Matrix LSQR algorithm for structured solutions to quaternionic least squares problem

In this paper, we employ matrix LSQR algorithm to deal with quaternionic least squares problem in order to find the minimum norm solutions with kinds of special structures, and propose a strategy to accelerate convergence rate of the algorithm via right–left preconditioning of the coefficient matrices. We mainly focus on analyzing the minimum norm $\eta$-Hermitian solution and the minimum norm $\eta$-biHermitian solution to the quaternionic least squares problem, $\eta \in \{\mathbf{i},\mathbf{j},\mathbf{k}\}$. Other structured solutions also can be obtained using the proposed technique. A number of numerical experiments are performed to show the efficiency of the preconditioned matrix LSQR algorithm.     

An accelerated cyclic-reduction-based solvent method for solving quadratic eigenvalue problem of gyroscopic systems

The quadratic eigenvalue problem (QEP) $({\lambda }^{2}M+\lambda G+K)x=0$, with $M^T=M$ being positive definite, $K^T=K$ being negative definite and $G^T=?G$, is associated with gyroscopic systems. In Guo (2004), a cyclic-reduction-based solvent (CRS) method was proposed to compute all eigenvalues of the above mentioned QEP. Firstly, the problem is converted to find a suitable solvent of the quadratic matrix equation (QME) $M{X}^2+GX+K=0$. Then using a Cayley transformation and a proper substitution, the QME is transformed into the nonlinear matrix equation (NME) $Z+A^T{Z}^{?1}A=Q$ with $A=M+K+G$ and $Q=2(M?K)$. The problem finally can be solved by applying the CR method to obtain the maximal symmetric positive definite solution of the NME as long as the QEP has no eigenvalues on the imaginary axis or for some cases where the QEP has eigenvalues on the imaginary axis. However, when all eigenvalues of the QEP are far away from or near the origin, the Cayley transformation seems not to be the best one and the convergence rate of the CRS method proposed in Guo (2004) might be further improved. In this paper, inspired by using a doubling algorithm to solve the QME, we use a Möbius transformation instead of the Cayley transformation to present an accelerated CRS (ACRS) method for solving the QEP of gyroscopic systems. In addition, we discuss the selection strategies of optimal parameter for the ACRS method. Numerical results demonstrate the efficiency of our method.     

Quasi gradient-based inversion-free iterative algorithm for solving a class of the nonlinear matrix equations

Inspired by the gradient-based and inversion-free iterations, a new quasi gradient-based inversion-free iterative algorithm is proposed for solving the nonlinear matrix equation $X+A^{\mathrm{T}}{X}^{?n}A=I$. The convergence proof of the suggested algorithm is given. Several matrix norm inequalities are established to depict the convergence properties of this algorithm. Three numerical examples are given to illustrate the effectiveness of the suggested algorithms.     

A remark concerning divergence accuracy order for H(div)-conforming finite element flux approximations

The construction of finite element approximations in $\mathbf{H}(\text{div},\Omega )$ usually requires the Piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region $\Omega$. It is known that degradation may occur in convergence order if non affine geometric mappings are used. On this point, we revisit a general procedure for the improvement of two-dimensional flux approximations discussed in a recent paper of this journal (Comput. Math. Appl. 74 (2017) 3283–3295). The starting point is an approximation scheme, which is known to provide $L^2$-errors with accuracy of order $k+1$ for sufficiently smooth flux functions, and of order $r+1$ for flux divergence. An example is $R{T}_k$ spaces on quadrilateral meshes, where $r=k$ or $k?1$ if linear or bilinear geometric isomorphisms are applied. Furthermore, the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions. The goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy $r+n+1$ as desired, for any $n\ge 1$. The enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level $k+n$, while keeping the original border fluxes at level $k$. The case $n=1$ has been discussed in the mentioned publication for two particular examples. General stronger enrichment $n>1$ shall be analyzed and applied to Darcy’s flow simulations, the global condensed systems to be solved having same dimension and structure of the original scheme.     

Ground state solutions of Pohoz̆aev type for fractional Choquard equations with general nonlinearities

In this paper, we study the fractional Choquard equation

${(?\Delta )}^{s}u+u=\left(\right|x{|}^{?\mu }1F\left(u\right))f\left(u\right),\text{in}{\mathbb{R}}^N,$

where $N\ge 3$, $0<s<1$, $0<\mu <min\{N,4s\}$, and $f\in C(\mathbb{R},\mathbb{R})$ satisfies the general Berestycki–Lions conditions. Combining constrained variational method with deformation lemma, we obtain a ground state solution of Pohoz?aev type for the above equation. The result improves some ones in Shen et al. (2016).     

Existence of multiple solutions for nonhomogeneous Schrödinger–Kirchhoff system involving the fractional p-Laplacian with sign-changing potential

In this paper, we prove the existence of multiple solutions for the following Schrödinger–Kirchhoff system involving the fractional $p$-Laplacian

$\begin{array}{c}\hfill \left\{\begin{array}{c}M\left({?}_{{\mathbb{R}}^{2N}}\frac{{|u\left(x\right)?u\left(y\right)|}^p}{{|x?y|}^{N+ps}}dxdy\right){(?\Delta )}_p^{s}u+V\left(x\right){\left|u\right|}^{p?2}u=F_u(x,u,v)+\lambda g\left(x\right),x\in {\mathbb{R}}^N,\hfill \\ M\left({?}_{{\mathbb{R}}^{2N}}\frac{{|v\left(x\right)?v\left(y\right)|}^p}{{|x?y|}^{N+ps}}dxdy\right){(?\Delta )}_p^{s}u+V\left(x\right){\left|v\right|}^{p?2}v=F_v(x,u,v)+\lambda h\left(x\right),x\in {\mathbb{R}}^N,\hfill \\ u\left(x\right)\to 0,v\left(x\right)\to 0,\text{as}\left|x\right|\to +\infty ,\hfill \end{array}\right.\end{array}$

where ${(?\Delta )}_p^s$ denotes the fractional $p$-Laplacian of order $s\in (0,1)$, $2\le p<\infty$, $ps<N$, $F_u=\frac{?F}{?u}$, $F_v=\frac{?F}{?v}$, $V\left(x\right)$ is allowed to be sign-changing, $\lambda >0$ and $g,h:{\mathbb{R}}^N\to \mathbb{R}$ is a perturbation. Under some certain assumptions on $f$, we obtain the existence of multiple solutions for this problem via Ekeland’s variational principle and mountain pass theorem.     

On the absence of global solutions for quantum versions of Schrödinger equations and systems

We, first, consider the quantum version of the nonlinear Schrödinger equation

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |u(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

where $0<q<1$, $i^q$ is the principal value of $i^q$, $D_{q|t}$ is the $q$-derivative with respect to $t$, $\Delta$ is the Laplacian operator in ${\mathbb{R}}^N$, $\lambda \in ??\left\{0\right\}$, $p>1$, and $u(t,x)$ is a complex-valued function. Sufficient conditions for the nonexistence of global weak solution to the considered equation are obtained under suitable initial data. Next, we study the system of nonlinear coupled equations

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |v(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

$i^q{D}_{q|t}v(t,x)+\Delta v(qt,x)=\lambda |u(qt,x){|}^m,t>0,x\in {\mathbb{R}}^N,$     

Global solution of a diffusive predator–prey model with prey-taxis

We consider the prey-taxis system:

$\left\{\begin{array}{cc}{u}_t=d_1\Delta u?\chi ??(u?v)+u(a?\mu u)+buf\left(v\right),\hfill & x\in \Omega ,t>0,\hfill \\ v_t=d_2\Delta v+v(c?\beta v)?uf\left(v\right),\hfill & x\in \Omega ,t>0\hfill \end{array}\right.$

in a smoothly bounded domain $\Omega ?{\mathbb{R}}^n$, with zero-flux boundary condition, where $a,d_1,d_2,\chi ,\mu ,b,c$ are positive constants and $\beta$ is a non-negative constant. We first investigate the global existence and local boundedness of solution for the case $\beta =0$. Moreover, when $\beta >0$, we show that the solution exists globally and is uniformly bounded provided $\mu$ is large enough.     

Existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional Kirchhoff-type systems

This paper is concerned with the following linearly coupled fractional Kirchhoff-type system

$\left\{\begin{array}{c}\left(a+b{\int }_{{\mathbb{R}}^3}|{(?△)}^{\frac{\alpha }{2}}u{|}^2\mathrm{d}x\right){(?△)}^{\alpha }u+\lambda u=f\left(u\right)+\gamma v,\text{in}{\mathbb{R}}^3,\hfill \\ \left(c+d{\int }_{{\mathbb{R}}^3}|{(?△)}^{\frac{\alpha }{2}}v{|}^2\mathrm{d}x\right){(?△)}^{\alpha }v+\mu v=g\left(v\right)+\gamma u,\text{in}{\mathbb{R}}^3,\hfill \\ u,v\in H^{\alpha }\left({\mathbb{R}}^3\right),\hfill \end{array}\right.$

where $a,c,\lambda ,\mu >0$, $b,d\ge 0$ are constants, $\alpha \in [\frac{3}{4},1)$ and $\gamma >0$ is a coupling parameter. Under the general Berestycki–Lions conditions on the nonlinear terms $f$ and $g$, we prove the existence of positive vector ground state solutions of Poho?aev type for the above system via variational methods. Moreover, the asymptotic behavior of these solutions as $\gamma \to 0^{+}$ is explored as well. Recent results from the literature are generally improved and extended.     

Global well-posedness of the incompressible fractional Navier–Stokes equations in Fourier–Besov spaces with variable exponents

We study the Cauchy problem of the fractional Navier–Stokes equations in critical variable exponent Fourier–Besov spaces $F{\stackrel{?}{B}}_{p\left(?\right),q}^{4?2\alpha ?\frac{3}{p\left(?\right)}}$. We discuss some properties of variable exponent Fourier–Besov spaces and prove a general global well-posedness result which covers some recent works about classical Navier–Stokes equations.     

Stability and convergence of semi-implicit time-stepping algorithm for stationary incompressible magnetohydrodynamics

For the stationary incompressible magnetohydrodynamics (MHD) equations, we provide a new uniqueness assumption (A0) and show the exponential stability of the solution. Then, the semi-implicit time-stepping algorithm is used to solve the stationary MHD equations. The algorithm is proved to be unconditionally stable. The discrete velocity and magnetic field are bounded in $L^{\infty }(0,+\infty ;{\mathit{L}}^{\infty }\left(\Omega \right))$ for any space and time step sizes. The error estimates for the algorithm are deduced under the uniqueness conditions. Finally, numerical experiments are performed to testify our theoretical analysis.     

Phase diagrams of the thermoelectric Bi–Sb–Se system

Bi–Sb–Se is an important thermoelectric material system. However, a phase diagram of the entire compositional range is not available in the literature. This study determines the Bi–Sb–Se liquidus projection and its phase equilibria isothermal section at 400 °C. Ternary Bi–Sb–Se alloys are prepared. Their primary solidification phases and invariant reactions are determined. There are seven primary solidification phases, (Se), Bi₂Se₃, Sb₂Se₃, (Bi,Sb), ${\left({\mathrm{B}\mathrm{i}}_2\right)}_{\mathrm{m}}{\left({\mathrm{B}\mathrm{i}}_2{\mathrm{S}\mathrm{e}}_3\right)}_{\mathrm{n}}$, Bi₃Sb₅Se₂ and Bi₃Sb₁₂Se₅. Both Bi₃Sb₅Se₂ and Bi₃Sb₁₂Se₅ are newly found ternary compounds. In the Bi–Sb–Se isothermal section at 400 °C, there are eight three-phase regions. Besides these two ternary compounds, the other single phases are, Sb₂Se₃, Bi₂Se₃, ${\left({\mathrm{B}\mathrm{i}}_2\right)}_{\mathrm{m}}{\left({\mathrm{B}\mathrm{i}}_2{\mathrm{S}\mathrm{e}}_3\right)}_{\mathrm{n}}$, Liquid (Bi,Sb), Liquid (Se), and (Bi,Sb) phases. It has been found the solubilities of Sb in the ${\left({\mathrm{B}\mathrm{i}}_2\right)}_{\mathrm{m}}{\left({\mathrm{B}\mathrm{i}}_2{\mathrm{S}\mathrm{e}}_3\right)}_{\mathrm{n}}$ and Bi₂Se₃ compounds are significant.