An Improved Wavelet Approximation Method And Its Application To Spatio-Temporal Dynamics

Nonlinear differential equations are mathematical models of many nonlinear problems in sciences and engineering. Method of solving nonlinear differential equations become extremely important in the studying of nonlinear science. We introduce an expansion formula of Coiflets approximations for square-integrable functions. The expansion coefficient of nonlinear term can be expressed explicitly, and the precision of the expansion is independent of truncation error. Consruct a new extansion method through the basis of the Taylor series expansion on boundarys. We obtained a wavelet tmethod for uniformly solving nonlinear differential equations according to the Galerkin method.Firstly, we verified the accuracy of Coiflets wavelet approximation formula and proposed a new boundary treatment method in this paper. By substituting Taylor expansion into scale function expansion and representing derivative values at the boundary by function values within the interval, we use the unknown values at the nodes expressed the boundary derivative values explicitly. The numerical results show that the new expression form able to inhibition the undesired jump or wiggle phenomenon near the boundary points more effectively. Finally, by comparison with the original approximation formula, the improved wavelet approximation has higher accuracy.Secondly, we solve the Klein-Gordon equations with wavelet-Galerkin method and choose equations with standing wave solutions, traveling wave solutions and the classical sine-Gordon equation as examples. The Runge-Kutta method is used in the time dimension. Finally, by comparison with the exact soltion, show that the wavelet-Galerkin method has good numerical accuracy, and can meet the practical requirement of solving problems.Finally, according to composite laminated plate theory and the piezoelectric control theory, we introduced the dynamic control process of the beam-plate based on wavelet theory. Piezoelectric patches are pasted on the both side of the beam, as sensors and actuators. The displacement and speed of the beam can be obtained according to the charge and current in piezoelectric patches, further the control voltage applied on the beam can be got through the negative feedback rate. A dynamic control model for the piezoelectric plates is formulated by the improved Coiflets wavelet approximation. Under the same conditions, the numerical results show that the control method in this paper can make the amplitude of the plate decay to zero in a shorter time.