查看“︁冯·卡门方程”︁的源代码

[[File:Von Karman equation U Maple plot.png|thumb|Von Karman equation U Maple plot]]
[[File:Von Karman equation w Maple plot.png|thumb|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[[非线性偏微分方程列表|非线性偏微分方程]]组：<ref>Theodore von Karman,Enklopedie der mathematischen Wissenshaften, vol 4, p349,1910</ref>

<math>\Delta\Delta(u)=a((w_{xy})^2-w_{xx}w_{yy})</math>

<math>\Delta\Delta(w)=b(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy})+c</math>

其中
<math>\Delta=\frac{\partial}{\partial x^2}+\frac{\partial}{\partial y^2}</math>

==通解==
卡门方程有下列解析解<ref>Andre Polyanin,Valentin  Zaitsev  Handbook of Nonlinear Partial Differential Equations, 2nd edition, p1192-1196</ref>

<math>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</math>

<math>w := \int((x-t)*f(t), t=0..x)+x</math>

其中
<math>f''(x) = b*(A[3]*x^3+A[2]*x^2+A[1]*x+A[0])*f(x)+c</math>
==特解==
当<math>A[2]=A[3]=0</math>时

<math> f(x) = AiryAi(-1.3200061217959123977*x+2.0087049679503014748)</math>
<math>*_C2+AiryBi(-1.3200061217959123977*x+2.0087049679503014748)</math>
<math>*_C1-2.2727167324939371067*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))</math>

因此

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


<math>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</math>

==参考文献==
<references/>


[[Category:材料力學]]
[[Category:非线性偏微分方程]]