Seakweng Vong’s research while affiliated with University of Macau and other places

We consider the conservation laws, regarding the energy and mass, for the time-fractional nonlinear Schrödinger (TFNLS) equation on both continuous and discrete levels. By introducing a class of fractional Leibniz formulas on the complex-field, a local energy conservation law and a local mass conservation law are established for the TFNLS equation. The two conservative laws are asymptotically compatible with the associated conservation laws of the NLS equation in the fractional order limit $\alpha \rightarrow 1^-$. On the discrete level, by establishing the discrete counterparts of fractional Leibniz formulas, we build up a unified framework for the construction and analysis of conservative variable-step methods, which is suitable for many discrete Caputo formulas, including the variable-step L1, L1-2, L2, L2-1$_\sigma$ and L1$^+$ formulas. These variable-step methods are showed to admit a discrete local energy conservation and a discrete local mass conservation laws without restrictions. The discrete energy and the discrete mass conservation laws are all asymptotically compatible with the corresponding conservation laws of the associated variable-step scheme, respectively, for the NLS problem. Numerical experiments are provided to verify the accuracy and efficiency of the conservative methods together with an adaptive time-stepping strategy in long time simulations. So far as we know, this is the first work on the energy and the mass variation laws as well as their asymptotic compatibility for the TFNLS equation.

In this paper, a new method for solving the vertical linear complementarity problems is established by the nonsmooth Newton iteration. With the row $\mathcal {W}$-property of the set of the system matrices, the nonsingularity of the generalized Jacobian is proved and the nearly one-step convergence is presented. Furthermore, a hybrid nonsmooth Newton method is constructed to obtain global convergence. Finally, numerical examples are given to demonstrate the efficiency of the proposed methods.

This work investigates the exponential stability of neural networks (NNs) systems with time delays. By considering orthogonal polynomials with weighted terms, a new weighted integral inequality is presented. This inequality extend several recently established results. Additionally, based on the reciprocally convex inequality, this study focuses on analyzing the exponential stability of systems with time-varying delays that include an exponential decay factor, a weighted version of the reciprocally convex inequality is first derived. Utilizing these inequalities and the suitable Lyapunov-Krasovskii functionals (LKFs) within the framework of linear matrix inequalities (LMIs), the new criteria for the exponential stability of NNs system is obtained. The effectiveness of the proposed method is demonstrated through multiple numerical examples.

In the present paper, to fill the gap of the effect of singularity arising from multiple fractional derivatives on numerical analysis, the regularity and high order difference scheme for time fractional telegraph equations are taken into consideration. Firstly, the analytic solution is obtained by employing Laplace transform, and its regularity is then deduced. Secondly, by the technic of decomposition, the improved regularity of solution is derived. Furthermore, to overcome the weak singularity and enhance convergence precision, a second-order fitted scheme based on L2-$1_\sigma$ approximation and order reduction method is applied to such problems, which is an improvement for the work [6]. Ultimately, examples are presented to verify the effectiveness of our theoretical results.

In this paper, we conduct a further analysis of the modulus-based matrix splitting iteration method proposed in J-W He, S Vong. (Appl Math Lett 134: 108344, 2022) for vertical linear complementarity problems. Our study extends and enhances the existing theoretical theorems, which are also validated through numerical examples.

We consider a high accuracy numerical scheme for solving the two‐dimensional time‐fractional advection‐diffusion equation including mixed derivatives, where the variable‐step Alikhanov formula and a fourth‐order compact approximation are employed to time and space derivatives, respectively. Under mild assumptions on the time step‐sizes, we obtain the unconditional stability and high‐order convergence (second‐order in time and fourth‐order in space) of the proposed scheme by energy method. The theoretical statements are justified by the numerical experiments.

In this paper, based on the weighted alternating direction implicit method, we investigate a second-order scheme with variable steps for the two-dimensional time-fractional telegraph equation (TFTE). Firstly, we derive a coupled system of the original equation by the symmetric fractional-order reduction (SFOR) method. Then the renowned L2-$1_\sigma$ formula on graded meshes is employed to approximate the Caputo derivative and a weighted ADI scheme for the coupled problem is constructed. In addition, with the aid of the Grönwall inequality, the unconditional stability and convergence of the weighted ADI scheme are analyzed. Finally, the numerical experiments are shown to verify the effectiveness and correctness of theoretical results.