This work aims at focusing on modifying the moving least squares (MMLS) methods for solving two-dimensional linear and nonlinear systems of integral equations and system of differential equations. The modified shape function is our main aim, so for computing the shape function based on the moving least squares method (MLS), an efficient algorithm is presented. In this modification, additional terms is proposed to impose based on the coefficients of the polynomial base functions on the quadratic base functions (m = 2). So, the MMLS method is developed for solving the systems of two-dimensional linear and nonlinear integral equations at irregularly distributed nodes. This approach prevents the singular moment matrix in the context of MLS based on meshfree methods. Also, determining the best radius of the support domain of a field node is an open problem for MLS-based methods. Therefore, the next important thing is that the MMLS algorithm can automatically find the best neighborhood radius for each node. Then, numerical examples are presented that determine the main motivators for doing this so. These examples enable us to make comparisons of two methods: MMLS and classical MLS.
Part of the book: Nonlinear Systems