Advection-diffusion heat-kernel solution
= Advection-diffusion heat-kernel solution
The Cauchy problem with initial value $u_0$ has
$$
u(t,x)=\int_{-\infty}^{\infty}K_t(x-at-y)u_0(y)\,dy.
$$
= Advection-diffusion heat-kernel solution
The Cauchy problem with initial value $u_0$ has
$$
u(t,x)=\int_{-\infty}^{\infty}K_t(x-at-y)u_0(y)\,dy.
$$