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The nonlinear diffusion-convection equation f(x)ut=(g(x)D(u)ux)x+h(x)P(u)ux+q(x)Q(u) with variable coefficients and source term has been studied. This equation is symmetrically reduced by the generalized conditional symmetry method. Some exact solutions to the resulting equations are constructed, with the diffusion terms D(u)=um (m≠-1,0,1) and D(u)=eu. These exact solutions are also the generalized functional separable solutions. Solutions to the equation with constant coefficients are covered by those exact solutions to the equation with variable coefficients.
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Keywords:
- generalized conditional symmetry /
- exact solution /
- the nonlinear diffusion-convection equation
[1] Olver P J 1993 Applications of Lie Groups to Differential Equations (New York: Springer) p75
[2] Saied E A 1994 J. Phys. A: Math. Gen. 27 4867
[3] King J R 1990 J. Phys. A: Math. Gen. 23 3681
[4] Sophocleous C 1998 J. Phys. A: Math. Gen. 31 6293
[5] Gandarias M L, Bruzón M S 2008 Commun. Nonlinear Sci. Numer. Simul. 13 508
[6] Qu C Z 1999 IMA J. Appl. Math. 62 283
[7] Qu C Z, Estévez P G 2004 Nonlinear Anal. TMA 37 549
[8] Qu C Z, Ji L N 2009 Nonlinear Analysis 71 243
[9] Lou S Y 1996 J. Phys. A: Math. Gen. 29 4209
[10] Lou S Y 2000 Phys. Lett. A 277 94
[11] Ji F Y, Zhang S L 2012 Acta Phys. Sin. 61 080202 (in Chinese) [吉飞宇, 张顺利 2012 61 080202]
[12] Galaktionov V A 1995 Proc. Roy. Soc. Edinburgh 125 225
[13] Galaktionov V A, Posashkov S A 1996 Physica D 99 217
[14] Goard J M 2000 Eur. J. Appl. Math. 11 215
[15] Ivanova N M 2008 Dynamics of PDE 5 139
[16] Ivanova N M, Popovych R O, Sophocleous C 2010 Lobachevskii Journal of mathematics 31 100
[17] Crank J 1979 Mathematics of Diffusion (2nd ed.) (London: Oxford)
[18] Peletier L A 1981 Applications of Nonlinear Analysis in the Physical Sciences (London: Pitman)
[19] Sophocleous C 2003 Physica A 320 169
[20] Tao G T S, Si R D E J 2009 Acta Phys. Sin. 58 2121 (in Chinese) [套格图桑, 斯仁道尔吉 2009 58 2121]
[21] Ma Y L, Li B Q 2009 Acta Phys. Sin. 58 2121 (in Chinese) [马玉兰, 李帮庆 2009 58 4373]
[22] Zhang S L, Qu C Z 2006 Chin. Phys. Lett. 23 527
[23] Zhang H Q, Fan E G, Lin G 1998 Chin. Phys. 7 649
[24] Mo J Q 2009 Acta Phys. Sin. 58 2930 (in Chinese) [莫嘉琪 2009 58 2930]
[25] Fokas A S, Liu Q M 1994 Phys. Rev. Lett. 72 3293
[26] Zhdanov R Z 1995 J. Phys. A: Math. Gen. 128 3841
[27] Qu C Z 1997 Stud. Appl. Math. 99 107
[28] Ji L N 2012 J. Math. Anal. Appl. 389 979
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[1] Olver P J 1993 Applications of Lie Groups to Differential Equations (New York: Springer) p75
[2] Saied E A 1994 J. Phys. A: Math. Gen. 27 4867
[3] King J R 1990 J. Phys. A: Math. Gen. 23 3681
[4] Sophocleous C 1998 J. Phys. A: Math. Gen. 31 6293
[5] Gandarias M L, Bruzón M S 2008 Commun. Nonlinear Sci. Numer. Simul. 13 508
[6] Qu C Z 1999 IMA J. Appl. Math. 62 283
[7] Qu C Z, Estévez P G 2004 Nonlinear Anal. TMA 37 549
[8] Qu C Z, Ji L N 2009 Nonlinear Analysis 71 243
[9] Lou S Y 1996 J. Phys. A: Math. Gen. 29 4209
[10] Lou S Y 2000 Phys. Lett. A 277 94
[11] Ji F Y, Zhang S L 2012 Acta Phys. Sin. 61 080202 (in Chinese) [吉飞宇, 张顺利 2012 61 080202]
[12] Galaktionov V A 1995 Proc. Roy. Soc. Edinburgh 125 225
[13] Galaktionov V A, Posashkov S A 1996 Physica D 99 217
[14] Goard J M 2000 Eur. J. Appl. Math. 11 215
[15] Ivanova N M 2008 Dynamics of PDE 5 139
[16] Ivanova N M, Popovych R O, Sophocleous C 2010 Lobachevskii Journal of mathematics 31 100
[17] Crank J 1979 Mathematics of Diffusion (2nd ed.) (London: Oxford)
[18] Peletier L A 1981 Applications of Nonlinear Analysis in the Physical Sciences (London: Pitman)
[19] Sophocleous C 2003 Physica A 320 169
[20] Tao G T S, Si R D E J 2009 Acta Phys. Sin. 58 2121 (in Chinese) [套格图桑, 斯仁道尔吉 2009 58 2121]
[21] Ma Y L, Li B Q 2009 Acta Phys. Sin. 58 2121 (in Chinese) [马玉兰, 李帮庆 2009 58 4373]
[22] Zhang S L, Qu C Z 2006 Chin. Phys. Lett. 23 527
[23] Zhang H Q, Fan E G, Lin G 1998 Chin. Phys. 7 649
[24] Mo J Q 2009 Acta Phys. Sin. 58 2930 (in Chinese) [莫嘉琪 2009 58 2930]
[25] Fokas A S, Liu Q M 1994 Phys. Rev. Lett. 72 3293
[26] Zhdanov R Z 1995 J. Phys. A: Math. Gen. 128 3841
[27] Qu C Z 1997 Stud. Appl. Math. 99 107
[28] Ji L N 2012 J. Math. Anal. Appl. 389 979
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