Representing PDE surfaces in terms of B-splines
The paper discusses a technique for expressing PDE surfaces in terms of B-splines. The surface is obtained not as exact solution of the original boundary value problem but they have the virtue of being genuine tensor- product surfaces, which have B-splines as their basis functions. The collocation method for the solution of ordinary and partial differential equations is used: The approximate solution of the PDE \((\frac{\partial^ 2}{\partial u^ 2}+a^ 2\frac{\partial}{\partial v^ 2})X=0\) over the region \(\Omega =(0,1)\times (0,1)\) is searched in the following form, allowing local control: \(\tilde X=\sum^{N+2}_{i=-2}\sum^{M+2}_{j=-2}a_{ij}B_ i(u)B_ j(v)\) where, \(a_{ij}\) are the ``smoothing vectors''. To illustrate the principles of the method the authors concentrate their attention to the particularly simple surfaces with constrained boundary conditions.
scientific article; zbMATH DE number 4159948