Hyperbolic Boundary Value Problems of Mathematical Physics in a Piecewise Homogeneous Wedge-Shaped Cylindrical-Circular Half-Space

The unique exact analytical solutions of hyperbolic boundary value problems of mathematical physics in piecewise homogeneous by the radial variable r, wedge-shaped by the angular variable φ, cylindrical-circular half-space were constructed at first time by the method of classical integral and hybrid integral transforms in combination with method of main solutions (matrices of influence and Green matrices) in the proposed article. The cases of assigning on the verge of the wedge the boundary conditions of the 1st kind (Dirichlet) and the 2nd kind (Neumann) and their possible combinations (Dirichlet – Neumann, Neumann – Dirichlet) are considered. Finite integral Fourier transform by an angular variable φ, an integral Fourier transform on the Cartesian semiaxis (0; +∞) by an applicative variable z and hybrid Fourier-Bessel-type integral transform on the polar axis (0; +∞) with n conjugate points by the radial variable were used to construct solutions of investigated boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for an ordinary linear inhomogeneous 2nd order differential equation whose unique solution is written in a closed form. The application of inverse integral transforms to the obtained solution in the space of images restores in an explicit form in the space of the originals the solutions of the considered hyperbolic boundary value problems of mathematical physics through their integral image. At the same time, the main solutions of the problems are obtained in an explicit form.