Order Reduction and μ-Conservation Law for the Non-Isospectral KdV Type Equation a with Variable-Coefficients

Theory of Approximation and Applications

مقاله 3، دوره 12، شماره 1، مرداد 2018، صفحه 29-41   

نوع مقاله: Research Articles

نویسنده

Khodayar Goodarzi^*

Department of Mathematics, Broujerd Branch, Islamic Azad University, Broujerd, Iran.

چکیده

The goal of this paper is to calculate of order reduction of the KdV type
equation and the non-isospectral KdV type equation using the μ-symmetry
method. Moreover we obtain μ-conservation law of the non-isospectral KdV
type equation using the variational problem method.

کلیدواژه‌ها

Symmetry؛ μ-Symmetry؛ μ-Conservation Law؛ Variational problem؛ Order reduction

مراجع

[1]H. Airault, Rational solutions of Painleve equation, Studies in Applied
Mathematics , 1979, 61, P.31-53.

[2]T. B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long
waves in nonlinear dispersive systems, Trans. R. Soc. (Lond) ser.A, 1992,
272, P.234-356.

[3]W. Bluman, F. Cheviakov, C. Anco, Construction of conservation law:
how the direct method generalizes Nother's theorem, Group analysis of
differential equations and integrability, 2009, 12 P.1-23.

[4] J.L. Bona, Bryant, P.J., A mathematical model for long waves generated
by wave makers in nonlinear dispersive systems, Proc. Cambridge Phil.
Soc., 1973, 4, P.12-34.

[5]G. Cicogna, G. Gaeta, P. Morando, On the relation between standard and
\mu-symmetries for PDEs, J. Phys. A. , 2004, 37, P. 9467-9486.

[6]G. Cicogna, G. Gaeta, Norther theorem for \mu-symmetries, J. Phys. A.,
2007, 40, P.11899-11921.

[7] G. Gaeta, P. Morando, On the geometry of lambda-symmetries and PDEs
reduction, J. Phys. A., 2004, 37, P.6955-6975.

[8] G. Gaeta, Lambda and mu-symmetries, SPT2004 , World Scientic,
Singapore, 2005.

[9] KH. Goodarzi, M.Nadjakhah, mu-symmetry and mu-conservation law for
the extended mKdV equation, JNMP (2014),3, P.371-381.

[10]D.J.Korteweg, G. de Vries, On the change of form of long waves advancing
in a rectangular canal and on a new type of long stationary waves, Phil.
Mag. 1895, 39, P.422-443.

[11]A. Kudryashov, I. Sinelshchikov, A note on the Lie symmetry analysis and
exact solutions for the extended mKdV equation, Acta. Appl.Math., 2011,
113, 41-44.

[12]C. Muriel, J.L. Romero, New methods of reduction for ordinary dierential
equation, IMA J. Appl. Math., 2001, 66, P.111-125.

[13]C. Muriel, J.L. Romero, C-symmetries and reduction of equation
without Lie point symmetries, J. Lie Theory, 2003, 13, P.167-188.

[14]C. Muriel, J.L. Romero, P.J. Olver, Variationl C-symmetries and EulerLagrange
equations, J. Di. Eqs, 2006, 222, P.164-184.

[15]C. Muriel, J.L. Romero, Prolongations of vector fields and the invariantsby-derrivation
property, Theor. Math. Phys., 2002, 133,P.1565-1575.

[16]P.J. Olver, Applications of Lie Groups to Differential Equations, New
York, 1986.

[17]C. Vasconcellos, P. N. da Silva, Stabilization of the linear Kawahara
equation, Applied and Computational Mathematics, 2015, 3, P. 45-67.