Solving a nonlinear PDE numerically

I want to solve numerically the following PDE: $$u_x + u_t - (u_{xt})^2 = u(x,t)$$ The boundary conditions are no concern of mine, pick the ones that work.

So which numerical method will work for this nonlinear PDE? Any suggestions? I tried expanding the Taylor expansion for two dimensions, but I don't see or remember how to continue.

Thanks. P.S. $u\in C^1(x,t)$.

Not an answer, but I'll just note that there are some special cases that reduce to ODE's.

With $u(x,t) = v(x+at)$, the differential equation becomes

$$(1+a) v' - a^2 (v'')^2 = v \tag{1}$$

In particular, for $a=0$ we have solutions $u(x,t) = c e^x$ (by symmetry, $u(x,t) = ce^t$ is also a solution), and for $a=-1$ we have $u(x,t) = -(x-t+c)^4/144$ as well as some solutions that can be expressed in terms of elliptic integrals: $$x-t = F\left(\sqrt{-u(x,t)}\right)\ \text{where}\ F(z) = \pm \int \dfrac{3 z\; dz}{\sqrt{c \pm 3 z^3}}$$ Standard numerical ODE solvers should work for (1) if you decide on the sign of $v''$, i.e. choose the $\pm$ in

$$a v'' = \pm \sqrt{(1+a) v' - v}$$

although you are likely to run into trouble if the right side hits $0$.

• which numerical method do you suggest me to use? I read the tutorial here: mathworks.com/matlabcentral/fileexchange/… but it include examples where you guess your approximate solution, are there methods where you don't guess your approximate solution? – Alan Aug 12 '15 at 6:23
• the last comment was addressed to you, I don't understand why the ping didn't work. – Alan Aug 12 '15 at 6:48
• Initial value ODE problems may be easier than boundary value problems. But e.g. in Maple's dsolve you don't generally need to guess an approximate solution. – Robert Israel Aug 12 '15 at 15:35