On Wikipedia, https://en.wikipedia.org/wiki/Green%27s_function#Table_of_Green's_functions, there is a Green's function for the telegrapher's equation. The differential operator L is $${\displaystyle \partial _{t}^{2}+2\gamma \partial _{t}-c^{2}\partial _{x}^{2}}.$$ The Green's function is $${\displaystyle {\frac {1}{2}}e^{-\gamma t}\left[\delta (ct-x)+\delta (ct+x)+\Theta (ct-|x|)\left({\frac {\gamma }{c}}I_{0}\left({\frac {\gamma u}{c}}\right)+{\frac {\gamma t}{u}}I_{1}\left({\frac {\gamma u}{c}}\right)\right)\right]}$$ with ${\displaystyle u={\sqrt {c^{2}t^{2}-x^{2}}}}$.
I am attempting to verify this theorem:
Theorem There is one and only one solution u(x) that satisfies \begin{aligned}\operatorname {L} \,u&=f\\{\vec {\operatorname {D} }}\,u&={\vec {0}}\end{aligned} and it is given by $${u(x)=\int_{0}^{\ell }f(s)\,G(x,s)\,ds~,}$$ where $${\displaystyle G(x,s)}$$ is a Green's function satisfying the following conditions: ${\displaystyle G(x,s)}$ is continuous in x and s. For $${\displaystyle x\neq s~}, {\displaystyle \quad \operatorname {L} \,G(x,s)=0~}.$$
Using Mathematica
G[x,t]= 1/2*e^(-\[Gamma]t)(DiracDelta[c*t-x]+DiracDelta[c*t+x]+HeavisideTheta[ct-Abs[x]]*(\[Gamma]/c*Bessel[0,\[Gamma]*Sqrt[c^2t^2-x^2]/c]+\[Gamma]t/Sqrt[c^2t^2-x^2]*Bessel[1,\[Gamma]*Sqrt[c^2t^2-x^2]/c]))
PDETele[u_,x_,t_] := D[u[x,t],{t,2}]+2\[Gamma]D[u[x,t],t] -c^2D[u[x,t],{x,2}]==0
Simplify[PDETele[G,x,t]]
I am not getting output True. What's wrong with my attempts for simplifying this pde with the Green's Function? The operator and Green's Function look input correctly, is there something else I am missing?