带源项的变系数非线性反应扩散方程的精确解

万晖

doi:10.7498/aps.62.090203

摘要

本文利用广义条件对称方法对带源项的变系数非线性反应扩散方程 f(x)ut=(g(x)D(u)ux)x+h(x)P(u)ux+q(x)Q(u)进行研究. 当扩散项D(u)取um (m≠-1,0,1)和eu两种重要情形时，对该方程进行对称约化，得到了具有广义泛函分离变量形式的精确解. 这些精确解包含了该方程对应常系数情况下的解.

关键词:

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.

Keywords:

generalized conditional symmetry /
exact solution /
the nonlinear diffusion-convection equation

作者及机构信息

万晖

1.
西北大学数学系非线性科学研究中心, 西安 710069

基金项目: 国家自然科学基金(批准号:11001220)和陕西省教育厅基金(批准号:2010jk866)资助的课题.

Authors and contacts

Wan Hui

1.
Center for Nonlinear Studies, Department of Mathematics, Northwest University, Xi'an 710069, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11001220), and the Educational Science Foundation of Shaanxi Province, China (Grant No. 2010JK866).

参考文献

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[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 物理学报 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 物理学报 58 2121]
[21]	Ma Y L, Li B Q 2009 Acta Phys. Sin. 58 2121 (in Chinese) [马玉兰, 李帮庆 2009 物理学报 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 物理学报 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

施引文献

[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 物理学报 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 物理学报 58 2121]
[21]	Ma Y L, Li B Q 2009 Acta Phys. Sin. 58 2121 (in Chinese) [马玉兰, 李帮庆 2009 物理学报 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 物理学报 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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