A quasi-linear technique applied to the method of fundamental solution

Abstract

This paper proposes the use of a quasi-linear technique for the method of fundamental solution (MFS) to treat the non-linear Poisson-type equations. The MFS, which is a fully meshless method, often deals with the linear and non-linear poisson equations by approximating a particular solution via employing radial basis functions (RBFs). The interpolation in terms of RBFs often leads to a badly conditioned problem which demands special cares. The current work suggests a linearization scheme for the non-homogeneous term in terms of the dependent variable resulting in Helmholtz-type equations whose fundamental solutions are available. Consequently, the MFS can be directly applied to the new linearized equation. The numerical examples illustrate the effectiveness of the presented method.