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I am new to Mathematica and trying to learn its language. I want to solve a non-linear PDE but don't know how to insert it. Would you please help me or give me some tutorial about it?

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ClearAll[h, x, n, t]
hx = D[h[x, t], x];
ht = D[h[x, t], t];
sqrtTerm = Sqrt[1 + 4*n*Abs[hx]];
ode = -Sign[hx]*1/2*D[ h[x, t]*(sqrtTerm - 1), x] == -ht

Mathematica graphics

Just do not try to solve this analytically. It wont work (Abs and Sign do not play well with analytic solvers).

May be numerical solution might work (with right IC/BC and values for any parameters.

see DSolve and NDSovle

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  • $\begingroup$ Thank you. it makes sense and I think everything is correct except that the eq is PDE not ODE. you wrote ode =.... thanks for your suggestions, I do not want to solve it analytically its impossible, I wanted to use NDSOLVE but I could not introduce the equation in Mathematica $\endgroup$
    – AWer
    Feb 6 at 13:34

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