Hybrid integral transform of Fourier-Fourier-(Kontorovych-Lebedev) type on the polar axis (Q2768787)

Using the method of delta-shaped sequence the authors introduce the integral transform generated on the set \(I_{2}^{+}=\{r: r\in(R_0,R_1)\cup(R_1,R_2)\cup(R_2,\infty); R_0\geq 0\}\) by the hybrid differential operator \(M_{\alpha}=a_1^2\Theta(r-R_0)\Theta(R_1-r)d^2/dr^2+ a_2^2\Theta(r-R_1)\Theta(R_2-r)d^2/dr^2+a_3^2\Theta(r-R_2)B_{\alpha}\), where \(a_{j}>0, \Theta(x)=\begin{cases} 0,&x<0,\\ 1,&x\geq 0,\end{cases}\) NEWLINE\[CARRIAGE_RETURNNEWLINEB_{\alpha}=r^2{d^2\over dr^2}+(2\alpha+1)r{d\over dr}+\alpha^2-\lambda^2r^2,\;2\alpha+1\geq 0,\;\lambda\in (0,\infty).CARRIAGE_RETURNNEWLINE\]NEWLINE A theorem on the fundamental identity is proved. The proposed integral transform is applied to solve in the domain \(\{(t,r):\;t\in(0,\infty), r\in I^{+}_2\}\) the system of equations NEWLINE\[CARRIAGE_RETURNNEWLINE\begin{aligned} {\partial^2 u_{j}\over\partial t^2}+\gamma_{j}^2u_{j}-a_{j}^2{\partial^2 u_{j}\over\partial r^2} =f_{j}(t,r), r\in (R_{j-1},R_{j}),\;j & =1,2; \\ {\partial^2 u_{3}\over\partial t^2}+\gamma_{3}^2u_{3}-a_{3}^2 B_{\alpha}[u_3] =f_{3}(t,r),\;r\in (R_{2},\infty)\end{aligned} CARRIAGE_RETURNNEWLINE\]NEWLINE with the initial conditions \(u_{m}(t,r)|_{t=0}=g_{m1}(r),\;\partial u_{m}/\partial t|_{t=0}=g_{m2}(r),\;m=1,2,3\), the conjunction conditions NEWLINE\[CARRIAGE_RETURNNEWLINE[(\alpha_{j1}^{k}{\partial\over\partial r}+\beta_{j1}^{k})u_{k}(t,r)-(\alpha_{j2}^{k}{\partial\over\partial r}+\beta_{j2}^{k})u_{k+1}(t,r)]|_{r=R_{k}}=\omega_{jk}(t);\;j,k=1,2CARRIAGE_RETURNNEWLINE\]NEWLINE and the boundary conditions NEWLINE\[CARRIAGE_RETURNNEWLINE(\alpha_{11}^{0}{\partial\over\partial r}+\beta_{11}^{0})u_{1}(t,r)|_{r=R_0}=g_0(t),\;|u_3(t,r)||_{r=\infty}<\infty.CARRIAGE_RETURNNEWLINE\]