# Initial condition trouble with NDSolve for a 2nd order PDE

In general, when solving a 2nd order PDE (such as the wave equation below) for $$u(x,t), \quad x \in(-\infty,\infty), \: t\in (0,\infty)$$ it should be sufficient to provide initial conditions $u(x,0)$ and $\partial_t u(x,0)$. However when I try solve this numerically in Mathematica

NDSolve[{D[u[x, t], {t, 2}] == D[u[x, t], {x, 2}],
u[x, 0] == E^(-x^2), Derivative[0, 1][u][x, 0] == 1}, u, {x, -7, 7}, {t, 0, 4}]


I get the error

NDSolve::bcart: Warning: an insufficient number of boundary conditions
have been specified for the direction of independent variable x.
Artificial boundary effects may be present in the solution.


The only thing I can think of is that when solving numerically I have introduced fictitious new boundaries to the problem by simply specifying a finite range {x,-7,7}. If this is the problem, how do I get around it? (NB the real PDE I'm trying to solve is more complicated and does not describe localized waves).

Also, oddly, if I take out the second I.C.

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


It doesn't complain but it seems to invent its own boundary conditions and gives the wrong solution.

Edit: the actual PDE I'm solving is of the form $$-\partial_t^2\phi + \sqrt{\frac{2M}{r}} \left( \frac{3}{2r}\partial_t\phi + 2\partial_t \partial_r \phi \right) + \frac{1}{r^2} \partial_r \left[ r (r-2M)\partial_t \phi\right] -m^2\phi =0,$$

with the initial conditions $\phi(r>>1, t) = 10 - t^{1/3}$ and \partial_r\phi(r>>1,t) = 0$. Any tips regarding the best way to get around the issues above in this case would be amazing. • "It doesn't complain but it seems to invent its own boundary conditions and gives the wrong solution. " Yes, for more information, check this: mathematica.stackexchange.com/q/73961/1871 Then how to circumvent? It really depends. For your specific problem, absorbing boundary condition (ABC) can be used: mathematica.stackexchange.com/a/128524/1871 But do notice there's no universal technique for dealing with this type of problem, so, please be more specific if ABC isn't suitable for you. You can also have a look at those posts tagged with [boundary-condition-at-infinity] – xzczd Jun 6 '18 at 15:47 • Thank you for the links, I'm looking at them now. My specific problem is solving the PDE:$-\partial_t^2\phi + \sqrt{\frac{2M}{r}} \left( \frac{3}{2r}\partial_t\phi + 2\partial_t \partial_r \phi \right) + \frac{1}{r^2} \partial_r \left[ r (r-2M)\partial_t \phi\right] -m^2 =0,$with the initial conditions$\phi(r>>1, t) = 10 - t^{1/3} $and$\partial_r\phi(r>>1,t) = 0$(and$M$and$m$are specified numbers). – Rudyard Jun 7 '18 at 16:50 • Well, these are not initial conditions (i.c.), they're boundary conditions (b.c.) at infinity in$r$direction. What's the condition in$t\$ direction then? Is this an equation already studied in literature? Can you show us some background information? – xzczd Jun 8 '18 at 6:20