查普里金方程

恰普雷金方程（英語：Chaplygin equation）是俄羅斯物理學家謝爾蓋·阿列克謝耶維奇·恰普雷金在1904年發表的描寫噴氣流的非線性偏微分方程：^[1]

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

解析解

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

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

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

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

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

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

行波圖

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