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I am trying to solve a set of partial differential equations numerically:

NDSolve[
 D[f[x, y], x, x]^2 + D[g[x, y], x, x]^2 + D[f[x, y], x, y]^2 == 0 && 
 D[f[x, y], y, y]^2 + D[g[x, y], y, y]^2 + D[f[x, y], x, y]^2 == 0 &&
 f[x, 0] == 0 && g[x, 0] == 0 && 1 == (D[f[x, q], q] /. q -> 0) && 
 1 == (D[g[x, q], q] /. q -> 0) && 1 == (D[f[x, 0], x]) && 1 == (D[g[x, 0], x])
 ,{f, g}, {x, 0, 1}, {y, 0, 1}]

Mathematica 12 returns the following error message: enter image description here

what does this error message mean? How can I fix the input?

user21
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Kagaratsch
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  • What does this model describe? – Alex Trounev Apr 25 '19 at 22:35
  • @AlexTrounev This is a dummy minimal input example that reproduces the same Mathematica error as my actual PDE of interest, which would not fit onto the page. – Kagaratsch Apr 25 '19 at 22:51
  • It is necessary to bring the system to a quasilinear form. – Alex Trounev Apr 25 '19 at 23:52
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    @AlexTrounev Hmm, now I'm confused about the statement "A PDE which is neither linear nor quasi-linear is said to be nonlinear." from https://reference.wolfram.com/language/tutorial/DSolveLinearAndQuasiLinearFirstOrderPDEs.html . How can we bring a nonlinear PDE into quasi-linear form? Since studying a nonlinear one is kind of the objective... – Kagaratsch Apr 25 '19 at 23:59
  • Usually differentiated equations by x or y and introduce new variables. – Alex Trounev Apr 26 '19 at 00:23

1 Answers1

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If you click on the three dots in front of the error message

enter image description here

and follow the link to the reference page you will find some information on this error message.

Next, you have to bring your PDE into the coefficient form:

$$ \nabla \cdot (-c(t,X,u,\nabla _Xu) \nabla u-\alpha (t,X,u,\nabla _Xu) u $$ $$ + \gamma (t,X,u,\nabla _Xu)) + \beta (t,X,u,\nabla _Xu)\cdot \nabla u+a(t,X,u,\nabla _Xu) u$$ $$ - f(t,X,u,\nabla _Xu)=0.$$

Details are here. And you have to get your equation into that form otherwise you are out of FEM luck. Also see this answer.

user21
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