I'm solving the PDE problem using separation of variations method. Let $a$ and $b$ be positive constant satisfying the following PDE problem $$w_t-a\Delta w-bw=0$$ for $t>0$ and $0<x<L$ where $L$ is a real number. with boundary and initial conditions : $$w(0,t)=g(t), \quad w(L,t)=0,\quad\text{and}\quad w(0,x)=h(x)$$
I try to begin with a usual strategy, i.e., let $w(x,t)=X(x)T(t)$. Then $$XT'-a(XT''+X''T)-bXT=0$$ Dividing through by $XT$, we have $$\frac{T'(t)}{T(t)}-a\frac{T''(t)}{T(t)}=a\frac{X''(x)}{X(x)}+b$$ This holds for all $x$ and $t$ which means that the above equality equals to some constant, say $\lambda$. Thus $$aX''+(b-\lambda)X=0\quad\text{and}\quad aT''-T'+\lambda T=0$$ Together with initial conditions : $X(0)=g(t), X(L)=0$, and $T(0)=h(x)$.
Note that different values of $\lambda$ contributes various possible solutions to the PDE :
Case 1 : $\lambda=b$
$X''=0 \Longrightarrow X(x)=\alpha x+\beta \Longrightarrow X(x)=-\frac{g(t)}{L}+g(t)$ after passing to initial conditions. Similarly, we get $T''-\frac{1}{a}T'=0 \Longrightarrow T(t)=c+(h(x)-c)e^{\frac{1}{a}t}$, where $c$ is a constant. Thus the solution to PDE is $w(x,t)=\left(-\frac{g(t)}{L}+g(t)\right)\left(c+(h(x)-c)e^{\frac{1}{a}t}\right)$
Case 2: $\lambda >b\Longrightarrow -k^2=\frac{b-\lambda}{a}<0 \Longrightarrow k=\pm\sqrt{-\frac{b-\lambda}{a}}$
Thus $X(x)=s_1\cos kx+s_2\sin kx\Longrightarrow X(L)=s_1\cos kL+s_2\sin kL=0$ and $X(0)=s_1=g(t)$. It follows that $s_2=-\frac{g(t)\cos kL}{\sin kL}$. This means that $$X(x)=g(t)\cos kx-\frac{g(t)\cos kL}{\sin kL}\sin kx$$
I'm struggling finding $T(t)$. Plus, I'm not even sure whether I'm on the right track.
Case 3: $\lambda <b\Longrightarrow k^2=\frac{b-\lambda}{a}>0 \Longrightarrow k=\pm\sqrt{\frac{b-\lambda}{a}}$
We have $X(x)=c_1e^{kx}+c_2e^{-kx} \Longrightarrow X(0)=c_1+c_2=g(t)$ and $X(L)=c_1e^{Lx}+c_2e^{-Lx}=0$. Solving these equations, it yields $$X(x)=\frac{g(t)}{1-e^{2kL}}e^{kx}-\frac{g(t)}{e^{-2kL}-1}e^{-kx}$$
The same issue seeking for $T(t)$ carries over here.
I would really appreciate that anyone can point out whether I come this far the right direction or not. Furthermore, you're welcome to suggest me an alternative idea to tackle it but based on the idea of separation of variables.