Solving Poisson equation with Robin boundary condition with DSolve

Question

I have the following BVP: $-\Delta u = f \;\;\;\text{in}\;\Omega=(0,1)^{2}$

$u+\varepsilon\nabla u\cdot n = 0, \;\;\;\text{on}\;\partial\Omega$

where $f(x,y) = 2x(1-x) + 2y(1-y)$ and $\varepsilon>0$. I would like to use DSolve to obtain a solution but I'm not sure how to implement the BC, as all examples I've found so far only deal with pure Dirichlet or Neumann conditions. The Mathematica documentation states that NeumannCondition handles both Neumann and Robin BCs but I didn't see any examples that do it.

Answer 1

Mathematica DSolve (V 12) can not solve this. You did not show your attempt. Here is what I tried.

ClearAll[x, y];
eps = 1;
L = 1; (*length of x side*)
H = 1; (*Length of y side*)
f = 2 x (1 - x) + 2 y (1 - y);
pde = -Laplacian[u[x, y], {x, y}] == f;
bc = {u[x, 0] + eps* Derivative[0, 1][u][x, 0] == 0,
   u[x, H] + eps* Derivative[0, 1][u][x, H] == 0,
   u[0, y] + eps* Derivative[1, 0][u][0, y] == 0,
   u[L, y] + eps* Derivative[1, 0][u][L, y] == 0
   };
DSolve[{pde, bc}, u[x, y], {x, y}, Assumptions -> {0 <= x <= L && 0 <= y <= H}]

But Maple 2019.2.1 can solve it

restart;
eps:=1;
L:=1;H:=1;
f:=2*x*(1 - x) + 2*y*(1 - y);
pde := diff(u(x,y),x$2)+diff(u(x,y),y$2)=-f;
bc:=u(x,0)+eps*D[2](u)(x,0)=0,
    u(x,H)+eps*D[2](u)(x,H)=0,
    u(0,y)+eps*D[1](u)(0,y)=0,
    u(L,y)+eps*D[1](u)(L,y)=0;
pdsolve([pde,bc],u(x,y)) assuming 0<=x , x<=L , 0<=y , y<=H

$$ u \left( x,y \right) =\sum _{n=1}^{\infty }4\,{\frac { \left( \pi\,n \cos \left( n\pi\,x \right) -\sin \left( n\pi\,x \right) \right) \left( \left( \left( \pi\,n+1 \right) ^{2} \left( -1 \right) ^{n}+{ \pi}^{2}{n}^{2}-1 \right) {{\rm e}^{-\pi\,n \left( y-2 \right) }}- \left( \pi\,n+1 \right) \left( \left( \pi\, \left( {y}^{2}-y+1 \right) n-2\,y+1 \right) \left( -1 \right) ^{n}-\pi\, \left( {y}^{2} -y-1 \right) n+2\,y-1 \right) {{\rm e}^{2\,\pi\,n}}+ \left( \left( - \pi\,n \left( -1 \right) ^{n}-\pi\,n+ \left( -1 \right) ^{n}-1 \right) {{\rm e}^{n\pi\,y}}+ \left( \pi\, \left( {y}^{2}-y+1 \right) n+2\,y-1 \right) \left( -1 \right) ^{n}-\pi\, \left( {y}^{2}-y-1 \right) n-2\,y+1 \right) \left( \pi\,n-1 \right) \right) }{{n}^{3}{ \pi}^{3} \left( {\pi}^{4}{{\rm e}^{2\,\pi\,n}}{n}^{4}-{\pi}^{4}{n}^{4} -{{\rm e}^{2\,\pi\,n}}+1 \right) }} $$