A new approach is developed to solve the Green＇s function that satisfies the Hehmholtz equation with complex refractive index. Especially, the Green＇s function for the Helmholtz equation can be expressed in terms of...A new approach is developed to solve the Green＇s function that satisfies the Hehmholtz equation with complex refractive index. Especially, the Green＇s function for the Helmholtz equation can be expressed in terms of a onedimensional integral, which can convert a Helmholtz equation into a Schrodinger equation with complex potential. And the Schrodinger equation can be solved by Feynman path integral. The result is in excellent agreement with the previous work.展开更多
Quasicrystals have additional phason degrees of freedom not found in conventional crystals. In this paper, we present an exact solution for time-harmonic dynamic Green＇s function of one-dimensional hexagonal quasicry...Quasicrystals have additional phason degrees of freedom not found in conventional crystals. In this paper, we present an exact solution for time-harmonic dynamic Green＇s function of one-dimensional hexagonal quasicrystals with the Laue classes 6/mh and 6/mhmm. Through the introduction of two new functions φ and ψ, the original problem is reduced to the determination of Green＇s functions for two independent Helmholtz equations. The explicit expressions of displacement and stress fields are presented and their asymptotic behaviors are discussed. The static Green＇s function can be obtained by letting the circular frequency approach zero.展开更多
The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and ...The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson＇s equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition （SVD） technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric （MQ） are fully analyzed to provide useful reference.展开更多
We propose a theorem for the quantum operator that corresponds to the solution of the Helmholtz equation, i.e., ∫∫∫V （x1 ,x2,x3）〈X1 ,x2,x3〉〈x1 ,x2,x3〉d3x = V （X1 ,X2,X3） = e-λ2/4 ：V （X1 ,X2,X3）：,where...We propose a theorem for the quantum operator that corresponds to the solution of the Helmholtz equation, i.e., ∫∫∫V （x1 ,x2,x3）〈X1 ,x2,x3〉〈x1 ,x2,x3〉d3x = V （X1 ,X2,X3） = e-λ2/4 ：V （X1 ,X2,X3）：,where V （X1 ,X2,X3） is the solution to the Helmholtz equation △2V ＋λ2V = 0, the symbol ： ： denotes normal ordering, and X1, X2, X3 are three-dimensional coordinate operators. This helps to derive the normally ordered expansion of Dirac＇s radius operator functions. We also discuss the normally ordered expansion of Bessel operator functions.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10574010 and 10974010)Beijing Commission of Education (Grant No. 1010005466903)
文摘A new approach is developed to solve the Green＇s function that satisfies the Hehmholtz equation with complex refractive index. Especially, the Green＇s function for the Helmholtz equation can be expressed in terms of a onedimensional integral, which can convert a Helmholtz equation into a Schrodinger equation with complex potential. And the Schrodinger equation can be solved by Feynman path integral. The result is in excellent agreement with the previous work.
基金Project supported by Shanghai Leading Academic Discipline Project (No.Y0103).
文摘Quasicrystals have additional phason degrees of freedom not found in conventional crystals. In this paper, we present an exact solution for time-harmonic dynamic Green＇s function of one-dimensional hexagonal quasicrystals with the Laue classes 6/mh and 6/mhmm. Through the introduction of two new functions φ and ψ, the original problem is reduced to the determination of Green＇s functions for two independent Helmholtz equations. The explicit expressions of displacement and stress fields are presented and their asymptotic behaviors are discussed. The static Green＇s function can be obtained by letting the circular frequency approach zero.
文摘The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson＇s equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition （SVD） technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric （MQ） are fully analyzed to provide useful reference.
基金supported by the National Natural Science Foundation of China(Grant No.11175113)
文摘We propose a theorem for the quantum operator that corresponds to the solution of the Helmholtz equation, i.e., ∫∫∫V （x1 ,x2,x3）〈X1 ,x2,x3〉〈x1 ,x2,x3〉d3x = V （X1 ,X2,X3） = e-λ2/4 ：V （X1 ,X2,X3）：,where V （X1 ,X2,X3） is the solution to the Helmholtz equation △2V ＋λ2V = 0, the symbol ： ： denotes normal ordering, and X1, X2, X3 are three-dimensional coordinate operators. This helps to derive the normally ordered expansion of Dirac＇s radius operator functions. We also discuss the normally ordered expansion of Bessel operator functions.
