Suppose I have Green`s function of initial-boundary value problem $$ \frac{\partial^4w}{\partial x^4}+\alpha^2\frac{\partial^2w}{\partial t^2}=f(x,t),~ \alpha\neq 0,~ 0<x<l,~ t>0, $$ $$ w = \frac{\partial w}{\partial x}=0,~ x=0,l,~ t\geq0, $$ $$ w=w_0(x),~~ \frac{\partial w}{\partial t}=w_0^1(x),~ t=0,~ 0\leq x\leq l. $$
How to find Green`s function for corresponding initial-boundary value problem for equation $$ \frac{\partial^4w}{\partial x^4}+\alpha^2\frac{\partial^2w}{\partial t^2}+\beta\frac{\partial w}{\partial t}=f(x,t),~ \beta>0,~ 0<x<l,~ t>0. $$
Substituting $$ u(x,t)=\exp\left[\frac{\beta}{2}t\right]w(x,t), $$ into $$ \frac{\partial^4w}{\partial x^4}+\alpha^2\frac{\partial^2w}{\partial t^2}+\beta\frac{\partial w}{\partial t}=f(x,t),~ \beta>0,~ 0<x<l,~ t>0. $$ we arrive at $$ \frac{\partial^4u}{\partial x^4}+\alpha^2\frac{\partial^2u}{\partial t^2}-\frac{\beta^2}{4}u=\exp\left[\frac{\beta}{2}t\right]f(x,t),~ \beta>0,~ 0<x<l,~ t>0. $$ Green`s function of that equation is known: $$ G\left(x,\xi,t\right)=\sum_{k=1}^\infty\frac{\varphi_k\left(x\right)\varphi_k\left(\xi\right)}{\left|\left|\varphi_k\right|\right|^2}\cdot\frac{\sin\left(t\sqrt{\alpha^2\lambda_n^2-\frac{\beta^2}{4}}\right)}{\sqrt{\alpha^2\lambda_n^2-\frac{\beta^2}{4}}}, $$ in which $\varphi_n$ and $\lambda_n$ are the eigenfunctions and eigenvalues of the equation $$ \varphi''''-\lambda^4\varphi=0, $$ subject to appropriate boundary conditions.