Chinese Quarterly Journal of Mathematics

Applications of Matrix Equations in Linear Time-Invariant Systems

ZHOU Yan-ping, CHEN Yan-ping, ZHANG Juan

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.

A Posteriori Error Estimate of Multiphysics Discontinuous Galerkin Method for a Poroelasticity#br#

GE Zhi-hao, HE Wen-long, MA Meng-xia

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.

A High-Order Scalar Auxiliary Variable Approach for Nonlinear Parabolic Integro-Differential Equations

YAN Li-na, ZHANG Gen-gen, HUANG Qiong-ao

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.

Optimal Error Estimates of Multiphysics Finite Element Method for a Nonlinear Poroelasticity Model with Nonlinear Stress-Strain Relation

GE Zhi-hao, LI Hai-run, LI Ting-ting

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.

Numerical Methods for Boundary Value Problems in Variable Coefficient Ordinary Differential Equations

ZHAO Ting-ting, CAI Wei-yun

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.

Blow-Up Phenomena for a Non-Homogeneously Strongly Damped Wave Equation with Riemann-Liouville Fractional Integral

XIANG Chang-yong, DUAN Ji-song, LONG Qun-fei

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.

The Gauss Circle Problem Related to the Fourier Coefficients of Cusp Forms

CHEN Feng

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.,...

On an Open Problem in Bottleneck Algebra

TAN Yi-jia

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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