# Numerical solutions of a fourth order PDE

Let $B_R$ be a ball of radius $R>0$ centered at the origin in $\mathbb{R}^2$.

Consider the PDE $u_t + \nabla \cdot (u^2 \nabla \Delta u) = 0$, in $(0,T)\times B_R,$ with boundary conditions $u =\frac{\partial}{\partial\nu} \Delta u = 0$, in $(0,T)\times B_R,$ and initial data $u(0,\cdot) = g,$ in $B_R$, where for instance

g[x] = 1


in $B_r$ and $0$ outside $B_r$, with $r < R$. A reference on numerical analysis for this problem is https://dml.cz/handle/10338.dmlcz/134511.

Is it possible to use Mathematica (for example NDSolve) to obtain and plot solutions to such equation?

• What have you tried? Have you looked into NDSolve and friends? – MarcoB May 11 '18 at 15:00
• @MarcoB I don't know how to implement the boundary conditions on NDSolve or if using NDSolve is actually the correct approach. – user58151 May 11 '18 at 15:11
• That's a nonlinear PDE and Mathematica's FEM capacities don't support that at the moment (version 11.3). Moreover, it is a fourth order PDE and I doubt that these are supported at all.You could try reformulate this as a coupled system of order 2 and use NDSolve for solving linearized PDEs in order to implement a semi-implicit time integrator... – Henrik Schumacher May 11 '18 at 15:23
• @HenrikSchumacher We can rewrite the PDE as $u_t + \nabla \cdot (u^2 \nabla w) = 0$ and $w = \Delta u$, but I don't know how to approach systems with Mathematica. – user58151 May 11 '18 at 15:39
• for fourth order PDE problems, We can use Isogeometric Analysis to solve it, but I am not sure, does Mathematica support Isogeometric Analysis? – ABCDEMMM May 13 '18 at 20:05