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

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

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

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

通解

卡门方程有下列解析解^[2]

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

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

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

特解

当 $A[2]=A[3]=0$ 时

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

因此

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

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

参考文献

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

[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

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