Use the heat kernel to solve the initial value problem: $$u_t(x, t) − ku_{xx}(x, t) = 0$$ $$∀x ∈ \mathbb{R}, t > 0$$
subject to-
$u(x, 0) = x^2 − 3x − 1$ $∀x ∈ \mathbb{R}.$
Ive started by using the "magic rule" for the heat kernel which gives me:
$$u(x,t)=\int_{\mathbb{R}}e^\frac{-(x-y)^2}{4kt}(y^2-3y-1)dy$$
and then I made $σ=y-x$ therefore I get
$$u(x,t)=\int_{\mathbb{R}}e^\frac{-σ^2}{4kt}((σ+x)^2-3(σ+x)-1)dσ$$
that simplifies to:
$$u(x,t)=\int_{\mathbb{R}}e^\frac{-σ^2}{4kt}(σ^2+2σx+x^2-3σ-3x-1)dσ$$
I don't really know where to go from here so if anyone could help me that would be appreciated.