Acceleration function of position and time

Velocity $vec v(vec r,t)$ is the integral of acceleration $vec a(vec r, t)$ with respect to time, and position $vec r(t)$ is the integral of velocity with respect to time. So if the object starts at the origin with velocity zero then at time $t$

$vec v(vec r(t) ,t) = int_0^t vec a(vec r(t), t) dt \ vec r(t) = int_0^t vec v(vec r(t), t) dt$

Unless your acceleration function is something simple, the best you can do here is numerical integration. The simplest method of numerical integration is Euler's method. With this method, you take a small time step $delta t$ and assume that velocity and acceleration are constant over that time step. Then you update position and velocity for the beginning of the next time step as follows:

$vec r(t + delta t) = vec r(t) + vec v(vec r(t), t) delta t \vec v(t + delta t) = vec v(t) + vec a(vec r(t), t) delta t$

There are other methods of numerical integration which are more complicated, but give more accurate results.