# NDSolve error: what does “It may help to rewrite the PDE in inactive form” mean?

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

• 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
• @AlexTrounev Hmm, now I'm confused about the statement "A PDE which is neither linear nor quasi-linear is said to be nonlinear." from reference.wolfram.com/language/tutorial/… . 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 0:23 $$\nabla \cdot (-c(t,X,u,\nabla _Xu) \nabla u-\alpha (t,X,u,\nabla _Xu) u$$ $$+ \gamma (t,X,u,\nabla _Xu)) + \beta (t,X,u,\nabla _Xu)\cdot \nabla u+a(t,X,u,\nabla _Xu) u$$ $$- f(t,X,u,\nabla _Xu)=0.$$