Table of Content

  

20 October 2024
Volume 59 Issue 10

Block triangular splitting and its preconditioning iterative algorithms for a class of complex symmetric linear systems

Yang WANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  1-9.  doi:10.6040/j.issn.1671-9352.0.2023.225

Abstract ( 278 )   HTML ( 1 )   PDF (1013KB) ( 360 )   Save

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Based on block triangular splitting for block 2×2 matrix, the block triangular splitting(BTS) iteration method and the preconditioned block triangular splitting (PBTS) iteration method for a class of complex symmetric linear system are proposed. Theoretical analysis shows that the BTS and PBTS methods converge under certain conditions. The optimal iteration parameters of these two methods are obtained. Numerical experiments demonstrate the effectiveness and superiority of the BTS method and the PBTS iterative methods.

An ultra-weak discontinuous Galerkin method for drift-diffusion model of semiconductor problem

Lulu AI,Yunxian LIU

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  10-21.  doi:10.6040/j.issn.1671-9352.0.2023.089

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An ultra-weak discontinuous Galerkin (UWDG) method is developed for the drift-diffusion model of the semiconductor problem with the error analysis. The UWDG method has the advantage of the classical discontinuous Galerkin (DG) method. Compared to the local discontinuous Galerkin method, this method can solve partial differential equations with higher-order spatial derivatives without introducing auxiliary variables, which is simpler in scheme and more direct in calculation. The main technical difficulty is to select the appropriate projection for error analysis. A numerical simulation is performed to validate the numerical stability of the UWDG method.

 

MNSGA-Ⅱ algorithm based on bi-objective for solving nonlinear equation systems

Zhenai LI,Hui WEI,Xin CHEN

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  22-29.  doi:10.6040/j.issn.1671-9352.0.2023.236

Abstract ( 259 )   HTML ( 1 )   PDF (2217KB) ( 181 )   Save

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MONES transformation technique is applied to transform the problem of solving nonlinear equation systems into a bi-objective optimization problem, and a dynamic crowding distance strategy of MNSGA-Ⅱ algorithm is included to dynamically calculate individual crowding distance in the process of population selection, which improves the diversity of Pareto front. In order to verify the performance of algorithm, thirty nonlinear equation systems are selected for testing NSGA-Ⅱ, dynamic NSGA-Ⅱ and MNSGA-Ⅱ algorithm based on MONES transformation technique. Experimental results show that MNSGA-Ⅱ algorithm based on MONES transformation technique has a better root-found ratio and success rate. Finally, the Pareto front of three algorithms mentioned above is compared, and the uniformity and convergence of Pareto front of the proposed algorithm performs better than others'.

Differential quadrature method for solving the generalized Burgers-Fisher equations

ALI Adil,RAHMAN Kaysar

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  30-39.  doi:10.6040/j.issn.1671-9352.0.2023.112

Abstract ( 273 )   HTML ( 2 )   PDF (2761KB) ( 278 )   Save

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In this paper, a high accuracy numerical scheme is constructed for the generalized Burgers-Fisher equation with Dirichlet boundary or Neumann boundary conditions. Firstly, the Lagrange interpolation polynomial differential quadrature method with uniform grid and Chebyshev-Gauss-Lobatto grid is used in space, and the third-order strong stability-preserving Runge-Kutta scheme is used in time. Secondly, the stability of the scheme is analyzed by using the matrix method. Finally, two numerical examples with different boundary conditions are calculated, and the results are compared with other numerical methods to verify the effectiveness of the proposed scheme.

Structured backward error for a class of generalized saddle point problems

Yuling LIU

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  40-45.  doi:10.6040/j.issn.1671-9352.0.2023.046

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In this paper, an explicit expression of the structured backward error of a generalized saddle point system of a structure is derived, and the structured and unstructured backward errors are compared, and it is found that they can be arbitrarily far apart in some cases.

Sufficient maximum principle for one kind of nonzero-sum stochastic differential game involving noisy memory

Feng ZHANG,Jiawei LIANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  46-52.  doi:10.6040/j.issn.1671-9352.0.2023.281

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One nonzero-sum stochastic differential game is considered, whose main feature is that several kinds of delays of the state and the control are involved. The state process can contain distributed delays, discrete delays, and noisy memory, and control processes can contain distributed delays and discrete delays. The control domains are convex sets. Sufficient conditions for the equilibrium point of the game are established by means of the stochastic maximum principle. Finally, an illustrative example is considered for which the equilibrium point is obtained in explicit form.

Optimal local and remote control for multiplicative noise stochastic systems with packet loss and delay

Yujiao SONG,Qingyuan QI

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  53-63.  doi:10.6040/j.issn.1671-9352.0.2023.235

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The optimal local and remote control problem for a multiplicative noise stochastic system consisting of a remote controller and a local controller is investigated. It is assumed that the state information is prone to packet loss and delay when passing through the uplink channel, while the downlink channel is perfect. Firstly, the necessary and sufficient solvability conditions are developed by using the convex variational skill, which shows that the solvability of the optimal local and remote control problem is equivalent to the solvability of the forward and backward stochastic difference equations. Moreover, a novel hierarchical control algorithm is proposed, and the optimal feedback control strategy is obtained by using the complete square method. Finally, a numerical example is shown to illustrate the effectiveness of the obtained results.

On ramp control of ARZ traffic flow model based on time-delay feedback

Yiyan WANG,Dongxia ZHAO,Caixia GAO

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  64-73, 88.  doi:10.6040/j.issn.1671-9352.0.2023.328

Abstract ( 248 )   HTML ( 1 )   PDF (1102KB) ( 137 )   Save

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For the Aw-Rascle-Zhang (ARZ) non-equilibrium traffic flow model, if the traffic flow at the entrance is constant and the density of the traffic flow at the exit is constant, the system is in critical stability and there will be continuous oscillations near the equilibrium state. This article proposes the design of a time-delay feedback control strategy at the entrance ramp, and characterizes the time-delay term with the solution of the initial value problem of the first-order transportation equation, establishing the form of an infinite dimensional coupled closed-loop system for PDE-PDE. The operator semigroup theory is used to prove the well posedness of the system. The conclusion of exponential stability of the system is obtained by constructing a weighted strict Lyapunov function. The results indicate that when the feedback gain and delay values satisfy certain inequality constraints, the system energy reaches exponential decay. Finally, through numerical simulation, the effectiveness of the designed time-delay controller and the feasibility of parameter conditions are verified.

Stationary distribution and probability density function of a stochastic predation system

Yufeng ZHAO,Guirong LIU

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  74-88.  doi:10.6040/j.issn.1671-9352.0.2023.547

Abstract ( 263 )   HTML ( 1 )   PDF (3927KB) ( 265 )   Save

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A class of stochastic predator-prey models with predator-stage structure and rate-dependent Holling III type functional responses are developed. Firstly, the existence and uniqueness of global positive solutions for stochastic model are obtained. Secondly, the existence and uniqueness of the ergodic stationary distribution are studied by constructing a suitable Lyapunov function and using the ergodic theory of Has′Minskii. Next, by solving the corresponding three-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic predator-prey model near the positive equilibrium point is derived. Finally, the rationality of the theoretical results is verified by numerical simulation.

Shadow wave solutions for ternary simplified chromatography equations

Fan YANG,Lihui GUO

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  89-100.  doi:10.6040/j.issn.1671-9352.0.2023.165

Abstract ( 207 )   HTML ( 1 )   PDF (3361KB) ( 165 )   Save

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This paper mainly considers the overall structure of Riemann solution and the existence and convergence of shadow wave solutions of the ternary simplified chromatography equations. According to Riemann initial data, and Riemann problem is divided into six different cases, Riemann solutions for the ternary simplified chromatography equations are obtained. When -1 < p_-≤0≤p₊, the existence and convergence of shadow wave solutions in the sense of Schwartz distributions are proved. Finally, numerical simulation is given.

Spectra of product operation signed graphs

Ziqiang FANG,Longjie LI,Haizhen REN

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  101-106.  doi:10.6040/j.issn.1671-9352.0.2023.048

Abstract ( 216 )   HTML ( 1 )   PDF (684KB) ( 243 )   Save

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A graph whose edges are labeled either as positive or negative is called a signed graph. The product operations, i.e. symmetric product, direct product, semi-strong product and strong product, of signed graphs are given, respectively. The adjacency matrices of these product operation signed graphs in tensor form are obtained, and some relations on the eigenvalues of signed graphs on product operations (direct product, semi-strong product, strong product) are also formulated.

Vertex reducible edge coloring of the Lexicographic product of graphs

Fei LEI,Fei WEN,Zepeng LI,Muchun LI

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  107-114.  doi:10.6040/j.issn.1671-9352.0.2023.032

Abstract ( 215 )   HTML ( 2 )   PDF (1511KB) ( 201 )   Save

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Let $f: E(G) \rightarrow\{1, 2, \cdots, k\}$ be a non-proper k-edge coloring of G, and 1≤k≤Δ. If for any two adjacent vertices u, v ∈ V(G) with d(u)=d(v) satisfy C(u)=C(v), f is called a k-vertex-reducible edge coloring, where C(u) denotes the set of colors of edges incident with u. The maximum positive integer k is called vertex-reducible edge chromatic number of G. According to the characters of the lexicographic product graphs, we apply combinatorial analysis to give a lower bound of the vertex reducible edge chromatic number of the lexicographic product G[H] for simple graphs G and H. As applications, the vertex-reducible edge chromatic numbers of $K_{n}\left[\overline{K_{2 m}}\right], K_{n}[H]$ and P_n[H] are obtained.

r-wide semigroups with right regular medial idempotents

Yang LIU,Chunmei GONG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  115-121.  doi:10.6040/j.issn.1671-9352.0.2023.158

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The concepts of right regular medial idempotents over the r-wide semigroups is defined. The property characteristics of such idempotents are studied. A construction of r-wide semigroups with a right regular medial idempotents is established.

On the rank of semigroup $\mathscr{C} \mathscr{F}_{(n, r)}$

Heyu GONG,Qin SHU,Ping ZHAO

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(10):  122-126.  doi:10.6040/j.issn.1671-9352.0.2023.111

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Let $\mathscr{T}_{n}$ be the full transformation semigroup on $X_{n}=\{1, 2, \cdots, n\}$. Let $\leqslant r \leqslant n$, put$\mathscr{F}_{(n, r)}=\left\{\alpha \in \mathscr{T}_{n}: i \alpha=i, \quad \forall i \in\{1, 2, \cdots, r\}\right\},$it is obvious that the semigroup $\mathscr{F}_{(n, r)}$ is subsemigroup of $\mathscr{T}_{n}$. In the paper, we study the core $\left(\mathscr{C} \mathscr{F}_{(n, r)}\right)=\left\langle E\left(\mathscr{F}_{(n, r)}\right)\right\rangle$ of the semigroup $\mathscr{F}_{(n, r)}$, where $\mathscr{C}\left(\mathscr{F}_{(n, r)}\right)=\left\{\alpha \in \mathscr{F}_{(n, r)}: \alpha^{2}=\alpha\right\}$, by analyzing idempotents of the semigroup $\mathscr{F}_{(n, r)}$, we prove that the rank and idempotent rank of semigroup $\mathscr{C} \mathscr{F}_{(n, r)}$ are both equal to $\frac{(n-r)(n-r-1)}{2}+r(n-r)+1$.