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
pde1 := u[t]-nuu[x, x]+auu[x] = bu(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：
p[1] := .50000000000000000000+.50000000000000000000tanh(3+.25000000000000000000x-.62500000000000000000*t)
p[2] := 1-tanh(3+x-t)
p[3] := .50000000000000000000+.50000000000000000000coth(3+.50000000000000000000x-.65000000000000000000*t)
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
sol6 := u = 1/2+(1/2)tanh(_C1+(1/8)(-a+sqrt(a^2+8bnu))x/nu+(1/8)(-a^14b-2880b^6a^4c^3nu^5+4196b^6a^4cnu^5+7840b^6a^4c^2nu^5+64c^5b^6a^4nu^5+96a^10b^3cnu^2+8a^10b^3c^2nu^2+208a^8nu^3b^4c^2+840a^8nu^3b^4c+4b^2a^12cnu-32a^8nu^3b^4c^3-16a^6nu^4b^5c^4+3152a^6b^5nu^4c+1952a^6b^5nu^4c^2-544a^6nu^4b^5c^3+11880b^7a^2nu^6c^2+648c^5b^7a^2nu^6-2160c^4b^7a^2nu^6-4536c^3b^7a^2nu^6-352b^6a^4c^4nu^5-432b^7a^2nu^6c-6081a^6b^5nu^4-2064a^8nu^3b^4-3348b^7a^2nu^6-1296nu^7b^8c+3240nu^7b^8c^2-3240c^4b^8nu^7+1296c^5b^8nu^7-8106b^6a^4nu^5-354a^10b^3nu^2-30b^2a^12nu+(1/4)(-a+sqrt(a^2+8bnu))a^15/nu+(8(-a+sqrt(a^2+8bnu)))ba^13-584nu^3(-a+sqrt(a^2+8bnu))b^4a^7c^2-540nu^6(-a+sqrt(a^2+8bnu))ab^7c^5-3456nu^6(-a+sqrt(a^2+8bnu))ab^7c^2+4nu^3(-a+sqrt(a^2+8bnu))b^4a^7c^4-192nu^5(-a+sqrt(a^2+8bnu))b^6a^3c^5+696nu^5(-a+sqrt(a^2+8bnu))b^6a^3c^4-16nu^4(-a+sqrt(a^2+8bnu))b^5a^5c^5+96nu^4(-a+sqrt(a^2+8bnu))b^5a^5c^4+1512nu^6(-a+sqrt(a^2+8bnu))ab^7c^4-2760nu^4(-a+sqrt(a^2+8bnu))b^5a^5c^2+2160nu^5(-a+sqrt(a^2+8bnu))c^3b^6a^3+972nu^6(-a+sqrt(a^2+8bnu))c^3ab^7+152nu^3(-a+sqrt(a^2+8bnu))c^3b^4a^7+960nu^4(-a+sqrt(a^2+8bnu))c^3b^5a^5+8nu^2(-a+sqrt(a^2+8bnu))c^3b^3a^9-1128nu^3(-a+sqrt(a^2+8bnu))b^4a^7c-2089nu^4(-a+sqrt(a^2+8bnu))b^5a^5c-26nu(-a+sqrt(a^2+8bnu))b^2a^11c-254nu^2(-a+sqrt(a^2+8bnu))b^3a^9c-726nu^5(-a+sqrt(a^2+8bnu))b^6a^3c+864nu^6(-a+sqrt(a^2+8bnu))ab^7c-2nu(-a+sqrt(a^2+8bnu))b^2a^11c^2-56nu^2(-a+sqrt(a^2+8bnu))b^3a^9c^2-5610nu^5(-a+sqrt(a^2+8bnu))b^6a^3c^2-(-a+sqrt(a^2+8bnu))ba^13c+(9193/4)nu^3(-a+sqrt(a^2+8bnu))b^4a^7+3931nu^4(-a+sqrt(a^2+8bnu))b^5a^5+(205/2)nu(-a+sqrt(a^2+8bnu))b^2a^11+667nu^2(-a+sqrt(a^2+8bnu))b^3a^9+2673nu^5(-a+sqrt(a^2+8bnu))b^6a^3+324nu^6(-a+sqrt(a^2+8bnu))ab^7)t/(nu(-a^12b-8a^8b^3cnu^2+8a^8b^3c^2nu^2+144nu^3b^4a^6c^2-144nu^3b^4a^6c-16nu^4a^4b^5c^4-848a^4b^5nu^4c+832a^4b^5nu^4c^2-1728cb^6nu^5a^2+32nu^4a^4b^5c^3-162a^2b^6c^4nu^5+324a^2b^6c^3nu^5+1566a^2b^6c^2nu^5+324b^7nu^6c^2-324c^4b^7nu^6+648c^3b^7nu^6-254a^8b^3nu^2-26a^10nub^2-648nu^6b^7c-2217b^5a^4nu^4-1350b^6a^2nu^5-1136nu^3a^6b^4+7a^11b(-a+sqrt(a^2+8bnu))+(1/4)a^13(-a+sqrt(a^2+8bnu))/nu-2b^2a^9nu(-a+sqrt(a^2+8bnu))c^2-272b^4a^5nu^3(-a+sqrt(a^2+8bnu))c^2-40b^3a^7nu^2(-a+sqrt(a^2+8bnu))c^2-8b^4a^5nu^3(-a+sqrt(a^2+8bnu))c^3-459nu^5(-a+sqrt(a^2+8bnu))ab^6c^2-270nu^5(-a+sqrt(a^2+8bnu))ab^6c^3+135nu^5(-a+sqrt(a^2+8bnu))ab^6c^4+4b^4a^5nu^3c^4(-a+sqrt(a^2+8bnu))+2b^2a^9nu(-a+sqrt(a^2+8bnu))c+594nu^5(-a+sqrt(a^2+8bnu))ab^6c+744nu^4(-a+sqrt(a^2+8bnu))a^3b^5c+276b^4a^5nu^3(-a+sqrt(a^2+8bnu))c+40b^3a^7nu^2(-a+sqrt(a^2+8bnu))c-696nu^4(-a+sqrt(a^2+8bnu))a^3b^5c^2-96nu^4(-a+sqrt(a^2+8bnu))a^3b^5c^3+48nu^4(-a+sqrt(a^2+8bnu))a^3b^5c^4+(151/2)b^2a^9nu(-a+sqrt(a^2+8bnu))+162nu^5(-a+sqrt(a^2+8bnu))ab^6+918nu^4(-a+sqrt(a^2+8bnu))a^3b^5+(3809/4)b^4a^5nu^3(-a+sqrt(a^2+8bnu))+389b^3a^7nu^2(-a+sqrt(a^2+8bnu)))))
代人参数params1 := {a = 1, b = 1, c = 1.5, nu = 1} 得
q2 := .50000000000000000000-.50000000000000000000tanh((.12500000000000000000-.33071891388307382381I)x+(1.1875000000000000000+.16535945694153691190I)*t)
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