应用数学和力学

An Improved Precise Integration Method for Fractional Ordinary Differential Equations

BAO Siyuan, SHEN Feng

2019, 40(12): 1309-1320. doi: 10.21656/1000-0887.390355

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Abstract:
Based on the definition of the Mittag-Leffler function, the precise iteration computation scheme for the Mittag-Leffler matrix function was constructed. Compared with the normal iteration scheme for exponential functions, the constructed scheme has additional correction items. The expression of the correction item is related to the order of the fractional derivative. For dynamic fractional ordinary differential equation D^(α)v=Hv with the Caputo fractional definition, the solution function value at the endpoint of the time phase can be obtained with the precise iteration method. The numerical examples demonstrated effectiveness and efficiency of the presented method.

A Timoshenko Micro-Beam Model and Its Size Effects Based on the Modified Couple Stress Theory

ZHOU Bo, ZHENG Xueyao, KANG Zetian, XUE Shifeng

2019, 40(12): 1321-1334. doi: 10.21656/1000-0887.400056

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Abstract:
Based on the modified couple stress theory, the basic variables of the Timoshenko microbeam, such as the stress, couple stress, strain and curvature, were described as the functions of partial derivatives of displacement components. According to the principle of minimum total potential energy, the governing differential equation was derived to determine the displacement field of the Timoshenko microbeam. The series method was utilized to solve the governing differential equation for the simply supported Timoshenko microbeam under arbitrary load, and the theoretical couple stress solutions of the deflection, rotation angle and stress, which can reflect the size effects, were obtained. The size effects of the deflection, rotation angle and stress of the Timoshenko microbeam subjected to a cosine distribution load were investigated in detail, and the influence of Poisson’s ratio on the mechanical behaviors of the Timoshenko microbeam and the size effects were analyzed. The results show that, both the stiffness and the strength of the Timoshenko microbeam improve clearly with the decreased cross section height and their size effects are obvious when the ratio of the cross section height to the material characteristic length is less than 5. However, both the stiffness and the strength of the Timoshenko microbeam tend to be stable and their size effects can be neglected when the ratio of the cross section height to the material characteristic length is greater than 10. Poisson’s ratio is an important factor influencing the mechanical behaviors of the Timoshenko microbeam and the size effects. The smaller Poisson’s ratio is, the more significant the size effects of the stiffness and the strength will be. The developed model can effectively describe the mechanical behaviors of Timoshenko microbeams and their size effects, and makes a theoretical basis and technical reference for the design and analysis of microstructures in the micro electro mechanical systems (MEMS).

A Simplified Analysis Method for Seismic Responses of Floating-System Cable-stayed Bridges With Viscous Dampers

SHI Jun, XU Lueqin, LU Xiaoluo

2019, 40(12): 1335-1344. doi: 10.21656/1000-0887.400045

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Abstract:
In view of the deficiencies of the existing viscous damper parameter design methods for floating-system cable-stayed bridges, a more efficient and effective analysis method was proposed. Based on the pendulum principle, a double-mass model was used for simplified simulation of the dynamic response characteristics of floating-system cable-stayed bridges. Meanwhile, an equivalent linear model for viscous dampers was proposed according to the principle of energy equivalence. Finally, a simplified analysis method for seismic responses of floating-system cable-stayed bridges with viscous dampers was established based on the principle of structural dynamics. On this basis, a full-bridge numerical model was established for a cable-stayed bridge with a main span of 392 m. Under the action of sine waves, calculation errors of the full-bridge numerical model, the double-mass analytical model and the double-mass numerical model were compared in detail. The results show that, the double-mass numerical solution has high calculation accuracy, and can replace the full-bridge numerical solution; the double-mass analytical solution agrees well with the double-mass numerical solution, which verifies the theoretical reliability of the simplified double-mass analysis method. Calculation errors of the 3 models meet the engineering accuracy requirements under different ground motion characteristics and system periods, indicating that the proposed simplified analysis method has good applicability and provides a more efficient way for damper parameter optimization.

Dynamical Analysis and Solutions for (3+1)-Dimensional Time Fractional KdV-Zakharov-Kuznetsov Equations

ZHANG Xue, SUN Yuhuai

2019, 40(12): 1345-1355. doi: 10.21656/1000-0887.390352

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Abstract:
By means of the ansatz method and the bifurcation analysis, the singular soliton solution, the bright soliton solution, the topological soliton solution, the periodic explosive solution and the solitary wave solution of the (3+1)-dimensional time fractional KdV-Zakharov- Kuznetsov equations were constructed. In addition, the phase portraits of KdV-Zakharov-Kuznetsov equations were obtained for various cases with the MAPLE software. Finally, the relationships among travelling wave solutions were discussed.

A Class of Fractional Nonlinear Singularly Perturbed Problems With Time Delays

ZHU Hongbao

2019, 40(12): 1356-1363. doi: 10.21656/1000-0887.400195

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Abstract:
A class of fractional nonlinear singularly perturbed problems with time delays were considered. Firstly, the outer solution was constructed by means of the singular perturbation method. Then, a stretched variable was introduced to obtain 2 boundary layer correction items for the solution, and the asymptotic analytic expansion solution to the problem was also acquired. Finally, under suitable conditions, the theory of differential inequalities was applied to prove the uniformly valid asymptotic expansion of the solution to the original problem, and the conclusion with the future research directions was given.

Relations Between Robust Efficient Solutions and Properly Efficient Solutions to Multiobjective Optimization Problems

YANG Ming, LI Linting, GAO Ying

2019, 40(12): 1364-1372. doi: 10.21656/1000-0887.400032

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Abstract:
The relations between the robust efficient solutions and properly efficient solutions to multiobjective optimization problems were studied, and the optimality conditions for the robust efficient solutions were discussed. Firstly, the concept of weakly robust efficient solutions to multiobjective optimization problems was given. Then, the relations between the (weakly) robust efficient solutions and the properly efficient solutions were made clear. Several examples were given to illustrate the main results. Finally, the necessary and sufficient optimality conditions for the robust efficient solutions were established under the subconvexity and pseudoconvexity assumptions.

Dynamic Behaviors of Stochastically Delayed SIRS Epidemic Models With Standard Incidence Rates Under Information Intervention

ZHAO Yingying, HU Hua

2019, 40(12): 1373-1388. doi: 10.21656/1000-0887.400031

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Abstract:
A class of stochastic-time-delay SIRS infectious disease models with standard incidence under information intervention were considered. A stopping time was defined. Then the existence of a unique global positive solution was proved through construction of a suitable Lyapunov function to prove the stopping time is infinite. The asymptotic behaviors of the model solution around the disease-free equilibrium point and the endemic equilibrium point of the deterministic model were studied with suitable Lyapunov functions respectively. The results show that, the solution of the stochastic system involves random vibration around the 2 equilibrium points under certain conditions respectively.

Synchronization of Master-Slave Systems With Multiple Time-Varying Delays Based on the Event-Triggered Mechanism

ZHOU Jun, TONG Dongbing, CHEN Qiaoyu

2019, 40(12): 1389-1398. doi: 10.21656/1000-0887.400152

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Abstract:
The synchronization problem of master-slave systems with multiple time-varying delays based on the event-triggered mechanism was studied. The proposed event-triggered mechanism can effectively reduce data transmission and mitigate the pressure of network bandwidth during the synchronization process of master-slave systems. With the Lyapunov stability theory and the linear matrix inequality method, the sufficient conditions for the synchronization of master-slave systems with multiple time-varying delays were obtained. Finally, the effectiveness of the proposed synchronization criterion was verified through the MATLAB simulation.

Numerical Solution to the Second Kind of Fredholm Integral Equation Based on the Adaptive Wavelet Neural Network

JIANG Wei, HAN Huili, LI Fengjun

2019, 40(12): 1399-1408. doi: 10.21656/1000-0887.400029

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Abstract:
A 3-layer feedforward adaptive wavelet neural network model was constructed. The fitting of the translation factor and the scaling factor were combined in the wavelet analysis. The result of combination was set as the weight and bias of the hidden layer. The wavelet basis function was used as the hidden layer activation function and the parameters could be adaptively adjusted according to the gradient descent algorithm. Numerical solution to the second kind of Fredholm integral equation was solved with the adaptive wavelet neural network, and the feasibility and validity of the method were verified through numerical examples.

Impulse Control of Financial Systems With Probabilistic Delay Feedback

AZI Aying, RAO Ruofeng, ZHAO Feng, HUANG Hongyan, WANG Xue, LIU Hao

2019, 40(12): 1409-1416. doi: 10.21656/1000-0887.400059

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Abstract:
The global asymptotic stability of equilibrium points of impulsive financial systems with probabilistic delays was studied. Firstly, through definition of the random variables on the appropriate time-delay intervals, the mathematical model for the impulsive financial system with probabilistic time delay was given. According to the characteristics of impulsive differential inequalities, a simple and suitable Lyapunov function was constructed. By means of the impulsive differential inequality lemma, the technique of controlling impulse interval & impulse quantity and the probabilistic time delay analysis, the global exponential stability of the equilibrium point in the permissible category of large delays was obtained. Finally, the feasibility of the proposed method and the advantages of probabilistic delay were verified with a numerical example. In particular, the increase of the time delay allowable upper limit to the stability criterion enlarges the practicability of the criterion.

2019 Vol. 40, No. 12