The advection diffusion equation is the partial differential equation $$\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2} - v \frac{\partial C}{\partial x}$$ with the boundary conditions $$\lim_{x \to \pm \infty} C(x,t)=0$$ and initial condition $$C(x,0)=f(x).$$ How can I transform the advection diffusion equation into a linear diffusion equation by introducing new variables $x^\ast=x-vt$ and $t^\ast=t$?

-

We could simply apply the chain rule, to avoid some confusions we let $C(x,t) = C(x^* + vt,t^*) = C^*(x^*,t^*)$: $$\frac{\partial C}{\partial x} = \frac{\partial C^*}{\partial x^{\phantom{*}}}= \frac{\partial C^*}{\partial x^*} \frac{\partial x^*}{\partial x^{\phantom{*}}} + \frac{\partial C^*}{\partial t^*} \frac{\partial t^*}{\partial x^{\phantom{*}}} = \frac{\partial C}{\partial x^*}$$ remember here in chain rule, the partial derivative is being taken wrt the first and second variable if not to confuse this wrt the total derivative, similary we could have $\displaystyle \frac{\partial^2 C}{\partial x^2} = \frac{\partial^2 C^*}{\partial {x^*}^2}$, $$\frac{\partial C}{\partial t} = \frac{\partial C^*}{\partial t} = \frac{\partial C^*}{\partial x^*} \frac{\partial x^*}{\partial t^{\phantom{*}}} + \frac{\partial C^*}{\partial t^*} \frac{\partial t^*}{\partial t^{\phantom{*}}} = -v\frac{\partial C^*}{\partial x^*} + \frac{\partial C^*}{\partial t^*}$$ Plugging back to the original equation you will see the convection term is gone if we have done this velocity cone rescaling, you could think the original equation like a diffusion on a car with velocity $v$ measured by a standing person, after the change of variable it is just a pure diffusion measured on a car: $$\frac{\partial C^*}{\partial t^*} = D\frac{\partial^2 C^*}{\partial {x^*}^2}$$ and the initial condition changes to $C^*(x^*,0) = C(x^*+vt^*,t^*)\Big\vert_{t^*=0}= f(x^*)$, the boundary condition remains the same.
However, if the boundary conditions are for finite $x$, then for the transformed equation we have boundary conditions depending on time!! For example if $C$ denotes a poluttant concentration, we see his value at some points.