I am interested in solving the linear PDE for $f(r,t)$ $$ (\partial_{tt}+a\partial_t-b\nabla^2)f(r,t)=0 $$ $$ \nabla^2\equiv \frac{1}{r}\partial_r(r\partial_r)-\frac{1}{r^2}=\partial_{rr}+\frac{1}{r}\partial_r-\frac{1}{r^2} $$ with conditions $$ \frac{\partial f(0,t)}{\partial t}=0,\quad \frac{\partial f(R,t)}{\partial t}=d\cos (\omega t) $$ where $a,b,d,R>0$. You can see the laplacian like term is written in a cylindrical basis, so I assume Bessel function will arise. I would like a complete solution which includes the steps to find $f(r,t)$ and details in evaluating the expansion coefficients of the fourier-bessel series that may arise in $f(r,t)$.
Thank you! enjoy the bounty!
In a previous topic, $(\partial_{tt}+\partial_t-\nabla^2)f(r,t)=0$ , the solution was explicitely expressed on the form : $$f(r,t)=g(r)\cos(\omega t)+h(r)\sin(\omega t)$$ where the functions $g(r)$ and $h(r)$ are complicated terms involving a Bessel function. But there is an hitch because the coefficients and the Bessel functions are in the complex range.
Instead of trying to express the solution with the Bessel functions, another approach consists in looking for the Taylor series of $g(r)$ and $h(r)$. An attempt is done below :
The recurence relationships allow to compute $A_{2n+1}$ and $B_{2n+1}$ as functions of $A_3$ and $B_3$ :
Then, the last step is the computation of $A_3$ and $B_3$ according to the boundary condition :