Consider the discretisation of the initial-value problem $D^αu(t) = f (t)$ for $0 < t \leq T$ with $u(0) = u_0$ , where $D^αu(t)$ is a Caputo derivative of order $α \in (0, 1)$, using the ${\rm L1}$ scheme on a graded mesh with $N$ points. It is well known that one can prove the maximum nodal error in the computed solution is at most $\mathscr{O} (N^{− min\{rα,2−α\}})$, where $r\geq 1$ is the mesh grading parameter ($r = 1$ generates a uniform mesh). Numerical experiments indicate that this error bound is sharp, but no proof of its sharpness has been given. In the present paper the sharpness of this bound is proved, and the sharpness of the analogous nodal error bounds for the $\overline{{\rm L1}}$ and Alikhanov schemes will also be proved, using modifications of the ${\rm L1}$ analysis.

Water waves are actively studied. A new method to generate new wave systems through making perturbation in matrix spectral problems for integrable couplings is presented, which is called the “completion process of integrable couplings”. As its application, we construct an integrable coupling hierarchy and show that each equation in the resulting hierarchy has a bi-Hamiltonian structure by taking use of the component-trace identity. Moreover, the self-consistent sources of integrable coupling is presented based on the theory of self-consistent sources.

The Sobolev polynomials, which are orthogonal with respect to an inner product involving derivatives, are considered. The theory about these nonstandard polynomials has been developed along the last 40 years. The local asymptotics of these polynomials can be described by the Mehler-Heine formulae, which connect the polynomials with the Bessel functions of the first kind. In recent years, the formulae have been computed for discrete Sobolev orthogonal polynomials in several particular cases. We improve various known results by unifying them. Besides, an algorithm to compute these formulae effectively is presented. The algorithm allows to construct a computer program based on Mathematica$^®$ language, where the corresponding Mehler-Heine formulae are automatically obtained. Applications and examples show the efficiency of the approach developed.

This paper considers the global existence and boundedness of classical solutions to a predator-prey-mutualist model with prey-taxis. In addition, by constructing the Lyapunov functionals, we proved when $\alpha < (ac/b)·r +a/b,$ the positive equilibrium point is globally asymptotic stable; and when $\alpha \in ((ac/b)·r + a/b, M_1),$ the semi-trivial equilibrium point is globally asymptotic stable. Finally, we give some numerical examples to validate our results.

A simple stabilizer free weak Galerkin (SFWG) finite element method for a one-dimensional second order elliptic problem is introduced. In this method, the weak function is formed by a discontinuous $k$-th order polynomial with additional unknowns defined on vertex points, whereas its weak derivative is approximated by a polynomial of degree $k+1.$ The superconvergence of order two for the SFWG finite element solution is established. It is shown that the elementwise lifted $P_{k+2}$ solution of the $P_k$ SFWG one converges at the optimal order. Numerical results confirm the theory.

The behaviour of platelet flows in plasma and the change of the platelet surfaces is studied by numerical simulations. In particular, the influence of the 2D platelets surface roughness and Reynolds number on the rotation of platelets is considered. Reynolds numbers $Re$ vary from 0.4 to 50 and it is noted that if $Re = 50,$ the 2D platelets stop rotating after 90$^{\circ}$ turn. If $0.4 ≤ Re ≤ 20,$ the rotation period increases along with the increase of Reynolds number. The maximum angular velocity, the minimum angular velocity and the rotation period of platelets depends on their roughness. However, these characteristics are little affected by the roughness if the Reynolds number is close to 1. It is also worth noting that the torque of 3D platelets is greater than the torque of 2D platelets.

Let $1 ≤ p < 2$, $\alpha > p$, $\{a_{ni}, 1 ≤ i ≤ n, n ≥ 1\}$ be a set of real numbers with the property ${\rm sup}_{n≥1} n^{−1} \sum _{i=1}^n |a_{ni}|^α < ∞$ and let $\{X, X_n , n ≥ 1\}$ be a sequence of $H$-valued $ρ^∗$-mixing random vectors coordinatewise stochastically upper dominated by a random vector $X$. We provide conditions such that for any $\epsilon > 0$ the following inequalities hold: $$\sum\limits_{n=1}^∞ n^{-1}p \left(\max\limits_{1 \leq k \leq n}\left\|\sum\limits_{i=1}^k a_{ni}X_{i}\right\|>\epsilon n^{\frac{1}{p}}\right)<∞,$$  $$\sum\limits_{n=1}^∞ n^{-1-\frac{1}{p}}E \left(\max\limits_{1 \leq k \leq n}\left\|\sum\limits_{i=1}^k a_{ni}X_{i}\right\|-\epsilon n^{\frac{1}{p}}\right)^+<∞.$$These results generalize the results of Chen and Sung (cf. J. Ineq. Appl. 121, 1–16 (2018)) to the $ρ^∗$-mixing random vectors in $H$. In addition, a Marcinkiewicz-Zygmund type strong law of $ρ^∗$-mixing random vectors in $H$ is presented.

Steady Navier-Stokes equations are solved by three different space iteration methods based on the lowest order nonconforming finite element pairs $\mathscr{P}_1\mathscr{N} \mathscr{C}-\mathscr{P}_1,$ including simple, Oseen, and Newton iterative methods. The stability and convergence of these methods are studied, and their CPU time and numerical convergence rate are discussed. Numerical results are in good agreement with theoretical findings. In particular, numerical experiments show that for large viscosity, the Newton method converges faster than to others, whereas the Oseen method is more suitable for the equations with small viscosity.

A multiple interval Chebyshev-Gauss-Lobatto collocation method for solving multi-order fractional differential equations is proposed. The $hp$-version error estimates of the Chebyshev spectral collocation method are obtained in $L^2$- and $L^∞$-norms. Numerical experiments illustrate the theoretical results.

We prove the well-posedness of a nonlinear hidden-memory variable-order fractional stochastic differential equation driven by a multiplicative white noise, in which the hidden-memory type variable order describes the memory of a fractional order. We then present a Euler-Maruyama scheme for the proposed model and prove its strong convergence rate. Numerical experiments are performed to substantiate the theoretical results.

A semi-tensor product of matrices is proposed as a generalization of usual matrix product in the case where the dimensions of two factor matrices do not match. The properties of the semi-tensor product of tensors and swap tensors based on the Einstein product are studied. Applications of this new tensor product in image restoration and in finite dimensional algebras are discussed.