By using methods of generalized functions theory, the analytical solutions of Maxwell’s equations are constructed in uniaxial crystals with joint electric and magnetic anisotropy, when the matrix of dielectric permittivity and magnetic permeability are diagonal. The case of stationary vibrations is considered, which generated by the currents, where the complex amplitudes belong to the class of generalized vector functions. For solutions constructing the direct and inverse Fourier transforms of generalized functions are used and fundamental solutions of the wave equations. The results of some numerical calculations are presented for illustration of influence of anisotropy to waveguide properties of uniaxial crystals. In particular, radiation patterns of Hertz vibrator in uniaxial crystals with electric and magnetic anisotropy have been researched.

We consider biquaternionic wave (biwave) equations. They are biquaternionic generalizations of the Maxwell and Dirac equations and are equivalent to a system of eight differential equations of hyperbolic type. Using the theory of generalized functions, we construct fundamental and generalized solutions of such equations, including discontinuous ones, describing shock waves and obtain conditions on the fronts. A solution of the Cauchy problem for a biwave equation is constructed, and so are analogs of Kirchhoff–Green’s formulas that permit one to determine the solution inside a bounded domain given the boundary and initial values of the solution in this domain.

A coupled thermoelasticity model is used to study the dynamics of a thermoelastic half-plane influenced by nonstationary body forces and heat sources. In space of Laplace transforms with...

In the present paper, solutions of the equations of uncoupled thermoelastodynamics of ther- moelastic rods are constructed for power and thermal effects. Based on the Fourier transform, the Green tensor and generalized solutions of the thermoelasticity equations are constructed in the original...

In this paper, the solution to the Dirichlet problem for the wave equation on the star graph is constructed. To begin, we solve the boundary value problem on the interval (on one edge of the graph). We use the generalized functions method to obtain the wave equation with a singular right-hand side. The solution to the Dirichlet problem is determined through the convolution of the fundamental solution with the singular right-hand side of the wave equation. Thus, the solution found on the interval is determined by the initial functions, boundary functions, and their derivatives (the unknown boundary functions). A resolving system of two linear algebraic equations in the space of the Fourier transform in time is constructed to determine the unknown boundary functions. Following inverse Fourier transforms, the solution to the Dirichlet problem of the wave equation on the interval is constructed. After determining all the solutions on all edges and taking the continuity condition and Kirchhoff joint condition into account, we obtain the solution to the wave equation on the star graph.

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.