## Abstract

We extended the moving least squares (MLS) and modified moving least squares (MMLS) methods for solving two-dimensional linear and nonlinear systems of integral equations. This modification is proposed on the quadratic base functions \((m=2)\) by imposing additional terms based on the coefficients of the polynomial base functions. This approach prevents the singular moment matrix in the context of MLS based on meshfree methods. Additionally, finding the optimum value for the radius of the domain influence is an open problem for MLS-based methods. So an efficient algorithm is introduced for computing a suitable value of dilatation parameter to determine the radius of the support domain. This algorithm able to prevent the singular matrix which is an outcome of adverse selection of the radius of influence domain. In numerical examples are provided to enable us to compare MMLS method and standard MLS method by the new proposed algorithm. Comparing the errors of MMLS and MLS method determines the capability and accuracy of applied techniques to solve systems of integral equation problems. This indicates the advantage of the proposed method respect to MLS method.

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Communicated by Abimael Loula.

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Matinfar, M., Pourabd, M. Modified moving least squares method for two-dimensional linear and nonlinear systems of integral equations.
*Comp. Appl. Math.* **37**, 5857–5875 (2018). https://doi.org/10.1007/s40314-018-0667-6

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DOI: https://doi.org/10.1007/s40314-018-0667-6

### Keywords

- Moving least squares
- Modified moving least squares
- Systems of integral equations
- Algorithm of shape function
- Numerical solutions

### Mathematics Subject Classification

- 45G15
- 45F05
- 45F35
- 65D15