The Meshless Analog Equation Method:A New Highly Accurate Truly Mesh-free MethodFor Solving Partial Differential Equations

Abstract

A new purely meshless method to solve PDEs is presented. The method is based on the concept of the analog equation of Katsikadelis, hence its name meshless analog equation method (MAEM), which converts the original equation into a simple solvable substitute one of the same order under a fictitious source. The fictitious source is represented by MQ-RBFS. Integration of the analog equation allows the approximation of the sought solution by new RBFs. Then inserting the solution into the PDE and BCs and collocating at the mesh-free nodal points yields a system of linear equations, which permit the evaluation of the expansion coefficients. The method exhibits key advantages of over other RBF collocation methods as it is highly accurate and the matrix of the resulting linear equations is always invertible. The accuracy is increased using optimal values of the shape parameters of the multiquadrics by minimizing the potential that produces the PDE. Without restricting its generality, the method is illustrated by applying it to the general second order elliptic PDE. The studied examples demonstrate the efficiency and high accuracy of the developed method.