非规范伯格斯方程

**非規範伯格斯方程** (Unnormalized Burgers equation) 是一個非線性偏微分方程：

$$
u_{t}-\alpha *u_{xx}-\beta *u*u_{x}=0
$$

## 解析解

　
: $u(x,t)=_{C}3/(\beta *_{C}2)+2*\alpha *_{C}2*cot(_{C}1+_{C}2*x+_{C}3*t)/\beta $

　
: ${u(x,t)=_{C}3/(\beta *_{C}2)+2*\alpha *_{C}2*coth(_{C}1+_{C}2*x+_{C}3*t)/\beta }$

　
: ${u(x,t)=_{C}3/(\beta *_{C}2)-2*\alpha *_{C}2*tan(_{C}1+_{C}2*x+_{C}3*t)/\beta }$

　
: ${u(x,t)=_{C}3/(\beta *_{C}2)+2*\alpha *_{C}2*tanh(_{C}1+_{C}2*x+_{C}3*t)/\beta }$

　
: ${u(x,t)=(_{C}5+tanh((1/2)*_{C}1*(_{C}3+_{C}4*x+_{C}5*t-_{C}2)/(\alpha *_{C}4^{2}))*_{C}1)/(\beta *_{C}4)}$

　
: ${u(x,t)=(_{C}5+tanh((1/2)*_{C}1*({\sqrt {(}}csc(_{C}3+_{C}4*x+_{C}5*t)-1)*{\sqrt {(}}csc(_{C}3+_{C}4*x+_{C}5*t)+1)*arctan(1/{\sqrt {(}}csc(_{C}3+_{C}4*x+_{C}5*t)^{2}-1))+_{C}2*{\sqrt {(}}csc(_{C}3+_{C}4*x+_{C}5*t)^{2}-1))/(\alpha *_{C}4^{2}*{\sqrt {(}}csc(_{C}3+_{C}4*x+_{C}5*t)^{2}-1)))*_{C}1)/(\beta *_{C}4)}$

　
: ${u(x,t)=(_{C}5-tan((1/2)*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta -_{C}5^{2})*(ln(cosh(_{C}3+_{C}4*x+_{C}5*t)+{\sqrt {(}}cosh(_{C}3+_{C}4*x+_{C}5*t)^{2}-1))+_{C}2)/(\alpha *_{C}4^{2}))*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta -_{C}5^{2}))/(\beta *_{C}4)}$

　
: ${u(x,t)=(_{C}5-tan((1/2)*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta -_{C}5^{2})*(_{C}3+_{C}4*x+_{C}5*t+_{C}2)/(\alpha *_{C}4^{2}))*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta -_{C}5^{2}))/(\beta *_{C}4)}$

　
: ${u(x,t)=(_{C}5+tanh((1/2)*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta +_{C}5^{2})*(arctanh(1/{\sqrt {(}}1+csch(_{C}3+_{C}4*x+_{C}5*t)^{2}))-_{C}2)/(\alpha *_{C}4^{2}))*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta +_{C}5^{2}))/(\beta *_{C}4)}$

　
: ${u(x,t)=(_{C}5-tanh((1/2)*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta +_{C}5^{2})*((1/2)*Pi-_{C}3-_{C}4*x-_{C}5*t+_{C}2)/(\alpha *_{C}4^{2}))*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta +_{C}5^{2}))/(\beta *_{C}4)}$

　
: ${u(x,t)=-(-_{C}5+tanh((1/2)*_{C}1*({\sqrt {(}}sec(_{C}3+_{C}4*x+_{C}5*t)-1)*{\sqrt {(}}sec(_{C}3+_{C}4*x+_{C}5*t)+1)*arctan(1/{\sqrt {(}}sec(_{C}3+_{C}4*x+_{C}5*t)^{2}-1))+_{C}2*{\sqrt {(}}sec(_{C}3+_{C}4*x+_{C}5*t)^{2}-1))/(\alpha *_{C}4^{2}*{\sqrt {(}}sec(_{C}3+_{C}4*x+_{C}5*t)^{2}-1)))*_{C}1)/(\beta *_{C}4)}$

　
: ${u(x,t)=(1/2)*({\sqrt {(}}2)*_{C}5+2*tanh((1/2)*{\sqrt {(}}\beta *_{C}1*_{C}4*\alpha )*(_{C}3+_{C}4*x+_{C}5*t+_{C}2)*{\sqrt {(}}2)/(_{C}4*\alpha ))*_{C}4*{\sqrt {(}}\beta *_{C}1*_{C}4*\alpha ))*{\sqrt {(}}2)/(\beta *_{C}4)}$

## 行波圖

## 參考文獻

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