冯·卡门方程

卡门方程是一个模拟平板变形的四阶椭圆型非线性偏微分方程组：^[1]

Von Karman equation U Maple plot

Von Karman equation w Maple plot

${\displaystyle \Delta \Delta (u)=a((w_{xy})^{2}-w_{xx}w_{yy})}$

${\displaystyle \Delta \Delta (w)=b(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy})+c}$

其中 ${\displaystyle \Delta ={\frac {\partial }{\partial x^{2}}}+{\frac {\partial }{\partial y^{2}}}}$

通解

卡门方程有下列解析解^[2]

${\displaystyle u:=(1/2*(A[3]*x^{3}+A[2]*x^{2}+A[1]*x+A[0]))*y^{2}+y-(1/10)*x^{5}*A[3]+x^{3}+x^{2}+x}$

${\displaystyle w:=\int ((x-t)*f(t),t=0..x)+x}$

其中 ${\displaystyle f''(x)=b*(A[3]*x^{3}+A[2]*x^{2}+A[1]*x+A[0])*f(x)+c}$

特解

当${\displaystyle A[2]=A[3]=0}$时

${\displaystyle f(x)=AiryAi(-1.3200061217959123977*x+2.0087049679503014748)}$ ${\displaystyle *_{C}2+AiryBi(-1.3200061217959123977*x+2.0087049679503014748)}$ ${\displaystyle *_{C}1-2.2727167324939371067*Pi*(-(Int(AiryBi(-1.3200061217959123977*x+2.0087049679503014748),x))*AiryAi(-1.3200061217959123977*x+2.0087049679503014748)+(Int(AiryAi(-1.3200061217959123977*x+2.0087049679503014748),x))*AiryBi(-1.3200061217959123977*x+2.0087049679503014748))}$

因此

${\displaystyle u=(1/2*(-2.3*x+3.5))*y^{2}+y+x^{3}+x^{2}+x}$

${\displaystyle v=\int ((x-t)*(AiryAi(-1.3200061217959123977*t+2.0087049679503014748)+AiryBi(-1.3200061217959123977*t+2.0087049679503014748)-2.2727167324939371067*Pi*(-(Int(AiryBi(-1.3200061217959123977*t+2.0087049679503014748),t))*AiryAi(-1.3200061217959123977*t+2.0087049679503014748)+(Int(AiryAi(-1.3200061217959123977*t+2.0087049679503014748),t))*AiryBi(-1.3200061217959123977*t+2.0087049679503014748))),t)+x}$

参考文献

1. Theodore von Karman,Enklopedie der mathematischen Wissenshaften, vol 4, p349,1910
2. Andre Polyanin,Valentin Zaitsev Handbook of Nonlinear Partial Differential Equations, 2nd edition, p1192-1196