Mathematica 12.0.0.0 happily solves the trivial coupled differential system as follows:
NDSolve[{D[u[x], x] == 1, D[v[x], x] == 1,
DirichletCondition[v[x] == 1, x == 0],
DirichletCondition[u[x] == 0, x == 0]}, {u, v}, {x, 0, 1}]
but complains when given differently stated (but equivalent) boundary conditions:
NDSolve[{D[u[x], x] == 1, D[v[x], x] == 1,
DirichletCondition[v[x] - u[x] == 1, x == 0],
DirichletCondition[v[x] + u[x] == 1, x == 0]}, {u, v}, {x, 0, 1}]
NDSolve::fembdcc "Cross-coupling of dependent variables in DirichletCondition ... is not supported in this version"
The error message seems clear -- it does not like when u
and v
simultaneously appear in a Dirichlet condition. But why not? Indeed, the documentation of DirichletCondition
seems to imply that a general equation is supported:
DirichletCondition[beqn,pred]
: "In general, the boundary equationbeqn
needs to be affine linear in the dependent variables, i.e.h1 u1+ … = r
, wherehi
andr
can depend on any of the independent variables{x1,x2,…}
."