数学季刊

周艳平, 陈艳萍, 张娟

2025, 40(3): 221-237. doi:10.13371/j.cnki.chin.q.j.m.2025.03.001

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With the development of science and technology, the design and optimization of control systems are widely applied. This paper focuses on the application of matrix equations in linear time-invariant systems. Taking the inverted pendulum model as an example, the algebraic Riccati equation is used to solve the optimal control problem, and the system performance and stability are achieved by selecting the closed-loop pole and designing the gain matrix. Then, the numerical methods for solving the stochastic algebraic Riccati equations are applied to practical problems, with Newton’s iteration method as the outer iteration and the solution of the mixed-type Lyapunov equations as the inner iteration. Two methods for solving the Lyapunov equations are introduced, providing references for related research.

多孔弹性问题多物理场间断有限元方法后验误差估计

葛志昊, 贺文龙, 马梦霞

2025, 40(3): 238-261. doi:10.13371/j.cnki.chin.q.j.m.2025.03.002

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In this paper, we design a new error estimator and give a posteriori error analysis for a poroelasticity model. To better overcome “locking phenomenon” on pressure and displacement, we proposed a new error estimators based on multiphysics discontinuous Galerkin method for the poroelasticity model. And we prove the upper and lower bound of
the proposed error estimators, which are numerically demonstrated to be computationally very efficient. Finally, we present numerical examples to verify and validate the efficiency of the proposed error estimators, which show that the adaptive scheme can overcome “locking phenomenon” and greatly reduce the computation cost.

非线性抛物型积分-微分方程的高阶标量辅助变量方法

闫丽娜, 张根根, 黄琼敖

2025, 40(3): 262-270. doi:10.13371/j.cnki.chin.q.j.m.2025.03.003

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An efficient and accurate scalar auxiliary variable (SAV) scheme for numerically solving nonlinear parabolic integro-differential equation (PIDE) is developed in this paper. The original equation is first transformed into an equivalent system, and the k-order backward differentiation formula (BDFk) and central difference formula are used to discretize the temporal and spatial derivatives, respectively. Different from the traditional discrete method that adopts full implicit or full explicit for the nonlinear integral terms, the proposed scheme is based on the SAV idea and can be treated semi-implicitly, taking into account both accuracy and effectiveness. Numerical results are presented to
demonstrate the high-order convergence (up to fourth-order) of the developed schemes and it is computationally efficient in long-time computations.

具有非线性应力-应变关系的非线性多孔弹性模型多物理场有限元法的最优误差估计

葛志昊, 李海润, 李婷婷

2025, 40(3): 271-294. doi:10.13371/j.cnki.chin.q.j.m.2025.03.004

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In this paper, we propose a multiphysics finite element method for a nonlinear poroelasticity model with nonlinear stress-strain relation. Firstly, we reformulate the original problem into a new coupled fluid system-a generalized nonlinear Stokes problem of displacement vector field related to pseudo pressure and a diffusion problem of other pseudo pressure fields. Secondly, a fully discrete multiphysics finite element method is performed to solve the reformulated system numerically. Thirdly, existence and uniqueness of the weak solution of the reformulated model and stability analysis and optimal convergence order for the multiphysics finite element method are proven theoretically. Lastly, numerical tests are given to verify the theoretical results.

变系数常微分方程边值问题的数值方法

赵婷婷, 蔡伟云

2025, 40(3): 295-303. doi:10.13371/j.cnki.chin.q.j.m.2025.03.005

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In order to solve the problem of the variable coefficient ordinary differential equation on the bounded domain, the Lagrange interpolation method is used to approximate the exact solution of the equation, and the error between the numerical solution and the exact solution is obtained, and then compared with the error formed by the difference method, it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.

具有Riemann-Liouville分数阶积分的强阻尼波动方程的爆破现象

向常永, 段霁松, 龙群飞

2025, 40(3): 304-312. doi:10.13371/j.cnki.chin.q.j.m.2025.03.006

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We investigate the blow-up effect of solutions for a non-homogeneous wave equation
u_tt−∆u−∆u_t=I₀^α+ (|u|^p)+ω(x),
where p >1, 0≤α<1 and ω(x) with \int_{\mathbb{R}^{N}} ω(x)dx >0. By a way of combining the argument by contradiction with the test function techniques, we prove that not only any non-trivial solution blows up in finite time under 0< α <1, N ≥1 and p >1, but also any non-trivial solution blows up in finite time under α= 0, 2≤ N ≤4 and p being the Strauss exponent.

带有尖形式Fourier系数的Gauss圆内格点问题

陈锋

2025, 40(3): 313-323. doi:10.13371/j.cnki.chin.q.j.m.2025.03.007

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Let f be a Hecke eigenform of even integral weight k for the full modular group SL₂(Z). Denote by λ_f (n) the nth normalized coefficient of f. The sum of Fourier coefficients of cusp form over the quadratic polynomial m² +n² is considered, i.e.,...

关于Bottleneck代数的一个公开问题

谭宜家

2025, 40(3): 324-330. doi:10.13371/j.cnki.chin.q.j.m.2025.03.008

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A bottleneck algebra is a linearly ordered set (B,≤) with two operations a⊕b=max{a,b} and a⊗b=min{a,b}. A finite nonempty set of vectors of order m over a bottleneck algebra B is said to be 2B-independent if each vector of order m over B can be expressed as a linear combination of vectors in this set in at most one way. In 1996, Cechl´arov´a and Pl´avka posed an open problem: Find a necessary and sufficient condition for a finite nonempty set of vectors of order m over B to be 2B-independent. In this paper, we derive some necessary and sufficient conditions for a finite nonempty set of vectors of order m over a bounded bottleneck algebra to be 2B-independent and answer this open problem.

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