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physics - Solving coupled Differential equations with matching condition


I am somewhat new to using Mathematica and was wondering if it can solve the following: $(a \rightarrow h)$ are constants


$a w^4[x] =-b + c(d-w[x]-e x),\hspace{2em} x_1 \leq x \leq x_g \\ a w^4[x] =-b + f(d-w[x]),\hspace{2em} x_g < x \leq x_2 $


With the boundary conditions:


$ w[x_1]=d - g + e x_1\\ w'[x_1]=e \\ [w]=[w']=[w'']=[w''']=0, \; x=x_g \; \textrm{ (jump conditions)} \\ w[x_g]=d+e x_g \\ w''[x_2]=w'''[x_2]=0 $


There is also a constraint: $\int_{x1}^{x2} \sqrt{1+w'^2} dx=h$



I believe this model is analogous to solving beam equations with matched conditions except it is nonlinear: basically a cantilevered beam partially resting on a foundation with a pinning point at the end of the foundation that is allowed to move.


I tried searching the site for using jump conditions, but I didn't have much luck with the answers from the top hits. I was thinking that I could decouple the equations at $x_g$ and include matching conditions based on 2 differential equations: $w_1[x], w_2[x]$. However, the length constraint doesn't appear to be applicable.


So, I tried the following:


a = 8.06*10^16;
b = 8.99*10^6;
c = 8.99*10^11;
d = 500.;
e = -.268;
f = 10074.4;
g = 10.^(-5);

x1 = 0.;
x2 = 10000.;

NDSolve[{
  a*w''''[x] == -b + c (d - w[x] + e x),
  a*w2''''[x] == -b + f (d - w2[x]),
  w[x1] == d - g + e x1,
  w'[x1] == e,
  w[xg] == w2[xg],
  w'[xg] == w2'[xg],

  w''[xg] == w2''[xg],
  w'''[xg] == w2'''[xg],
  w2[xg] == d + e xg,
  w2''[x2] == 0,
  w2'''[x2] == 0
  },
 {w, w2, xg}, {x, x1, x2}
 ]

I get the error:



NDSolve::ndsv: Cannot find starting value for the variable w''

I think this error is pointing to a missing boundary condition. If I specify xg, I get another error, which I'm not sure how to interpret. I would greatly appreciate any comments on solving these equations. Thanks!




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