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SYMPLECTIC SCHEMES FOR TELEGRAPH EQUATIONS

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Journal of Computational Mathematics 2016年第3期34卷 285-299页

作者： Yi Lu yaolin jiang School of Mathematics and Statistics Xi'an Jiaotong University Shaanxi 710049 China

A new numerical algorithm for telegraph equations with homogeneous boundary con- ditions is proposed. Due to the damping terms in telegraph equations, there is no royal conservation law according to Noether＇s theorem. The algorithm origins from the discovery of a transform applied to a telegraph equation, which transforms the telegraph equation to a Klein-Gordon equation. The Symplectic method is then brought in this algorithm to solve the Klein-Gordon equation, which is based on the fact that the Klein-Gordon equation with the homogeneous boundary condition is a perfect Hamiltonian system and the symplectic method works very well for Hamiltonian systems. The transformation itself and the inverse transformation theoretically bring no error to the numerical computation. Therefore the error only comes from the symplectic scheme chosen. The telegraph equation is finally explicitly computed when an explicit symplectic scheme is utilized. A relatively long time result can be expected due to the application of the symplectic method. Mean- while, we present order analysis for both one-dimensional and multi-dimensional cases in the paper. The efficiency of this approach is demonstrated with numerical examples.

关键词： Telegraph equation Klein-Gordon equation Symplectic method Explicitmethod.

Structured multi-way arrays and their applications

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Frontiers of Mathematics in China 2013年第2期8卷 345-369页

作者： Xu KONG yaolin jiang School of Mathematics and Statistics Xi'an Jiaotong University Xi'an 710049 China

Based on the structure of the rank-1 matrix and the different unfolding ways of the tensor, we present two types of structured tensors which contain the rank-1 tensors as special cases. We study some properties of the ranks and the best rank-r approximations of the structured tensors. By using the upper-semicontinuity of the matrix rank, we show that for the structured tensors, there always exist the best rank-r approximations. This can help one to better understand the sequential unfolding singular value decomposition （SVD） method for tensors proposed by J. Salmi et al. [IEEE Trans Signal Process, 2009, 57（12）： 4719-4733] and offer a generalized way of low rank approximations of tensors. Moreover, we apply the structured tensors to estimate the upper and lower bounds of the best rank-1 approximations of the 3rd-order and 4th-order tensors, and to distinguish the well written and non-well written digits.

关键词： Tensor rank singular value decomposition (SVD) higher-order singular value decomposition approximation

QUASI-NEWTON WAVEFORM RELAXATION BASED ON ENERGY METHOD

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Journal of Computational Mathematics 2018年第4期36卷 542-562页

作者： yaolin jiang Zhen Miao School of Mathematics and Statistics Xi'an Jiaotong University Xi'an 710049 China

A quasi-Newton waveform relaxation （WR） algorithm for semi-linear reaction-diffusion equations is presented at first in this paper. Using the idea of energy estimate, a general proof method for convergence of the continuous case and the discrete case of quasi-Newton WR is given, which appears to be the superlinear rate. The semi-linear wave equation and semi-linear coupled equations can similarly be solved by quasi-Newton WR algorithm and be proved as convergent with the energy inequalities. Finally several parallel numerical experiments are implemented to confirm the effectiveness of the above theories.

关键词： Waveform relaxation Quasi-Newton Energy method Superlinear Parallelism

AN ACCELERATED WAVEFORM RELAXATION APPROACH BASED ON MODEL ORDER REDUCTION FOR LARGE COUPLING SYSTEMS

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Journal of Computational Mathematics 2013年第2期31卷 190-208页

作者： Haibao Chen yaolin jiang College of Mathematics and Systems Science Xinjiang University Urumqi 830046 China School of Mathematics and Statistics Xi'an Jiaotong University Xi'an 710049 China

In this paper, we present an accelerated simulation approach on waveform relaxation using Krylov subspace for a large time-dependent system composed of some subsystems. This approach first allows these subsystems to be decoupled by waveform relaxation. Then the Arnoldi procedure based on Krylov subspace is provided to accelerate the simulation of the decoupled subsystems independently. For the new approach, the convergent conditions on waveform relaxation are derived. The robust behavior is also successfully illustrated via numerical examples.

关键词： Large coupling systems Waveform relaxation Model order reduction Krylovsubspace Convergence analysis Accelerating technique.

WAVEFORM RELAXATION METHODS FOR LIE-GROUP EQUATIONS

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Journal of Computational Mathematics 2022年第4期40卷 649-666页

作者： yaolin jiang Zhen Miao Yi Lu School of Mathematics and Statistics Xi’an Jiaotong UniversityXi’an 710049China

In this paper,we derive and analyse waveform relaxation(WR)methods for solving differential equations evolving on a *** present both continuous-time and discrete-time WR methods and study their convergence *** the discrete-time case,the novel methods are constructed by combining WR methods with Runge-KuttaMunthe-Kaas(RK-MK)*** obtained methods have both advantages of WR methods and RK-MK methods,which simplify the computation by decoupling strategy and preserve the numerical solution of Lie-group equations on a *** numerical experiments are given to illustrate the feasibility of the new WR methods.

关键词： Lie-group equations Waveform relaxation RK-MK methods Convergence analysis

NONNEGATIVITY OF SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL-ALGEBRAIC EQUATIONS

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Acta Mathematica Scientia 2018年第3期38卷 756-768页

作者： Xiaoli DING yaolin jiang Department of Mathematics Xi＇an Polytechnic Universit Department of Mathematics Xi＇an Jiaotong Universit

Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.

关键词： Fractional differential-algebraic equations nonnegativity of solutions waveform relaxation monotone convergence

Modeling and analysis of a predator-prey system with time delay

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International Journal of Biomathematics 2017年第3期10卷 19-40页

作者： Wei Liu yaolin jiang ＊School of Mathematics and Statistics Xi'an Jiaotong University Xi'an 710049 P. R. China School of Mathematics and Computer ScienceXinyu University Xinyu 338004 P. R. China

In this paper, a differential-algebraic predator prey system with time delay is investigated, where the time delay is regarded as a parameter. By analyzing the corresponding characteristic equations, the local stability of the positive equilibrium and the existence of Hopf bifurcation are demonstrated, Furthermore, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions are obtained by applying the normal form theory and the center manifold argument. At last, some numerical simulations are carried out to illustrate the feasibility of our main results.

关键词： Predator-prey time delay bifurcation periodic solutions stability normal form.

ON STRUCTURED VARIANTS OF MODIFIED HSS ITERATION METHODS FOR COMPLEX TOEPLITZ LINEAR SYSTEMS

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Journal of Computational Mathematics 2013年第1期31卷 57-67页

作者： Fang Chen yaolin jiang Qingquan Liu Department of Mathematical Sciences Xi'an Jiaotong University Xi'an 710049 China Key Laboratory of Environmental Mechanics Institute of Mechanics Chinese Academy of Sciences Beijing 100190 China

The Modified Hermitian and skew-Hermitian splitting （MHSS） iteration method was presented and studied by Bai, Benzi and Chen （Computing, 87（2010）, 93-111） for solving a class of complex symmetric linear systems. In this paper, using the properties of Toeplitz matrix, we propose a class of structured MHSS iteration methods for solving the complex Toeplitz linear system. Theoretical analysis shows that the structured MHSS iteration method is unconditionally convergent to the exact solution. When the MHSS iteration method is used directly to complex symmetric Toeplitz linear systems, the computational costs can be considerately reduced by use of Toeplitz structure. Finally, numerical ex- periments show that the structured MHSS iteration method and the structured MHSS preconditioner are efficient for solving the complex Toeplitz linear system.

关键词： Toeplitz matrix MHSS iteration method Complex symmetric linear system.

Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator prey model with harvesting

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International Journal of Biomathematics 2020年第1期13卷 1-37页

作者： Wei Liu yaolin jiang School of Mathematics and Statistics’an Jiaotong University Xi’an 710049P.R.China School of Mathematics and Computer Science Xiriyu University Xinyu 338004P.R.China

In this paper,a difference-algebraic predator prey model is proposed,and its complex dynamical behaviors are *** model is a discrete singular system,which is obtained by using Euler scheme to discretize a differential-algebraic predator-prey model with harvesting that we ***,the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system the***,by applying the new normal form of difference-algebraic equations,center manifold theory and bifurcation theory,the Flip bifurcation and Neimark-Sacker bifurcation around the interior equilibrium point are studied,where the step size is treated as the variable bifurcation ***,with the help of Matlab software,some numerical simulations are performed not only to validate our theoretical results,but also to show the abundant dynamical behaviors,such as period-doubling bifurcations,period 2,4,8,and 16 orbits,invariant closed curve,and chaotic *** particular,the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.

关键词： Predator prey harvesting Flip bifurcation Neimark Sacker bifurcation chaos.