Solving Fourth-Order PDEs using the LMAPS

To overcome the difficulty for solving fourth order partial differential equations (PDEs) using localized methods, we introduce and extend a recent method to decompose the particular solution of such equation into particular solutions of two second-order differential equations using radial basis functions (RBFs). In this way, the localized method of approximate particular solutions (LMAPS) can be used to directly solve a fourth-order PDE without splitting it into two second-order problems. The closed-form particular solutions for polyharmonic splines RBFs augmented with polynomial basis functions for Helmholtz-type equations are the cores of the solution process. Several novel techniques are proposed to further improve the accuracy and efficiency. Four numerical examples are presented to show the effectiveness of our approach.