This paper deals with off-diagonal operator matrices and their applications in elasticity theory. Two kinds of completeness of the system of eigenvectors are proven, in terms of those of the compositions of two block ...This paper deals with off-diagonal operator matrices and their applications in elasticity theory. Two kinds of completeness of the system of eigenvectors are proven, in terms of those of the compositions of two block operators in the off-diagonal operator matrices. Using these results, the double eigenfunction expansion method for solving upper triangular matrix differential systems is proposed. Moreover, we apply the method to the two-dimensional elasticity problem and the problem of bending of rectangular thin plates on elastic foundation.展开更多
The completeness theorem of the eigenfunction systems for the product of two 2 × 2 symmetric operator matrices is proved. The result is applied to 4 × 4 infinite-dimensional Hamiltonian operators. A modified...The completeness theorem of the eigenfunction systems for the product of two 2 × 2 symmetric operator matrices is proved. The result is applied to 4 × 4 infinite-dimensional Hamiltonian operators. A modified method of separation of variables is proposed for a separable Hamiltonian system. As an application of the theorem, the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained. Finally, a numerical test is analysed to show the correctness of the results.展开更多
Symplectic self-adjointness of Hamiltonian operator matrices is studied, which is important to symplectic elasticity and optimal control. For the cases of diagonal domain and off-diagonal domain, necessary and suffici...Symplectic self-adjointness of Hamiltonian operator matrices is studied, which is important to symplectic elasticity and optimal control. For the cases of diagonal domain and off-diagonal domain, necessary and sufficient conditions are shown. The proofs use Frobenius-Schur factorizations of unbounded operator matrices.Under additional assumptions, sufficient conditions based on perturbation method are obtained. The theory is applied to a problem in symplectic elasticity.展开更多
The eigenvalue problem for the Hamiltonian operator associated with the mathematical model for the deflection of a thin elastic plate is investigated.First,the problem for a rectangular plate with simply supported edg...The eigenvalue problem for the Hamiltonian operator associated with the mathematical model for the deflection of a thin elastic plate is investigated.First,the problem for a rectangular plate with simply supported edges is solved directly.Then,the completeness of the eigenfunctions is proved,thereby demonstrating the feasibility of using separation of variables to solve the problem. Finally,the general solution is obtained by using the proved expansion theorem.展开更多
A separable Hamiltonian system of Mindlin plate bending problems is obtained. Using the equivalence between the differen form and integral form of the separable Hamiltonian system, the biorthogonal relationships of th...A separable Hamiltonian system of Mindlin plate bending problems is obtained. Using the equivalence between the differen form and integral form of the separable Hamiltonian system, the biorthogonal relationships of the eigenfunctions are presen! Based on the biorthogonal relationships, a novel complete biorthogonal expansion of the Mindlin plate bending problems x~ two opposite sides simply supported is proposed through the products of operator matrices. The exact solutions to deflections bending moments for the Mindlin plate with fully simply supported sides are obtained. A numerical example is illustrated to ve~ the accuracy and validity of the expansion method.展开更多
In this paper, we study a bounded-below singular Hamiltonian system. Sufficient and necessary conditions are obtained for the existence and the number of eigenvalues on the left-axis. The main results of this paper ar...In this paper, we study a bounded-below singular Hamiltonian system. Sufficient and necessary conditions are obtained for the existence and the number of eigenvalues on the left-axis. The main results of this paper are the extension and improvement of Weyl spectral theorem for singular sccond order differential equations.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10962004 and 11061019)the Doctoral Foundation of Inner Mongolia(Grant Nos.2009BS0101 and 2010MS0110)+1 种基金the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20070126002)the Chunhui Program of the Ministry of Education of China(Grant No.Z2009-1-01010)
文摘This paper deals with off-diagonal operator matrices and their applications in elasticity theory. Two kinds of completeness of the system of eigenvectors are proven, in terms of those of the compositions of two block operators in the off-diagonal operator matrices. Using these results, the double eigenfunction expansion method for solving upper triangular matrix differential systems is proposed. Moreover, we apply the method to the two-dimensional elasticity problem and the problem of bending of rectangular thin plates on elastic foundation.
基金supported by the National Natural Science Foundation of China (Grant No. 10962004)the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 20080404MS0104)
文摘The completeness theorem of the eigenfunction systems for the product of two 2 × 2 symmetric operator matrices is proved. The result is applied to 4 × 4 infinite-dimensional Hamiltonian operators. A modified method of separation of variables is proposed for a separable Hamiltonian system. As an application of the theorem, the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained. Finally, a numerical test is analysed to show the correctness of the results.
基金supported by National Natural Science Foundation of China(Grant Nos.11371185,11101200 and 11361034)Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20111501110001)+1 种基金Major Subject of Natural Science Foundation of Inner Mongolia of China(Grant No.2013ZD01)Natural Science Foundation of Inner Mongolia of China(Grant No.2012MS0105)
文摘Symplectic self-adjointness of Hamiltonian operator matrices is studied, which is important to symplectic elasticity and optimal control. For the cases of diagonal domain and off-diagonal domain, necessary and sufficient conditions are shown. The proofs use Frobenius-Schur factorizations of unbounded operator matrices.Under additional assumptions, sufficient conditions based on perturbation method are obtained. The theory is applied to a problem in symplectic elasticity.
基金supported by the National Natural Science Foundation of China(Grant No.10962004)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20070126002)+1 种基金the Natural Science Foundation of Inner Mongolia(Grant No. 20080404MS0104)the Research Foundation for Talented Scholars of Inner Mongolia University(Grant No. 207066)
文摘The eigenvalue problem for the Hamiltonian operator associated with the mathematical model for the deflection of a thin elastic plate is investigated.First,the problem for a rectangular plate with simply supported edges is solved directly.Then,the completeness of the eigenfunctions is proved,thereby demonstrating the feasibility of using separation of variables to solve the problem. Finally,the general solution is obtained by using the proved expansion theorem.
基金supported by the National Natural Science Foundation of China (Grant No. 10962004)the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 2012MS0105)
文摘A separable Hamiltonian system of Mindlin plate bending problems is obtained. Using the equivalence between the differen form and integral form of the separable Hamiltonian system, the biorthogonal relationships of the eigenfunctions are presen! Based on the biorthogonal relationships, a novel complete biorthogonal expansion of the Mindlin plate bending problems x~ two opposite sides simply supported is proposed through the products of operator matrices. The exact solutions to deflections bending moments for the Mindlin plate with fully simply supported sides are obtained. A numerical example is illustrated to ve~ the accuracy and validity of the expansion method.
基金This work was supported by Ningbo Doctoral Science Foundation (No.2004A620018) National Natural Science Foundation of China (No.10471069).
文摘In this paper, we study a bounded-below singular Hamiltonian system. Sufficient and necessary conditions are obtained for the existence and the number of eigenvalues on the left-axis. The main results of this paper are the extension and improvement of Weyl spectral theorem for singular sccond order differential equations.
