Wavelet-DQ Method And Its Application

The traditional numerical methods of the differential equation include FEM, BEM, FDM and the weighted residual method. These techniques have some shortcomings in solving some special differential equations, such as the equations with the singularity etc, although they are applicable to the regular questions. With its good time-frequency localization merit, wavelet analysis has been introduced as a powerful tool for the numerical solution of differential equations. At the same time, the DQ method is widely used in mechanics to get the numerical solution of the differential equations as well. In this paper, combining the theory of the multi-resolution analysis and the idea of the DQ method, Wavelet-DQ as a new method is obtained with wavelet scale function as the interpolation basis function. The main contents as follows:1. Based on DQ method and multi-resolution analysis theory, the wavelet-DQ method is proposed in this paper, in which the interpolation basic function is formed by using scaling function.2. Using the new method, the problem of longitudinal vibration with variable cross-section and visco-elasticity was studied, and the complex frequency equation was given. At the same time, under different parameters, the natural frequency of straight rod and cone straight rod with the clamped-clamped,free-free wedge has been calculated. The results shown that this method is effective for such problems.3. The stability of simply supported Kelvin visco-elastic pipes conveying fluid is investigated by the Wavelet-DQ method. The curves between internal fluid velocity and complex frequencies of the pipe with different parameters are given, and the effect of external fluid velocity on vibration behaviors and stability of Kelvin visco-elastic fluid-conveying pipes with different delay time of the visco-elastic material are analyzed. The numerical results shown that the method has wide applicability.