The N-soliton solution and localized wave interaction solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation

In this paper, the $N$-soliton solution is constructed for the ($2+1$)-dimensional generalized Hirota–Satsuma–Ito equation, from which some localized waves such as line solitons, lumps, periodic solitons and their interactions are obtained by choosing special parameters. Especially, by selecting appropriate parameters on the multi-soliton solutions, the two soliton can reduce to a periodic soliton or a lump soliton, the three soliton can reduce to the elastic interaction solution between a line soliton and a periodic soliton or the elastic interaction between a line soliton and a lump soliton, while the four soliton can reduce to elastic interaction solutions among two line solitons and a periodic soliton or the elastic interaction ones between two periodic solitons. Detailed behaviours of such solutions are illustrated analytically and graphically by analysing the influence of parameters. Finally, an inelastic interaction solution between a lump soliton and a line soliton is constructed via the ansatz method, and the relevant interaction and propagation characteristics are discussed graphically. The results obtained in this paper may be helpful for understanding the interaction phenomena of localized nonlinear waves in two-dimensional nonlinear wave equations.     

Unsteady micropolar fluid flow in a thin domain with Tresca fluid–solid interface law

We consider a micropolar fluid flow in a two-dimensional domain. We assume that the velocity field satisfies a non-linear slip boundary condition of friction type on a part of the boundary while the micro-rotation field satisfies non-homogeneous Dirichlet boundary conditions. We prove the existence and uniqueness of a solution. Then motivated by lubrication problems we assume that the thickness and the roughness of the domain are of order $0<\epsilon <<1$ and we study the asymptotic behaviour of the flow as $\epsilon$ tends to zero. By using the two-scale convergence technique we derive the limit problem which is totally decoupled for the limit velocity and pressure $(v^0,p^0)$ on one hand and the limit micro-rotation $Z^0$ on the other hand. Moreover we prove that $v^0$, $p^0$ and $Z^0$ are uniquely determined via auxiliary well-posed problems.     

Global well-posedness of the d-D Oldroyd-B type models with fractional Laplacian dissipation

This paper focuses on the Cauchy problem of the d-dimensional incompressible Oldroyd-B type models for viscoelastic flow with fractional Laplacian dissipation, namely, with ${(?\Delta )}^{{\eta }_1}u$ and ${(?\Delta )}^{{\eta }_2}\tau$. For ${\eta }_1\ge \frac{1}{2}+\frac{d}{4}$, ${\eta }_2>0$ and ${\eta }_1+{\eta }_2\ge 1+\frac{d}{2}$, we obtain the global regularity of strong solutions when the initial data $(u_0,{\tau }_0)$ are sufficiently smooth.     

A finite volume method for Stokes problems on quadrilateral meshes

We present a finite volume method for Stokes problems using the isoparametric $Q_1$–$Q_0$ element pair on quadrilateral meshes. To offset the lack of the $inf$–$sup$ condition, a jump term of discrete pressure (stabilizing term) is added to the continuity approximation equation. Thus, we establish a stabilized finite volume scheme on quadrilateral meshes. Then, based on some superclose estimates, we derive the optimal error estimates in the $H^1$- and $L_2$-norms for velocity and in the $L_2$-norm for pressure, respectively. Numerical examples are provided to illustrate our theoretical analysis. We emphasize that our work is the first time to propose and analyze a finite volume method for Stoke problems using isoparametric elements on quadrilateral meshes.     

Parallel two-grid finite element method for the time-dependent natural convection problem with non-smooth initial data

In this paper, we consider the parallel two-grid finite element method for the transient natural convection problem with non-smooth initial data. Our numerical scheme involves solving a nonlinear natural convection problem on the coarse grid and solving a linear natural convection problem on the fine grid. The linear natural convection problem can be split into two subproblems which can be solved in parallel: a linearized Navier–Stokes problem and a linear parabolic problem. We firstly provide the stability and convergence of standard Galerkin finite element method with non-smooth initial data. Secondly, we develop optimal error estimates of two-grid finite element method for velocity and temperature in $H^1$-norm and for pressure in $L^2$-norm. In order to overcome the difficulty posed by the loss of regularity, some suitable weight functions are introduced in our stability and convergence analysis for the natural convection equations. Finally, some numerical results are presented to verify the established theoretical results.     

An efficient NFEM for optimal control problems governed by a bilinear state equation

In this paper, a nonconforming finite element method (NFEM) is proposed for the constrained optimal control problems (OCPs) governed by a bilinear state equation. The state and adjoint state are approximated by the nonconforming $E{Q}_1^{rot}$ element, and the control is approximated by the orthogonal projection through the state and adjoint state. Some superclose and superconvergence properties are obtained by full use of the distinguish characters of this $E{Q}_1^{rot}$ element, such as the interpolation operator equals the Ritz projection, and the consistency error is one order higher than its interpolation error in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.     

Superconvergence analysis of nonconforming finite element method for time-fractional nonlinear parabolic equations on anisotropic meshes

In this paper, we prove a novel result of the consistency error estimate with order $O\left(h^2\right)$ for $E{Q}_1^{rot}$ element (see Lemma 2) on anisotropic meshes. Then, a linearized fully discrete Galerkin finite element method (FEM) is studied for the time-fractional nonlinear parabolic problems, and the superclose and superconvergent estimates of order $O(\tau +h^2)$ in broken $H^1$-norm on anisotropic meshes are derived by using the proved character of $E{Q}_1^{rot}$ element, which improve the results in the existing literature. Numerical results are provided to confirm the theoretical analysis.     

Alzheimer’s disease multiclass diagnosis via multimodal neuroimaging embedding feature selection and fusion

Alzheimer’s disease (AD) will become a global burden in the coming decades according to the latest statistical survey. How to effectively detect AD or MCI (mild cognitive impairment) using reliable biomarkers and robust machine learning methods has become a challenging problem. In this study, we propose a novel AD multiclass classification framework with embedding feature selection and fusion based on multimodal neuroimaging. The framework has three novel aspects: (1) An $l_{2,1}$-norm regularization term combined with the multiclass hinge loss is used to naturally select features across all the classes in each modality. (2) To fuse the complementary information contained in each modality, an $l_p$-norm $(1<p<\infty )$ regularization term is introduced to combine different kernels to perform multiple kernel learning to avoid a sparse kernel coefficient distribution, thereby effectively exploiting complementary modalities. (3) A theorem that transforms the multiclass hinge loss minimization problem using the $l_{2,1}$-norm and $l_p$-norm regularizations to a previous solvable optimization problem and its proof are given. Additionally, it is theoretically proved that the optimization process converges to the global optimum. Extensive comparison experiments and analysis support the promising performance of the proposed method.     

Note on nonparametric spectral analysis of wideband spectrum with missing data via sample-and-hold interpolation and deconvolution

In [1] a procedure for bias-free estimation of the autocorrelation function is introduced for equidistantly sampled data with randomly occurring samples being invalid. The method incorporates sample-and-hold interpolation of the missing data points. The occurring dynamic error of the primary estimate of the correlation function is treated by a deconvolution procedure with two parameters $c_0$ and $c_1$ with $c_0+2c_1=1$, which are the on-diagonal and the aside-diagonal parameters of a specific correction matrix (at all lag times except zero). The parameters $c_0$ and $c_1$ were obtained as a function of the probability α of a sample to be valid by numerical simulation. However, explicit expressions for the parameters $c_0(\alpha )=1-\frac{2}{\alpha }+\frac{2}{{\alpha }^2}$ and $c_1(\alpha )=\frac{1}{\alpha }-\frac{1}{{\alpha }^2}$ can be derived, which might improve the usability of the deconvolution procedure in [1].