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In this paper, the propagation of pulse wave in a deformable elastic vessel filled with inviscid blood is studied. Starting from the stress–strain relationship of blood vessel wall, momentum conservation equation and the Naiver–Stokes equation, the basic equations describing the wall motion and blood flow are established. By utilizing reductive perturbation technique and long wave approximation theory, the basic equations are simplified into a classical third-order mKdV equation with variable coefficients. In order to describe the propagation characteristics of pulse wave more accurately, a fifth-order variable-coefficient mKdV equation is derived. Then, the tanh-function method is applied to find the localized traveling wave solutions of these equations. Based on these localized traveling wave solutions, we further investigate the effects of higher order terms and initial vessel deformation on the characteristics of pulse wave propagation, blood flow velocity and the volume of blood flow. The results show that the higher-order nonlinear and dispersion terms lead to the distortion of the wave, while the initial deformation of the tube wall will influence the wave amplitude and wave width. [ABSTRACT FROM AUTHOR]

The paper addresses the problem of determining the field of a reflected wave in a multiscale inhomogeneous medium, which consists of a background deterministic large-scale irregularity and multiscale random irregularities with scales both larger and smaller than the wavelength. A layer with a linear permittivity profile is taken as the background irregularity. The small-scale irregularities are responsible for wide-angle scattering including backscattering. The first approximation of the perturbation theory is used to account for scattering from these irregularities. As the zero approximation and Green's function in determining the field backscattered by small-scale irregularities, we utilize the integral representation of the field, obtained in our earlier work by combining the method of double weighted Fourier transform (DWFT) and the Fock proper-time method. Asymptotic methods are used to reduce the sevenfold integral representation of a field to lower-order integrals. Conditions for validity of such representations are obtained. Formulas of the frequency coherence functions of waves reflected and backscattered from the turbulent plasma layer are given. The paper presents the results of the simulation of pulse sounding of a randomly inhomogeneous reflecting plasma layer demonstrating the effect of large-scale irregularities on backscattering by small-scale irregularities. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of the initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space. The methods applied are the Faedo–Galerkin approximation and the general theory of weak compactness in appropriate weighted Sobolev spaces together with using the Poincaré-type operator for dealing nonlinear nonlocal conditions. Furthermore, the boundedness and positivity of solutions depending on the boundedness and positivity of given data are also discussed by using suitable test functions. [ABSTRACT FROM AUTHOR]

This paper investigates an adaptive cooperative fault-tolerant supervisory control problem for nonlinear strict-feedback leader-following systems with unknown control coefficients and actuator faults under the fixed directed graph. Radial basis function neural networks are used to approximate system uncertainties. The Nussbaum gain technique is introduced to address unknown signs of control gains. Then, based on the dynamic surface control method, a distributed adaptive fault-tolerant control scheme is presented to compensate for the actuator faults of followers. For neural networks approximation theory, the unknown smooth function can only be approximated on a compact set due to the approximation errors. By theoretic analysis, the supervisory-based adaptive controllers are proposed to guarantee that system state signals can converge to compact sets and the leader-following systems can achieve the practical output consensus. Finally, the simulation results are provided to show the validity of the proposed consensus scheme. [ABSTRACT FROM AUTHOR]

From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab u ˇ ska and J. Osborn, [Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991.] is h 2 k − 2 for k ≥ 2. In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming H 1 (Ω) × H 1 (Ω) formulation, following the variational formulation of Ciarlet–Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order h 2 for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order. [ABSTRACT FROM AUTHOR]