伯格斯-赫胥黎方程

伯格斯-赫胥黎方程（Burgers-Huxley equation) 是一個模擬物理學、生物學、經濟學和生態學等領域非線性波動現象的非線性偏微分方程^[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-赫胥黎方程後求解得^[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展開行波解，不存在csch展開行波解^[3] 解析失败 (转换错误。服务器（“https://wikimedia.org/api/rest_”）报告：“Cannot get mml. upstream request timeout”): {\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)))))}

代人參數params1 := {a = 1, b = 1, c = 1.5, nu = 1} 得

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

參考文獻

^ 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《孤立子理論中的達布變換及其幾何應用》上海科學技術出版社
*閻振亞著《複雜非線性波的構造性理論及其應用》科學出版社 2007年
李志斌編著《非線性數學物理方程的行波解》科學出版社
王東明著《消去法及其應用》科學出版社 2002
*何青王麗芬編著《Maple 教程》科學出版社 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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