I am trying to solve the following equation numerically:$$\dfrac{\partial u}{\partial t}=\dfrac 3 x\dfrac \partial{\partial x}\left(\sqrt x \frac \partial {\partial x}(\nu(u,u^2,\cdots,u^k)u\sqrt x)\right)$$Expressed in its expanded form, this is:$$u_t=3\nu(u,u^2,\cdots)u_{xx}+\left(6\nu_x(u,u^2,\cdots)+\dfrac{3\nu(u,u^2,\cdots)}{2x}\right)u_x+\left(\nu_x(u,u^2,\cdots)\left(3+\dfrac 1 {2x}\right)+\dfrac{3\nu(u,u^2,\cdots)}{x}\right)$$Where $u$ is a function of $u=u(t,x)$.  $\ \nu$ a is a nonlinear function of $u$ and another variable, but for a given $u$ I can obtain it. What is the best way in terms of stability, performance, and accuracy to solve this? I was trying method of lines in MATLAB, but I have not been able to make any progress.
Matlab: Solving $u_t = f(u) u_{xx} + g(u) u_x + h(u) u$
pde;matlab