Effective Solution of Classical Boundary Value Problems of a Thermoelastostatically Multi-Cavity Body by Means of the Theory of Hilbert Spaces

The isomorphism of Hilbert spaces of internal and boundary states for a multi-cavity thermoelastostatic 3D body is used to solve boundary value problems. The mathematical model is a system of four partial differential equations of the second order of elliptic type. Inhomogeneous components of solutions of equations with a regular nature of internal influences are written out strictly. Bases of Hilbert spaces of states are generated to solve a homogeneous problem. The solutions are constructed in the form of Fourier series based on the elements of orthonormal bases. Specific studies of the states of a two-cavity cylinder under external bending force and various variants of temperature influences from spherical cavities have been carried out. The results of the solutions are illustrated.