Consider heat equation
$$ u_{xx} = u_t $$
on infinite domain $-\infty< x < \infty$ and $t>0$ with initial condition $u(x,0) = g(x)$. We know can solve this and obtain
$$ u(x,t) = \frac{1}{\sqrt{4 \pi t} } \int\limits_{- \infty}^{\infty} g( \xi ) e^{ \frac{ (x- \xi)^2 }{4t }} d \xi $$
Im trying to solve the following:
For (a), it is easy to plot this on software, Im just looking for a simple way to graph without using a graphing device.
For part b), We have
$$ u(x,t) = \frac{1}{4 \pi \sqrt{t} } \int\limits_{- \infty}^{\infty} \exp \left( - \frac{ (\xi +2 )^2 }{4} + \frac{ (x - \xi)^2 }{4t}\right) - \exp \left( - \frac{ (\xi -2 )^2 }{4} + \frac{ (x - \xi)^2 }{4t}\right)$$
This is really cumbersome to compute. In what easier way can we compute this integral by hand?