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.

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Mathematical modeling for many problems in different disciplines, such as engineering, chemistry, physics and biology leads to integral equation, or system of integral equations. It's the reason of great interest for solving these equations. There are some analytical and numerical methods for solving Volterra integral equations, but extension of these methods to systems of such integral equations is not easy to employ. Adomian decomposition method, well address in [1,2] has been used to solved some of these systems such as systems of differential equations, systems of integral equations and even systems of integro-differential equation [3,4,5]. Applying this method needs some computations which is sometimes boring, having a program to do all computations would be interesting and helpful. In this article a maple program is prepared to solve a system of Volterra integral equations of the second kind, linear or non-linear.

The numerical method developed in the current paper is based on the moving least squares (MLS) method. To this end, the MLS approximation method has been used, and a program has been made which can solve the system of Volterra integral equations (VIEs) with any number of equations and unknown functions. And then the proposed method is implemented on the system of linear VIEs with variable coefficients. The numerical examples are given that show the acceptable accuracy and efficiency of the proposed scheme.