[1] T. Wolf, A comparison of four approaches to the calculation of conservation laws Eur. J. Appl. Math., 13(2) (2002), 129-152.
[2] P. D. Lax, Shock wave and entropy, in Contributions to Functional Analysis, ed. EA Zarantonello, Academic Press, New York, 1971.
[3] R. J. DiPerna, Decay of solutions of hyperbolic systems of conservation laws with a convex extension, Arch. Ration. Mech. An., 64(1) (1977), 1-46.
[4] B. A. Bilby, K. J. Miller, J. R. Willis, Fundamentals of Deformation and Fracture, Cambridge University Press, Cambridge, 1985.
[5] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1993.
[6] P. S. Laplace, Traite de Mecanique Celeste, Tome I, Paris, 1798.
[7] E. Noether, Invariante variations probleme, Nachr. Konig. Gesell. Wiss. Gottingen Math. Phys. Kl. Heft 2 (1918), 235-257, English translation in Transport Theory Statist. Phys. 1(3) (1971), 186-207.
[8] H. Steudel, Uber die Zuordnung zwischen lnvarianzeigenschaften und Erhaltungssatzen, Z. Naturforsch., 17A(2) (1962), 129-132.
[9] Ü. Göktaş, W. Hereman, Symbolic computation of conserved densities for systems of nonlinear evolution equations, J. Symb. Comput., 24(5) (1997), 591-622.
[10] P. J. Adams, W. Hereman, TransPDEDensityFlux.m: Symbolic computation of conserved densities and fluxes for systems of partial differential equations with transcendental nonlinearities, Scientific Software, 2002.
[11] L. D. Poole , W. Hereman, ConservationLawsMD.m: A Mathematica package for the symbolic computation of conservation laws of polynomial systems of nonlinear PDEs in multiple space dimensions, Scientific Software, 2009, available at http://inside.mines.edu/~whereman/.
[12] A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Comput. Phys. Commun., 176(1) (2007), 48-61.
[13] A. F. Cheviakov, Computation of fluxes of conservation laws, J. Eng. Math., 66(1-3) (2010), 153-173.
[14] I. M. Anderson, E. S. Cheb-Terrab, Differential geometry package, Maple Online Help, 2009.
[15] T. M. Rocha Filho, A. Figueiredo, [SADE] A Maple package for the symmetry analysis of differential equations, Comput. Phys. Commun., 182(2) 2011), 467-476.
[16] B. Xia, Z. Qiao, J. Li, An integrable system with peakon, complex peakon, weak kink, and kink peakon interactional solutions, Commun. Nonlinear Sci. Numer. Simul., 63 (2018), 292-306.
[17] G. W. Bluman, A. F. Cheviakov, S. C. Anco, Applications of symmetry methods to partial differential equations (First Edition), Springer, New York, 2010.
[18] R. Naz, Symmetry solutions and conservation laws for some partial differential equations in fluid mechanics, Ph.D. Thesis, University of the
Witwatersrand, Johannesburg, South Africa, 2008.
Conservation Laws for a Model with both Cubic and Quadratic Nonlinearity
are considered for the six cases of coefficients. By using a variational derivative approach, conservation laws were constructed. The computations to derive multipliers and conservation law fluxes are conducted by using a Maple-based package which is called GeM.
[1] T. Wolf, A comparison of four approaches to the calculation of conservation laws Eur. J. Appl. Math., 13(2) (2002), 129-152.
[2] P. D. Lax, Shock wave and entropy, in Contributions to Functional Analysis, ed. EA Zarantonello, Academic Press, New York, 1971.
[3] R. J. DiPerna, Decay of solutions of hyperbolic systems of conservation laws with a convex extension, Arch. Ration. Mech. An., 64(1) (1977), 1-46.
[4] B. A. Bilby, K. J. Miller, J. R. Willis, Fundamentals of Deformation and Fracture, Cambridge University Press, Cambridge, 1985.
[5] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1993.
[6] P. S. Laplace, Traite de Mecanique Celeste, Tome I, Paris, 1798.
[7] E. Noether, Invariante variations probleme, Nachr. Konig. Gesell. Wiss. Gottingen Math. Phys. Kl. Heft 2 (1918), 235-257, English translation in Transport Theory Statist. Phys. 1(3) (1971), 186-207.
[8] H. Steudel, Uber die Zuordnung zwischen lnvarianzeigenschaften und Erhaltungssatzen, Z. Naturforsch., 17A(2) (1962), 129-132.
[9] Ü. Göktaş, W. Hereman, Symbolic computation of conserved densities for systems of nonlinear evolution equations, J. Symb. Comput., 24(5) (1997), 591-622.
[10] P. J. Adams, W. Hereman, TransPDEDensityFlux.m: Symbolic computation of conserved densities and fluxes for systems of partial differential equations with transcendental nonlinearities, Scientific Software, 2002.
[11] L. D. Poole , W. Hereman, ConservationLawsMD.m: A Mathematica package for the symbolic computation of conservation laws of polynomial systems of nonlinear PDEs in multiple space dimensions, Scientific Software, 2009, available at http://inside.mines.edu/~whereman/.
[12] A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Comput. Phys. Commun., 176(1) (2007), 48-61.
[13] A. F. Cheviakov, Computation of fluxes of conservation laws, J. Eng. Math., 66(1-3) (2010), 153-173.
[14] I. M. Anderson, E. S. Cheb-Terrab, Differential geometry package, Maple Online Help, 2009.
[15] T. M. Rocha Filho, A. Figueiredo, [SADE] A Maple package for the symmetry analysis of differential equations, Comput. Phys. Commun., 182(2) 2011), 467-476.
[16] B. Xia, Z. Qiao, J. Li, An integrable system with peakon, complex peakon, weak kink, and kink peakon interactional solutions, Commun. Nonlinear Sci. Numer. Simul., 63 (2018), 292-306.
[17] G. W. Bluman, A. F. Cheviakov, S. C. Anco, Applications of symmetry methods to partial differential equations (First Edition), Springer, New York, 2010.
[18] R. Naz, Symmetry solutions and conservation laws for some partial differential equations in fluid mechanics, Ph.D. Thesis, University of the
Witwatersrand, Johannesburg, South Africa, 2008.
Taşkesen, H., & Alaloush, M. (2019). Conservation Laws for a Model with both Cubic and Quadratic Nonlinearity. Fundamental Journal of Mathematics and Applications, 2(2), 180-185. https://doi.org/10.33401/fujma.587740
AMA
Taşkesen H, Alaloush M. Conservation Laws for a Model with both Cubic and Quadratic Nonlinearity. Fundam. J. Math. Appl. December 2019;2(2):180-185. doi:10.33401/fujma.587740
Chicago
Taşkesen, Hatice, and Mohanad Alaloush. “Conservation Laws for a Model With Both Cubic and Quadratic Nonlinearity”. Fundamental Journal of Mathematics and Applications 2, no. 2 (December 2019): 180-85. https://doi.org/10.33401/fujma.587740.
EndNote
Taşkesen H, Alaloush M (December 1, 2019) Conservation Laws for a Model with both Cubic and Quadratic Nonlinearity. Fundamental Journal of Mathematics and Applications 2 2 180–185.
IEEE
H. Taşkesen and M. Alaloush, “Conservation Laws for a Model with both Cubic and Quadratic Nonlinearity”, Fundam. J. Math. Appl., vol. 2, no. 2, pp. 180–185, 2019, doi: 10.33401/fujma.587740.
ISNAD
Taşkesen, Hatice - Alaloush, Mohanad. “Conservation Laws for a Model With Both Cubic and Quadratic Nonlinearity”. Fundamental Journal of Mathematics and Applications 2/2 (December 2019), 180-185. https://doi.org/10.33401/fujma.587740.
JAMA
Taşkesen H, Alaloush M. Conservation Laws for a Model with both Cubic and Quadratic Nonlinearity. Fundam. J. Math. Appl. 2019;2:180–185.
MLA
Taşkesen, Hatice and Mohanad Alaloush. “Conservation Laws for a Model With Both Cubic and Quadratic Nonlinearity”. Fundamental Journal of Mathematics and Applications, vol. 2, no. 2, 2019, pp. 180-5, doi:10.33401/fujma.587740.
Vancouver
Taşkesen H, Alaloush M. Conservation Laws for a Model with both Cubic and Quadratic Nonlinearity. Fundam. J. Math. Appl. 2019;2(2):180-5.