恰普雷金方程

恰普雷金方程（英语：Chaplygin equation）是俄罗斯物理学家谢尔盖·阿列克谢耶维奇·恰普雷金在1904年发表的描写喷气流的非线性偏微分方程：^[1]

${\displaystyle 0.5*u_{tt}+u_{x}*u_{xt}-u_{t}*u_{xx}=0}$

解析解

${\displaystyle u(x,t)=_{C}1+_{C}2*(_{C}3+C[1]*x+_{C}4*t)}$

${\displaystyle u(x,t)=_{C}1+((1/2)*\pi -_{C}3-C[1]*x-_{C}4*t)*_{C}2}$

${\displaystyle u(x,t)=_{C}1+\arctan(1/{\sqrt {(}}csc(_{C}3+C[1]*x+_{C}4*t)^{2}-1))*_{C}2}$

${\displaystyle u(x,t)=_{C}1+\ln(\cos(_{C}3+C[1]*x+_{C}4*t)+{\sqrt {(}}\cos(_{C}3+C[1]*x+_{C}4*t)^{2}-1))*_{C}}$

${\displaystyle u(x,t)=_{C}1+\ln(\cosh(_{C}3+C[1]*x+_{C}4*t)+{\sqrt {(}}\cosh(_{C}3+C[1]*x+_{C}4*t)^{2}-1))*_{C}2}$

${\displaystyle u(x,t)=-(0.479e-2*(11400.+581.*\tan(1.02*t)+60300.*\exp(.833*x)+3060.*exp(.833*x)*\tan(1.02*t)+79000.*\exp(1.67*x)+4010.*\exp(1.67*x)*\tan(1.02*t)))*\exp(-.833*x)/(-10.+197.*\tan(1.02*t))}$

行波图

Chaplygin equation traveling wave plot

Chaplygin equation traveling wave plot

Chaplygin equation traveling wave plot

Chaplygin equation traveling wave plot

Chaplygin equation traveling wave plot

Chaplygin equation traveling wave plot

Chaplygin equation traveling wave plot

Chaplygin equation traveling wave plot

Chaplygin equation traveling wave plot

Chaplygin equation traveling wave plot