Hybrid integral transform of the type: Lebedev-Legendre of the second kind -- Legendre of the second kind (Q2753515)

A hybrid differential operator \({\mathcal M}_{\mu,\alpha} = a_1^2\theta(r) \theta(R_1-r)B_\alpha + a_2^2\theta(r-R_1)\theta(R_2-r)\Lambda_{\mu_1} + a_3^2\theta(r-R_2)\theta(R_3-r)\Lambda_{\mu_2}\) is considered. Here \(a_j>0\), \(\theta(x)\) is the Heaviside step function, \(B_\alpha =r^2 d^2/dr^2 + r(2\alpha+1)d/dr - \lambda^2 r+\alpha^2\), \(0\leq\lambda < \infty\), \(\alpha\geq -1/2\), is the Bessel differential operator, \(\Lambda_\mu=d^2/dr^2 + \coth r d/dr + 1/4 - \mu^2/\sinh^2r\), \(\mu \geq 0\), is the Legendre differential operator. Using the method of delta-like sequences, a hybrid transform of the type Lebedev-Legendre of the second kind -- Legendre of the second kind with two conjugation points in the interval \((0, R_3)\), \(R_3 < \infty\) is constructed. As the delta-like sequence the fundamental matrix of solutions of the Cauchy problem for the system of parabolic differential equations related to the differential operator \({\mathcal M}_{\mu,\alpha}\) is chosen.