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]

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

解析解

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

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

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

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}

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

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))

Chaplygin equation traveling wave plot

^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p765 CRC PRESS

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