Finite integral Hankel transformations on \([0,R]\) with spectral parameter in boundary value condition (Q2703310)

Using the method of delta-like sequences the author constructs the finite integral Hankel transformation of the first type generated on the interval \((0,R)\) by the differential Bessel operator \(B_{\nu,\alpha}={d^{2}\over dr^{2}}+{2\alpha+1\over r}{d\over dr}- {\nu^{2}-\alpha^{2}\over r^{2}}\), \(\nu\geq\alpha\geq-{1\over 2},\) with the boundary operator \(B=h_{1}{\partial\over\partial r}+h_{2}{\partial\over\partial t}+h_{3}\); \(h_{j}\geq 0\), \(\sum_{j=1}^{3}h_{j}\neq 0\). By this integral transformation the author solves in the region \(\{(t,r):t>0\), \(r\in(0,R)\); \(R<\infty\}\) the equation \({1\over a^{2}}{\partial T\over\partial t} +{\gamma^{2}\over a^{2}}T-B_{\nu,\alpha}[T]= f(t,r)\) with the boundary conditions NEWLINE\[NEWLINE\left.\left[{\partial \over\partial r}(r^{\alpha -\nu}T(t,r)) \right]\right|_{r=0}=0, \left.\left(\alpha_{11}^{1} {\partial\over\partial r}+\gamma_{11}^{1}{\partial\over\partial t}+ \beta_{11}^{1}\right)T\right|_{r=R}=f_{R}(t)NEWLINE\]NEWLINE and the initial condition \(T(0,r)=g(r)\), \(g(R)=0\).