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Approximate conservation laws of nonlinear perturbed heat and wave equations. (English) Zbl 1257.35017

Summary: We construct approximate conservation laws for non-variational nonlinear perturbed \((1+1)\) heat and wave equations by utilizing the partial Lagrangian approach. These perturbed nonlinear heat and wave equations arise in a number of important applications which are reviewed. Approximate symmetries of these have been obtained in the literature. Approximate partial Noether operators associated with a partial Lagrangian of the underlying perturbed heat and wave equations are derived herein. These approximate partial Noether operators are then used via the approximate version of the partial Noether theorem in the construction of approximate conservation laws of the underlying perturbed heat and wave equations.

MSC:

35B06 Symmetries, invariants, etc. in context of PDEs
35K05 Heat equation
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[1] Baikov, V. A.; Gazizov, R. K.; Ibragimov, N. H., (Ibragimov, N. H., CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3 (1996), CRC Press: CRC Press Boca Raton, Florida)
[2] Kara, A. H.; Mahomed, F. M.; Ünal, G., Approximate symmetries and conservation laws with applications, Int. J. Theor. Phys., 38, 2389-2399 (1999) · Zbl 0989.37076
[3] Johnpillai, A. G.; Kara, A. H., Variational formulation of approximate symmetries and conservation laws, Int. J. Theor. Phys., 40, 1501-1509 (2001) · Zbl 1006.81035
[4] Johnpillai, A. G.; Kara, A. H.; Mahomed, F. M., A basis of approximate conservation laws for PDEs with a small parameter, Int. J. Nonlinear Mech., 41, 830-837 (2006) · Zbl 1160.35318
[5] Johnpillai, A. G.; Kara, A. H.; Mahomed, F. M., Approximate Noether-type symmetries and conservation laws via partial Lagrangians for PDEs with a small parameter, J. Comp. Appl. Math., 223, 508-518 (2009) · Zbl 1158.35306
[6] Johnpillai, A. G.; Kara, A. H.; Mahomed, F. M., Conservation laws of some non-variational perturbed pdes via a partial variational approach, Int. J. Mod. Phys. B, 24, 4253-4267 (2010) · Zbl 1221.35034
[7] Carslaw, H. S.; Jaeger, J. C., Conduction of Heat in Solids (1959), Oxford University: Oxford University Oxford · Zbl 0972.80500
[8] Chester, M., Phys. Rev., 131, 5, 2013-2015 (1963)
[9] Catleno, C., A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Compt. Rend. Acad. Sci. Paris, 247, 431-433 (1958)
[10] Peshkov, V., J. Phys. USSR, 8, 131 (1944)
[11] Vedavarz, A.; Mitra, K.; Kumar, S., Hyperbolic temperature profiles for laser surface interactions, J. Appl. Phys., 76, 9, 5014-5021 (1994)
[12] Pakdemirli, M.; Sahin, A. Z., Approximate symmetries of hyperbolic heat conduction equation with temperature dependent thermal properties, Math. Comput. Appl., 10, 1, 139-145 (2005)
[13] Diatta, B.; Wafo Soh, C.; Khalique, C. M., Approximate symmetries and solutions of the hyperbolic heat equation, Appl. Math. Comp., 205, 1, 263-272 (2008) · Zbl 1159.35303
[14] Arrieta, J.; Carvalho, A. N.; Hal, J. K., A damped hyperbolic equation with critical exponent, Comm. Partial Differ. Equ., 17, 841-866 (1992) · Zbl 0815.35067
[15] Pata, V.; Zelik, Sergey, Comm. Pure Appl. Anal., 5, 3 (2006)
[16] Ahmad, Ferhana; Kara, A. H.; Bokhari, A. H.; Zaman, F. D., On approximate Lagrangian and invariants of scaling reductions of a non-linear wave equation with damping, Appl. Math. Comp., 206, 1, 16-20 (2008) · Zbl 1157.65454
[17] Ibragimov, N. H.; Kara, A. H.; Mahomed, F. M., Lie-Bäcklundand Noether symmetries with applications, Nonlinear Dyn., 15, 1, 15-136 (1998) · Zbl 0912.35011
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