伯格斯-赫胥黎方程

伯格斯-赫胥黎方程（Burgers-Huxley equation) 是一个模拟物理学、生物学、经济学和生态学等领域非线性波动现象的非线性偏微分方程^[1]

$pde1:=u[t]-nu*u[x,x]+a*u*u[x]=b*u*(1-u)*(u-c)$

其中 u=u(x,t),u[t]= ${\frac {\partial u}{\partial t}}$ 等等。

解析解

特解

以

  :{a = 1, b = 1, c = 1.5, nu = 1}
  :{a = 1, b = 1, c = 2, nu = 1}
  :{a = -1, b = 1, c = 2.3, nu = 1}

代人伯格斯-赫胥黎方程后求解得^[2]： $p[1]:=.50000000000000000000+.50000000000000000000*tanh(3+.25000000000000000000*x-.62500000000000000000*t)$

$p[2]:=1-tanh(3+x-t)$

$p[3]:=.50000000000000000000+.50000000000000000000*coth(3+.50000000000000000000*x-.65000000000000000000*t)$

Burgers Huxley eq animation1
Burgers Huxley eq animation2
Burgers Huxley eq animation3

通解

伯格斯-赫胥黎方程有tanh展开行波解，不存在csch展开行波解^[3] 解析失败 (转换错误。服务器（“https://wikimedia.org/api/rest_”）报告：“Cannot get mml. upstream connect error or disconnect/reset before headers. reset reason: connection termination”): {\displaystyle sol6:=u=1/2+(1/2)*tanh(_{C}1+(1/8)*(-a+sqrt(a^{2}+8*b*nu))*x/nu+(1/8)*(-a^{1}4*b-2880*b^{6}*a^{4}*c^{3}*nu^{5}+4196*b^{6}*a^{4}*c*nu^{5}+7840*b^{6}*a^{4}*c^{2}*nu^{5}+64*c^{5}*b^{6}*a^{4}*nu^{5}+96*a^{1}0*b^{3}*c*nu^{2}+8*a^{1}0*b^{3}*c^{2}*nu^{2}+208*a^{8}*nu^{3}*b^{4}*c^{2}+840*a^{8}*nu^{3}*b^{4}*c+4*b^{2}*a^{1}2*c*nu-32*a^{8}*nu^{3}*b^{4}*c^{3}-16*a^{6}*nu^{4}*b^{5}*c^{4}+3152*a^{6}*b^{5}*nu^{4}*c+1952*a^{6}*b^{5}*nu^{4}*c^{2}-544*a^{6}*nu^{4}*b^{5}*c^{3}+11880*b^{7}*a^{2}*nu^{6}*c^{2}+648*c^{5}*b^{7}*a^{2}*nu^{6}-2160*c^{4}*b^{7}*a^{2}*nu^{6}-4536*c^{3}*b^{7}*a^{2}*nu^{6}-352*b^{6}*a^{4}*c^{4}*nu^{5}-432*b^{7}*a^{2}*nu^{6}*c-6081*a^{6}*b^{5}*nu^{4}-2064*a^{8}*nu^{3}*b^{4}-3348*b^{7}*a^{2}*nu^{6}-1296*nu^{7}*b^{8}*c+3240*nu^{7}*b^{8}*c^{2}-3240*c^{4}*b^{8}*nu^{7}+1296*c^{5}*b^{8}*nu^{7}-8106*b^{6}*a^{4}*nu^{5}-354*a^{1}0*b^{3}*nu^{2}-30*b^{2}*a^{1}2*nu+(1/4)*(-a+sqrt(a^{2}+8*b*nu))*a^{1}5/nu+(8*(-a+sqrt(a^{2}+8*b*nu)))*b*a^{1}3-584*nu^{3}*(-a+sqrt(a^{2}+8*b*nu))*b^{4}*a^{7}*c^{2}-540*nu^{6}*(-a+sqrt(a^{2}+8*b*nu))*a*b^{7}*c^{5}-3456*nu^{6}*(-a+sqrt(a^{2}+8*b*nu))*a*b^{7}*c^{2}+4*nu^{3}*(-a+sqrt(a^{2}+8*b*nu))*b^{4}*a^{7}*c^{4}-192*nu^{5}*(-a+sqrt(a^{2}+8*b*nu))*b^{6}*a^{3}*c^{5}+696*nu^{5}*(-a+sqrt(a^{2}+8*b*nu))*b^{6}*a^{3}*c^{4}-16*nu^{4}*(-a+sqrt(a^{2}+8*b*nu))*b^{5}*a^{5}*c^{5}+96*nu^{4}*(-a+sqrt(a^{2}+8*b*nu))*b^{5}*a^{5}*c^{4}+1512*nu^{6}*(-a+sqrt(a^{2}+8*b*nu))*a*b^{7}*c^{4}-2760*nu^{4}*(-a+sqrt(a^{2}+8*b*nu))*b^{5}*a^{5}*c^{2}+2160*nu^{5}*(-a+sqrt(a^{2}+8*b*nu))*c^{3}*b^{6}*a^{3}+972*nu^{6}*(-a+sqrt(a^{2}+8*b*nu))*c^{3}*a*b^{7}+152*nu^{3}*(-a+sqrt(a^{2}+8*b*nu))*c^{3}*b^{4}*a^{7}+960*nu^{4}*(-a+sqrt(a^{2}+8*b*nu))*c^{3}*b^{5}*a^{5}+8*nu^{2}*(-a+sqrt(a^{2}+8*b*nu))*c^{3}*b^{3}*a^{9}-1128*nu^{3}*(-a+sqrt(a^{2}+8*b*nu))*b^{4}*a^{7}*c-2089*nu^{4}*(-a+sqrt(a^{2}+8*b*nu))*b^{5}*a^{5}*c-26*nu*(-a+sqrt(a^{2}+8*b*nu))*b^{2}*a^{1}1*c-254*nu^{2}*(-a+sqrt(a^{2}+8*b*nu))*b^{3}*a^{9}*c-726*nu^{5}*(-a+sqrt(a^{2}+8*b*nu))*b^{6}*a^{3}*c+864*nu^{6}*(-a+sqrt(a^{2}+8*b*nu))*a*b^{7}*c-2*nu*(-a+sqrt(a^{2}+8*b*nu))*b^{2}*a^{1}1*c^{2}-56*nu^{2}*(-a+sqrt(a^{2}+8*b*nu))*b^{3}*a^{9}*c^{2}-5610*nu^{5}*(-a+sqrt(a^{2}+8*b*nu))*b^{6}*a^{3}*c^{2}-(-a+sqrt(a^{2}+8*b*nu))*b*a^{1}3*c+(9193/4)*nu^{3}*(-a+sqrt(a^{2}+8*b*nu))*b^{4}*a^{7}+3931*nu^{4}*(-a+sqrt(a^{2}+8*b*nu))*b^{5}*a^{5}+(205/2)*nu*(-a+sqrt(a^{2}+8*b*nu))*b^{2}*a^{1}1+667*nu^{2}*(-a+sqrt(a^{2}+8*b*nu))*b^{3}*a^{9}+2673*nu^{5}*(-a+sqrt(a^{2}+8*b*nu))*b^{6}*a^{3}+324*nu^{6}*(-a+sqrt(a^{2}+8*b*nu))*a*b^{7})*t/(nu*(-a^{1}2*b-8*a^{8}*b^{3}*c*nu^{2}+8*a^{8}*b^{3}*c^{2}*nu^{2}+144*nu^{3}*b^{4}*a^{6}*c^{2}-144*nu^{3}*b^{4}*a^{6}*c-16*nu^{4}*a^{4}*b^{5}*c^{4}-848*a^{4}*b^{5}*nu^{4}*c+832*a^{4}*b^{5}*nu^{4}*c^{2}-1728*c*b^{6}*nu^{5}*a^{2}+32*nu^{4}*a^{4}*b^{5}*c^{3}-162*a^{2}*b^{6}*c^{4}*nu^{5}+324*a^{2}*b^{6}*c^{3}*nu^{5}+1566*a^{2}*b^{6}*c^{2}*nu^{5}+324*b^{7}*nu^{6}*c^{2}-324*c^{4}*b^{7}*nu^{6}+648*c^{3}*b^{7}*nu^{6}-254*a^{8}*b^{3}*nu^{2}-26*a^{1}0*nu*b^{2}-648*nu^{6}*b^{7}*c-2217*b^{5}*a^{4}*nu^{4}-1350*b^{6}*a^{2}*nu^{5}-1136*nu^{3}*a^{6}*b^{4}+7*a^{1}1*b*(-a+sqrt(a^{2}+8*b*nu))+(1/4)*a^{1}3*(-a+sqrt(a^{2}+8*b*nu))/nu-2*b^{2}*a^{9}*nu*(-a+sqrt(a^{2}+8*b*nu))*c^{2}-272*b^{4}*a^{5}*nu^{3}*(-a+sqrt(a^{2}+8*b*nu))*c^{2}-40*b^{3}*a^{7}*nu^{2}*(-a+sqrt(a^{2}+8*b*nu))*c^{2}-8*b^{4}*a^{5}*nu^{3}*(-a+sqrt(a^{2}+8*b*nu))*c^{3}-459*nu^{5}*(-a+sqrt(a^{2}+8*b*nu))*a*b^{6}*c^{2}-270*nu^{5}*(-a+sqrt(a^{2}+8*b*nu))*a*b^{6}*c^{3}+135*nu^{5}*(-a+sqrt(a^{2}+8*b*nu))*a*b^{6}*c^{4}+4*b^{4}*a^{5}*nu^{3}*c^{4}*(-a+sqrt(a^{2}+8*b*nu))+2*b^{2}*a^{9}*nu*(-a+sqrt(a^{2}+8*b*nu))*c+594*nu^{5}*(-a+sqrt(a^{2}+8*b*nu))*a*b^{6}*c+744*nu^{4}*(-a+sqrt(a^{2}+8*b*nu))*a^{3}*b^{5}*c+276*b^{4}*a^{5}*nu^{3}*(-a+sqrt(a^{2}+8*b*nu))*c+40*b^{3}*a^{7}*nu^{2}*(-a+sqrt(a^{2}+8*b*nu))*c-696*nu^{4}*(-a+sqrt(a^{2}+8*b*nu))*a^{3}*b^{5}*c^{2}-96*nu^{4}*(-a+sqrt(a^{2}+8*b*nu))*a^{3}*b^{5}*c^{3}+48*nu^{4}*(-a+sqrt(a^{2}+8*b*nu))*a^{3}*b^{5}*c^{4}+(151/2)*b^{2}*a^{9}*nu*(-a+sqrt(a^{2}+8*b*nu))+162*nu^{5}*(-a+sqrt(a^{2}+8*b*nu))*a*b^{6}+918*nu^{4}*(-a+sqrt(a^{2}+8*b*nu))*a^{3}*b^{5}+(3809/4)*b^{4}*a^{5}*nu^{3}*(-a+sqrt(a^{2}+8*b*nu))+389*b^{3}*a^{7}*nu^{2}*(-a+sqrt(a^{2}+8*b*nu)))))}

代人参数params1 := {a = 1, b = 1, c = 1.5, nu = 1} 得

$q2:=.50000000000000000000-.50000000000000000000*tanh((.12500000000000000000-.33071891388307382381*I)*x+(1.1875000000000000000+.16535945694153691190*I)*t)$

Burgers Huxley eq animation4

参考文献

^ Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple p13-25 Springer
^ Inna Shingareva, Carlos Lizarrage-Celaya p15
^ Inna Shingareva, Carlos Lizarrage-Celaya p15

*谷超豪《孤立子理论中的达布变换及其几何应用》上海科学技术出版社
*阎振亚著《复杂非线性波的构造性理论及其应用》科学出版社 2007年
李志斌编著《非线性数学物理方程的行波解》科学出版社
王东明著《消去法及其应用》科学出版社 2002
*何青王丽芬编著《Maple 教程》科学出版社 2010 ISBN 9787030177445
Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
Dongming Wang, Elimination Practice,Imperial College Press 2004
David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759

[1] Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple p13-25 Springer

[2] Inna Shingareva, Carlos Lizarrage-Celaya p15

[3] Inna Shingareva, Carlos Lizarrage-Celaya p15

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