Abstract:
In this study, a two-dimensional Haar wavelet expansion is developed and the biharmonic equation with functional boundary conditions are examined. The bending problem of an elastic thin plate with simply-supported edges subjected to external force is implemented based on the proposed methodology. This proposed method is an extension of the 1D Haar wavelet collocation method, in which expand the highest partial derivative in the differential equation into a 2D Haar series and then integrated it as the approximated solution. As a result, it successfully replaces the requirement of derivatives by adopting the integrations on the shape function in the desired differential equation. The grid points of the domain will be automatically generated due to the inherence of wavelet hierarchy. The boundary points will also be evaluated by using the wavelet expansion and then further adhered to the interior ones so that a so-called Haar wavelet operational matrix can be formed. By adopting the property of matrix operations, a simple form of the shape function without involving any derivatives can be achieved, and thus the approximation solution can be resolved by using the conventional matrix method. Here, numerical results for various resolution levels based on the two-dimensional Haar wavelet expansion are obtained; data are compared with existing literatures by one-dimensional Haar wavelet expansion. Furthermore, the bending problem of an elastic thin plate is also investigated. The validation and restriction of the proposed methodology are discussed, concise notion and precise notation are introduced for the first time, alternative form converting the partial differential equation and its boundary conditions into one single operational matrix is presented.
