Transient Boundary Conditions 

Solve a coupled system with a combination of free boundary conditions on the outside and transient boundary conditions on the inside.

In[1]:=

X s = 1/8; \[CapitalOmega] = ImplicitRegion[(x <= -s || x >= s || y <= -s || y >= s) && x^2 + y^2 <= 1, {x, y}];

In[2]:=

X Subscript[\[CapitalGamma], D] = {DirichletCondition[ u[t, x, y] == Sin[t]^2, -s <= x <= s && -s <= y <= s], DirichletCondition[ v[t, x, y] == D[Sin[t]^2, t], -s <= x <= s && -s <= y <= s]};

In[3]:=

X sysSol = NDSolveValue[{D[u[t, x, y], t] == v[t, x, y], D[v[t, x, y], t] == Inactive[Laplacian][u[t, x, y], {x, y}], Subscript[\[CapitalGamma], D], u[0, x, y] == 0, v[0, x, y] == 0}, u, {t, 0, 2 Pi}, {x, y} \[Element] \[CapitalOmega]];

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In[4]:=

X framesTFBC = Table[Plot3D[sysSol[t, x, y], {x, y} \[Element] \[CapitalOmega], PlotRange -> {-1, 2}, Boxed -> False, Axes -> False, Mesh -> None], {t, 0, 2 \[Pi], 2 \[Pi]/50}];

In[5]:=

X Manipulate[ framesTFBC[[i]], {{i, 25, "time"}, 1, Length[framesTFBC], 1}, SaveDefinitions -> True]

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