# PDE solution does not satisfy Neumann boundary conditions using NDSolve

I am trying to solve the free particle Schrodinger equation in 1D (hbar =1, Energy = 1, mass = 1), But specifying conditions only on x==0:

k = Sqrt;
sol = NDSolveValue[{op == NeumannValue[0, x == 0],
DirichletCondition[u[x] == Cos[k x], x == 0]}, u, {x, 0, 1}]

Plot[{sol[x] , Cos[k x]}, {x, 0, 1}] The blue line is the numeric solution, and it is obviously not the expected solution (in orange), since the derivative in x==0 is not zero. I understand that specifying the Neumann value is not needed since its default is zero, but it isn't zero here!

What is going on?

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– user9660
Feb 26 '15 at 13:35

Both NeumannValue and DirichletCondition are set to apply at x==0. In such cases the DirichletCondition will trump the NeumannValue. Besides the NeumannValue is set to zero which means that the natural boundary condition is 0, the boundary integral vanishes. I think you want the DirichletCondition at x==1:

k = Sqrt;
sol = NDSolveValue[{op == NeumannValue[0, x == 0],
DirichletCondition[u[x] == Cos[k x], x == 1]}, u, {x, 0, 1}];


But NeumannValue[0,...] is equivalent to not specifying anything at all:

k = Sqrt; • I am actually interested in solving using conditions only at x == 0. I want to use DiriclehtCondition and NeumannValue to get the equivalent solution to this: NDSolveValue[{-1/2 u''[x] - u[x] == 0, u == Cos[k 0] , u' == -k Sin [k 0]}, u, {x, 0, 1}] Is it not possible? Feb 26 '15 at 16:24
• @AmitAbir, note that the above NDSolve does a time integration, while the FEM is a spatial discretization method. Feb 26 '15 at 18:17