I am trying to solve the below-mentioned PDE that represents a damped heat diffusion in one-dimensional space. I am using the separation of variables to solve it, however when I try to find the solution. It results in $0$(trivial solution), whereas I am trying to find all the non-trivial solutions for this. Can someone help me with this.
Here considering this partial differential equation as,
$\frac{\partial{x}}{\partial t}(\zeta,t)=\alpha\frac{\partial^2{x}}{\partial^2 \zeta}(\zeta,t)+\beta\frac{\partial{x}}{\partial \zeta}(\zeta,t),~x(\zeta,0)=x_0(\zeta),\\x(0,t)=0=x(1,t)$
where $\zeta\in[0,~1]\forall~t\geq0$ and $\alpha,\beta\in\mathbb{R}^+$.
If we set
$$x(\xi, t) = f(\xi + \beta t, t) = f(z, t)$$
we find (by the chain rule)
\begin{align} x_{t} &= f_{z} \cdot z_{t} + f_{t} \\ &= \beta f_{z} + f_{t} \\ x_{\xi} &= f_{z} \cdot z_{\xi} \\ &= f_{z} \\ x_{\xi \xi} &= f_{zz} \cdot z_{\xi} \\ &= f_{zz} \end{align}
Substituting into our original PDE, we get
\begin{align} x_{t} &= \alpha x_{\xi \xi} + \beta x_{\xi} \\ \implies \beta f_{z} + f_{t} &= \alpha f_{zz} + \beta f_{z} \\ \implies f_{t} &= \alpha f_{zz} \end{align}
which is just the homogeneous heat equation, which is easily solved by the using the method of characteristics. You'll want to convert your boundary conditions to your new 'coordinates' $(z,t)$ at some point too.