Wavelet Solution Of Initial Boundary Value Problems With Variable Coefficients And Its Application In Structural Mechanics

In recent years,wavelet theory has been greatly developed and widely applied in mathematics and many scientific fields.Many mathematicians and scientists pay more and more attention to the solution of boundary value and initial boundary value problems,which is one of the important applications of wavelet theory.The numerical methods based on wavelet theory have unique advantages in boundary value and initial boundary value problems because of the good properties of smoothness,orthogonality and compact support of scale function.People have developed the wavelet method of boundary value and initial boundary value problems with constant coefficients,but the research on boundary value and initial boundary value problems with variable coefficients is not deep enough.In this paper,we propose a unified format to solve different kinds of differential equations with variable coefficients based on wavelet theory.The following are the main research results of this paper:(1)We propose a unified wavelet format to solve boundary value problems with variable coefficients based on the wavelet method in boundary value problems with constant coefficients,and two-dimensional or even multi-dimensional cases are considered.(2)Combining the time integration method based on wavelet,we propose a unified wavelet format to solve initial boundary value problems with variable coefficients,and two-dimensional or even multi-dimensional cases are considered.(3)By calculating some classical examples,we prove that the wavelet method has good calculation accuracy and efficiency in the solution of differential equations with constant coefficients.And we calculate the vibration problem of beam-plate structure with uniform section.(4)Based on the wavelet solution of boundary value problems we propose in this paper,we calculate the bending problem of beam-plate structure with variable section.(5)Based on the wavelet solution of initial boundary value problems we propose in this paper,we discuss the vibration problems of two kinds of beams with variable section,the deformation of different positions on beams and the vibration of beams at a certain time are given.In addition,we calculate the vibration problems of rectangular plates with variable thickness,the deformation of plates' midpoint and the vibration of plates at a certain time are given.Finally,we use this method to calculate the vibration problems of rectangular plates with variable thickness on elastic foundations,which contain nonlinear terms.The wavelet methods of solving differential equations with variable coefficients we propose in this paper make us avoid the calculation of complex connection coefficients and have high computational efficiency and accuracy.The unified wavelet format can solve different kinds of both linear and non-linear differential equations with variable coefficients in science and engineering.