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：

\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

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)
_C2+AiryBi(-1.3200061217959123977x+2.0087049679503014748)
_C1-2.2727167324939371067Pi(-(Int(AiryBi(-1.3200061217959123977x+2.0087049679503014748), x))AiryAi(-1.3200061217959123977x+2.0087049679503014748)+(Int(AiryAi(-1.3200061217959123977x+2.0087049679503014748), x))AiryBi(-1.3200061217959123977*x+2.0087049679503014748))

因此

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

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

参考文献