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In this article, a new high-precision boundary-type mesh-free method, called the virtual boundary mesh-free Galerkin method (VBMGM), is given for analyzing the heat conduction problem consisting of multiple media. A control equation of the proposed method is established by the the Galerkin method of the weighted residual methods. The virtual source function is approached through the radial basis function interpolation in the mesh-free method. Thus, the main feature of the proposed technique has the advantages of the mesh-free method, the boundary-element method, and the Galerkin method, a symmetrical coefficient matrix. The numerical meanings of the weighted values in the VBMGM equation, such as the partial derivatives the temperature, the heat flux, the temperature connection condition, and the heat flux connection condition, are clear. Numerical examples of a thick-walled cylinder consisting of two media and a two-layered rectangular plate are provided. The feasibility and the accuracy of the proposed technique are proved. [ABSTRACT FROM AUTHOR]

The virtual boundary meshfree Galerkin method (VBMGM) is presented to calculate the single-domain heat conduction with heat source problem. The virtual source function of the virtual boundary and the heat source in the computational domain both are interpolated by the radial basis function interpolation. The equation of the proposed method is obtained by the Galerkin method of the weighted residual methods. Consequently, VBMGM possesses the advantages of Galerkin method, boundary element method and meshfree method. And the numerical meanings of the weighted values in VBMGM, such as the partial derivative the temperature, the heat flux and the heat source expression, are definite. The detailed and specific equation of VBMGM is deducted. The numerical examples, namely the temperatures of the rectangular domain and the square domain, are computed. And the accuracy and the stability of the proposed method for the single-domain heat conduction with heat source problem are validated. [ABSTRACT FROM AUTHOR]

The dynamics equation of a viscoelastic shallow arch under lateral loads, in which fractional derivative model is introduced to simulate the material characteristics, is established. The simplified differential equations of the first and second order Galerkin systems are developed combining Galerkin method with boundary conditions. Nonlinear dynamic behaviors of the first order Galerkin system under harmonic loads are discussed by numerical calculations. The results show that the system may lead to chaotic motion via period--doubling bifurcations or intermittent routes; In addition, the coexistences of multiple periodic solutions and the coexistences of periodic solutions with chaos solutions are found in this system when the excitation amplitude and frequency, damping and other state parameters are fixed but the initial conditions are changed. [ABSTRACT FROM AUTHOR]