PDE list of equations, derivative for boundarry condition evaluates to true

Question

I'm trying to do a basic 2D wave equation PDE in mathematica (version 11). I have a set of equations

Edit: I have infact made sure I cleared u

Clear[u]

wave2deqs = {
D[u[t, x, y], t, t] - Laplacian[u[t, x, y] , {x, y}] == 0, 
u[0, x, y] == Exp[-((x - 1)^2 + (y - 1)^2)],
D[u[0, x, y], t] == 0}

This works fine except for

D[u[0, x, y], t] == 0

evaluates to 'True', and then when I put this into

wave2d = NDSolveValue[wave2deqs, u, {t, 0, 1}, {x, 0, 1}, {y, 0, 1} ]

I naturally get an error that eqlist is not a list of equations.

So my question is, why would

D[u[0, x, y], t] == 0

even evaluate to a boolean rather than being another equation?

I followed this examples for reference: https://www.wolfram.com/mathematica/new-in-10/pdes-and-finite-elements/solve-a-wave-equation-in-2d.html which is in mathematica 10, but seems ridiculous that this wouldn't work.

Answer 1

There is no way that D[u[0,x,y],t]==0 is not true. You are asking Mathematica to calculate the derivative of an expression of a parameter that the expression does not contain. u[0,x,y] is such an expression - no t involved. So the derivative is quite naturally and correctly equal to zero.

What you need to do is to tell Mathematica that you like to calculate the (first) derivative of u of its first parameter and that that derivative shall be zero.

Derivative[1, 0, 0][u][0, x, y] == 0

This is, by the way, the method used on the website you refer to.

Answer 2

Another way is, first take derivative w.r.t t using D[u[t, x, y], t] then replace the variable t by a specific value using /.value.

Thus the derivative condition can be describe alternatively like this,

(D[u[t, x, y], t] /. t -> 0) == 0