Solving for the kernels in a Sturm-Liouville transformation

Question

So in this paper on page 3 they try to model particle diffusion. In order to do this, they use a semi-analytical model, in which they apply a Sturm-Liouville Transformation $$\tau \{Y(\mathbf x,s)\} = \bar Y_n(\mu,s) = \iint \mathbf K_n^H(\mathbf x,\mu)\,\mathbf C\, \mathbf Y(\mathbf x, s)\,r\,dr\,d\phi $$ where $\mathbf C$ is a weight matrix, $\mathbf Y$ is the ouput matrix, $\mathbf x \in [r, \phi]$ and $\mathbf K_n^H$ is the kernel I am searching for. The kernels are solved and the solution is $$ \mathbf K_n^H = \begin{bmatrix} k_{\mu,n}J'(k_{\mu,n} r) \\ \frac{1}{r}(jn)J_n(k_{\mu,n}r) \\ J_n(k_{\mu,n} r) \end{bmatrix} e^{jn\phi}$$ where $J$ stands for the bessel function, $J'$ the derivative for the bessel function, $k$ stands for the wave number and $r$ is the radius of the cylinder, $j$ is the complex number. Sadly they skipped the part for brevity on how exactly to solve it. So my question is how to start to find an answer?