The inverse scattering transform (IST) for the defocusing Manakov system is developed with non-zero boundary conditions at infinity comprising non-parallel boundary conditions — i.e., asymptotic polarization vectors. The formalism uses a uniformization variable to map two copies of the spectral plane into a single copy of the complex plane, thereby eliminating square root branching. The “adjoint” Lax pair is also used to overcome the problem of non-analyticity of some of the Jost eigenfunctions. The inverse problem is formulated in term of a suitable matrix Riemann-Hilbert problem (RHP). The most significant difference in the IST compared to the case of parallel boundary conditions is the asymptotic behavior of the scattering coefficients, which affects the normalization of the eigenfunctions and the sectionally meromorphic matrix in the RHP. When the asymptotic polarization vectors are not orthogonal, two different methods are presented to convert the RHP into a set of linear algebraic-integral equations. When the asymptotic polarization vectors are orthogonal, however, only one of these methods is applicable. Finally, it is shown that, both in the case of orthogonal and non-orthogonal polarization vectors, no reflectionless potentials can exist, which implies that the problem does not admit pure soliton solutions.

The elastic mechanics of cuff-tissue-bone-artery systems is studied by using the phase-field method. It is shown that such factors as the Young modulus of the tissue, thickness and elasticity of the blood vessel wall, and blood and cuff pressures significantly influence the oscillation waveforms of the cuff pressure and cause difficulties in decoding the cuff measurements. In addition, our simulations point out at additional information in the oscillometric waveform, which can be of help in the blood pressure decoding. The development of the next-generation instruments for blood pressure measurements requires further studies of cuff-tissue-bone-artery systems and elastic properties of human tissue and blood vessel walls.

Novelty of this work is the development of a finite element method using discontinuous $P_k$ element, which has two-order higher convergence rate than the optimal order. The method is used to solve a one-dimensional second order elliptic problem. A totally new approach is developed for error analysis. Superconvergence of order two for the CDG finite element solution is obtained. The $P_k$ solution is lifted to an optimal order $P_{k+2}$ solution elementwise. The numerical results confirm the theory.

A novel sharp-interface model for the solid-state dewetting problem in the two-dimensional case is proposed. Instead of incorporating the Willmore energy or an $L_2$-curvature regularization term as is done in previous studies, we add an $L_1$-curvature regularization to the interfacial energy functional in order to make this problem well-posed. Experiments show that such regularization improves the computational efficiency. In strongly anisotropic case, we consider a new numerical scheme based on the convex-splitting idea. This approach remarkably relax the restriction on the time step. In addition, we present the theoretical analysis of the scheme. Numerical results demonstrate the high efficiency and accuracy of the method.

The stability and uniqueness of the solutions of time-fractional telegraph equations arising in the transmission line modeling are proved. The corresponding initial-boundary problems are then solved by a finite difference scheme. It is shown that the scheme is unconditionally stable and convergent. Computational efficiency of the method can be enhanced by transforming it into two finite volume schemes for solving two uncoupled time-fractional convection equations. Numerical experiments validate the theoretical results and show the efficiency of this approach even for the problems the solutions of which are not smooth at the initial moment.

The forager-exploiter model with gradient-dependent flux-limitation $$u_t=\Delta u-\chi\nabla \cdot (uk_f(1+|\nabla w|^2)^{-\frac{\alpha}{2}})\nabla w,$$$$v_t=\Delta v-\xi\nabla \cdot (vk_g(1+|\nabla u|^2)^{-\frac{\beta}{2}})\nabla u,$$$$w_t=\Delta w-(u+v)w-\mu w+r(x,t)$$is considered in smooth bounded domains $Ω ⊂ \mathbb{R}^N,$ $N ≥ 2.$ It is shown that if $α > (N − 2)/N(N − 1),$ $β > 0,$ then for any nonnegative functions $u_0,$ $v_0,$ $w_0∈ W^{2,∞}(Ω)$ such that $u_0 \not\equiv 0$ and $v_0 \not\equiv 0,$ the problem has a global classical solution $(u, v, w) ∈ (C^0 (\overline{Ω} × [0,∞))\cap C^{2,1}(\overline{Ω} × (0,∞)))^3$.

A greedy Gauss-Seidel based on the greedy Kaczmarz algorithm and aimed to find approximations of the solution $A^†b$ of systems of linear algebraic equations with a full column-rank coefficient matrix $A$ is proposed. Developing this approach, we introduce a partially greedy randomized extended Gauss-Seidel method for finding approximate least-norm least-squares solutions of column-rank deficient linear systems. The convergence of the methods is studied. Numerical experiments show that the proposed methods are robust and efficient.

The stability of high-order finite-difference schemes on a staggered-grid for two-dimensional poroelastic wave equations with spatially varying material parameters is studied. Using the energy method, we obtain sufficient stability conditions. This allows to find suitable time and spatial steps according to material parameters and the difference scheme coefficients. Two numerical examples verify the theoretical analysis and show that the corresponding range for the time step is close to that in the necessary condition. The perfectly matched layer is adopted in order to eliminate boundary reflections.

In this paper, the global momentum conservation laws and the global momentum evolution laws are presented for the two-dimensional stochastic nonlinear Schrödinger equation with multiplicative noise and the two-dimensional stochastic Klein-Gordon equation with additive noise, respectively. In order to preserve the global momenta or their changing trends in numerical simulation, the schemes are constructed by using a stochastic multi-symplectic formulation. It is shown that under periodic boundary conditions, the schemes have discrete global momentum conservation laws or the discrete global momentum evolution laws. Numerical experiments confirm global momentum-preserving properties of the schemes and their mean square convergence in the time direction.