A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a map...A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron, and can be taken as an extension of the 2-D Steiner domain. The polynomials can be viewed as Jacobi polynomials on such a domain. Three-term relations are derived explicitly. The number of the individual terms, involved in the recurrences relations, are shown to be independent on the total degree of the polynomials. The numbers now are determined to be five and seven, with respect to two conjugate variables z, $ \bar z $ and a real variable r, respectively. Three examples are discussed in details, which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds, and Legendre polynomials.展开更多
It is difficult to conduct shaking table tests that require large-displacement high-frequency seismic excitation due to the limited capacity of existing electrohydraulic servo systems.To address this problem,a double-...It is difficult to conduct shaking table tests that require large-displacement high-frequency seismic excitation due to the limited capacity of existing electrohydraulic servo systems.To address this problem,a double-layer shaking table(DLST)is proposed.The DLST has two layers of one table each(i.e.,an upper table and lower table)and aims at reproducing target seismic excitation on the upper table.The original signal is separated into two signals(i.e.,a high-frequency signal and low-frequency signal)through a fast Fourier transform/inverse fast Fourier transform process,and these signals are applied to the two tables separately.The actuators connected to different tables only need to generate large-displacement low-frequency or small-displacement high-frequency movements.The three-variable control method is used to generate large-displacement but low-frequency motion of the lower table and high-frequency but small-displacement motion of the upper table relative to the table beneath.A series of simulations are carried out using MATLAB/Simulink.The simulation results suggest that the DLST can successfully generate large-displacement high-frequency excitation.The control strategy in which the lower table tracks the low-frequency signal and the upper table tracks the original signal is recommended.展开更多
In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs wit...In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.展开更多
基金the Major Basic Project of China(Grant No.2005CB321702)the National Natural Science Foundation of China(Grant Nos.10431050,60573023)
文摘A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron, and can be taken as an extension of the 2-D Steiner domain. The polynomials can be viewed as Jacobi polynomials on such a domain. Three-term relations are derived explicitly. The number of the individual terms, involved in the recurrences relations, are shown to be independent on the total degree of the polynomials. The numbers now are determined to be five and seven, with respect to two conjugate variables z, $ \bar z $ and a real variable r, respectively. Three examples are discussed in details, which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds, and Legendre polynomials.
基金Scientific Research Fund of the Institute of Engineering Mechanics,China Earthquake Administration under Grant No.2019EEEVL0502Natural Science Foundation of China under Grant No.52078275+1 种基金the Institute for Guo Qiang,Tsinghua University under Grant No.2019GQC0001Beijing Natural Science Foundation under Grant No.JQ18029。
文摘It is difficult to conduct shaking table tests that require large-displacement high-frequency seismic excitation due to the limited capacity of existing electrohydraulic servo systems.To address this problem,a double-layer shaking table(DLST)is proposed.The DLST has two layers of one table each(i.e.,an upper table and lower table)and aims at reproducing target seismic excitation on the upper table.The original signal is separated into two signals(i.e.,a high-frequency signal and low-frequency signal)through a fast Fourier transform/inverse fast Fourier transform process,and these signals are applied to the two tables separately.The actuators connected to different tables only need to generate large-displacement low-frequency or small-displacement high-frequency movements.The three-variable control method is used to generate large-displacement but low-frequency motion of the lower table and high-frequency but small-displacement motion of the upper table relative to the table beneath.A series of simulations are carried out using MATLAB/Simulink.The simulation results suggest that the DLST can successfully generate large-displacement high-frequency excitation.The control strategy in which the lower table tracks the low-frequency signal and the upper table tracks the original signal is recommended.
基金This work was supported by the Collaborative Innovation Center of Taiyuan Heavy Machinery Equipment,Postdoctoral Startup Fund of Taiyuan University of Science and Technology(20152034)the Natural Science Foundation of Shanxi Province(201701D221135)National College Students Innovation and Entrepreneurship Project(201710109003)and(201610109007).
文摘In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.
