I have been trying to find the asymptotic (innner/outer) expansion of the follwing Heat Equation (with Dirichlet boundary condition): $$\begin{cases} u_t-\epsilon.u_{xx}=0 & x<0,\; t\in \mathbb{R^{+*}} \\ u(t,0)=0 & t\in \mathbb{R^{+*}} \\ u(0,x)=e^{-\frac{-x}{2\epsilon}} & x<0 \end{cases}$$ So the only boundary layer, is in the boudary condition at $\{x=0\}$.
My problem is that when I define the asymptotic outer expansion: $$u=u^0+\epsilon u^1+...$$ I can't find the initial conditions for $u^k$ for all $k\geq 0$. Because I can't write $e^{\frac{-x}{\epsilon}}$ as $\sum_{k\geq0}g^k(x)\epsilon^k$.
So that I can say for example: $u^0(0,x)=g^0(x)$