Discrete-continuous three-element model of impact device

There have been examined the mathematic model of the impact device provided for geological materials destruction. Basic elements of the impact device are variable cross-section tool, striker and impact device body. The interaction of these elements is described as a movement of two discrete mass and the rod in the presence of rigid and dissipative connections. One equation in partial derivatives and two ordinary differential equations associated by initial and boundary conditions represent the initial-boundary problem. The numerical method parameters of which are determined at tests problems solution by Fourier method is used for looking for solutions of mixed initial-boundary problem. Researches are made, and parameters determining the damping efficiency of tool, striker and impact device body oscillations are evaluated.

Fractional Dual-Phase Lag Equation—Fundamental Solution of the Cauchy Problem

In the paper, a fundamental solution of the fractional dual-phase-lagging heat conduction problem is obtained. The considerations concern the 1D Cauchy problem in a whole-space domain. A solution of the initial-boundary problem is determined by using the Fourier–Laplace transform technique. The final form of solution is given in a form of a series. One of the properties of the derived fundamental solution of the considered problem with the initial condition expressed be the Dirac delta function is that it is symmetrical. The effect of the time-fractional order of the Caputo derivatives and the phase-lag parameters on the temperature distribution is investigated numerically by using the method which is based on the Fourier-series quadrature-type approximation to the Bromwich contour integral.

Fujita blow-up phenomena of solutions for a Dirichlet problem of parabolic equations with space-time coefficients

This paper deals with a homogeneous Dirichlet initial-boundary problem of parabolic equations with different space-time coefficients, $$u_t =\Delta u + t^{\sigma_1} u^{\alpha} + \langle x\rangle^{n} v^{p},\quad v_t =\Delta v + \langle x\rangle^{m} u^{q} + t^{\sigma_2} v^{\beta},$$ where the eight exponents are nonnegative constants and $\langle x\rangle$ is the Japanese brackets. We obtain the Fujita exponents of solutions, which are determined by the eight exponents and the dimension of the space domain. Moreover, simultaneous or non-simultaneous blow-up of the two components of blow-up solutions is discussed with or without conditions on the initial data.

Zero viscosity and thermal diffusivity limit of the linearized compressible Navier–Stokes–Fourier equations in the half plane

We study the zero viscosity and thermal diffusivity limit of an initial boundary problem for the linearized Navier–Stokes–Fourier equations of a compressible viscous and heat conducting fluid in the half plane. We consider the case that the viscosity and thermal diffusivity converge to zero at the same order. The approximate solution of the linearized Navier–Stokes–Fourier equations with inner and boundary expansion terms is analyzed formally first by multiscale analysis. Then the pointwise estimates of the error terms of the approximate solution are obtained by energy methods. Thus establish the uniform stability for the linearized Navier–Stokes–Fourier equations in the zero viscosity and heat conductivity limit. This work is based on (Comm. Pure Appl. Math. 52 (1999), 479–541) and generalize their results from isentropic case to the general compressible fluid with thermal diffusive effect. Besides the viscous layer as in (Comm. Pure Appl. Math. 52 (1999), 479–541), the thermal layer appears and couples with the viscous layer linearly.

Longitudinal waves in an elastic rod caused by sudden damage to the foundation

We study the problem of propagating longitudinal waves in an elastic rod connected to a locally damaged foundation through a thin elastic layer. The motion of the rigid foundation blocks is considered predetermined. We formulated the initial-boundary problem for the Klein-Gordon equation with a discontinuous right-hand side. The nonstationary fields of displacements, velocities, and deformations were investigated by the Laplace integral transformation method. Examples of sudden divergence of fragments of the foundation by a given value and their mutual separation at a constant speed are considered.