computation. Basel: Birkh¨auser, 2005: 249–285

[10] Hereman W. Symbolic computation of conservation laws of nonlinear partial differential equations in multi-dimensions. Int J Quant Chem, 2006, 106(1): 278–299

[11] Ibragimov N H, ed. CRC Handbook of Lie Group Analysis of Differential Equations, Vol 1. Symmetries, Exact Solutions and Conservations Laws. Boca Raton: CRC, 1994

[12] Jhangeer A, Naeem I, Qureshi M N. Conservation laws of (1 + n)-dimensional heat equation on curved surfaces. Nonlinear Anal Real World Appl, 2011, 12(3): 1359–1370

[13] Johnpillai A G, Kara A H, Mahomed F M. Conservation laws of a non linear (1 + 1) wave equation. Nonlinear Anal B, 2010, 11(4): 2237–2242

[14] Kara A H, Mahomed F M. Realationship between symmetries and conservation laws. Int J Theort Phy, 2000, 39: 23–40

[15] Kara A H, Mahomed F M. Noether-type symmetries and conservation laws via partial Lagrangian. Nonl Dyn, 2006, 45(3/4): 367–383

[16] Naz R, Mahomed F M, Mason D P. Comparision of different approaches to conservation laws in fluid machanics. Appl Math Comp, 2008, 205: 212–230

[17] Nother E. Invariante variations probleme. Nacr K¨onig Gesell Wissen, G¨ottingen, Math-Phys Kl Heft, 1918, 2: 235–257 (English translation in Transport Theory and Statistical Physics, 1971, 1(3): 186–207)

[18] Olver P J. Applications of Lie Groups to Differential Equations. New York, Berlin, Heidelberg, Tokyo: Springer-Verlag, 1993

[19] Schoenborn O L. Phase ordering and kinetics on curved surfaces [D]. Canada: University of Toronto, 1998

[20] Steudel H. ¨Uber die Zuordnung zwischen invarianzeigenschaften und Erhaltungssatzen. Z Naturforsch, 1962, 17A: 129–132

[21] Stubbs D. Symmetries and analytic structure of phase seperation in curved geometries [D]. Canada: University
of Western Ontario, 2001

[22] Wolf T. A comparision of four approches to the calculation of conservation laws. Euor J Appl Math, 2002, 13: 129–152

[23] Wolf T, Brand A, Mohammadzadeh M. Computer algebra algorithems and routines for the computations of conservation laws and fixing of guage in differential expressions. J Symb Comput, 1999, 27: 221–238

[24] Valiakhmetov A Y. Membrane geometry and proteins functions (reviews). Biologicheskie Membrany, 2008, 25: 83–96