冯·卡门方程

Von Karman equation U Maple plot

Von Karman equation w Maple plot

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：

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

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

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

通解

卡门方程有下列解析解

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

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

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

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

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