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]

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

其中 u=u(x,t),u[t]=${\displaystyle {\frac {\partial u}{\partial t}}}$ 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]： ${\displaystyle p[1]:=.50000000000000000000+.50000000000000000000*tanh(3+.25000000000000000000*x-.62500000000000000000*t)}$

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

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

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  Burgers Huxley eq animation1
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  Burgers Huxley eq animation2
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  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] ${\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

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

Burgers Huxley eq animation4