The closed-form particular solutions for the Laplace operator using oscillatory radial basis functions in 2D

The closed-form particular solutions of the radial basis functions (RBFs) are essential for the implementation of several numerical methods to solve various partial differential equations (PDEs). Recently, a new class of oscillatory RBFs has been introduced. In this paper, we derive the closed-form particular solutions of the oscillatory RBFs for the Laplace operator in 2D so that it can be applied to particular solutions based numerical methods. We have successfully implemented the newly derived particular solutions in the method of particular solutions (MPS) for solving Poisson's equation as well as elliptic PDEs with variable coefficients.