Generalized polyharmonic multiquadrics

In this paper, we construct the two- and three-dimensional generalized polyharmonic multiquadrics (GPMQ) of order (K,L), which are the particular solution of the K-th order generalized multiquadrics (GMQ) associated with the L-th order polyharmonic operator for L>0. By observing the first few orders of the GPMQs, we construct methods of undetermined coefficients and determine the unknown coefficients by expanding the GPMQs into Laurent series. The derived GPMQs are hierarchically unique and infinitely differentiable. Then, the GPMQ definitions are extended for L<0 and the solutions are derived by similar methods. Both symbolic and floating-point implementations are performed for automatically obtaining the GPMQs of arbitrary orders, in which the former is explicitly provided and the later enables to implement numerical methods free from bookkeeping. The derived GPMQs are validated by numerical experiments, in which significant improvement on the accuracy can be observed.