Document Type : Research Paper

**Authors**

Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran.

**Abstract**

In this paper, the interpolating moving least-squares (IMLS) method is discussed. The interpolating moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data. Then we apply the IMLS method for the numerical solution of Volterra–Fredholm integral equations, and finally some examples are given to show the accuracy and applicability of the method.

**Keywords**

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[18] H. P. Ren, J. Cheng, and A. Huang, The complex variable interpolating moving least-squares method, Applied mathematics and computation., 219 (2012), 1724–1736.

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[20] F. Sun, J. Wang, and Y. Cheng, An improved interpolating element-free Galerkin method for elasticity, Chinese Physics B., 22 (2013), 120203.

[21] F. Sun, C. Liu, and Y.Cheng, An improved interpolating element-free Galerkin method based on nonsingular weight functions, Mathematical Problems in Engineering., 2014 (2014), 1–13.

[22] J. Wang, F. Sun, and Y. Cheng, An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems, Chinese Physics B., 21 (2012), 090204.

[23] J. Wang, J. Wang, F. Sun, and Y. Cheng, An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems, International Journal of Computational Methods., 10 (2013), 1350043.

[24] J. Wang, F. Sun, Y. Cheng, and A. Huang, Error estimates for the interpolating moving leastsquares method, Applied Mathematics and Computation., 245 (2014), 321–342.

[25] S. Yal¸cinba¸s, Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equations, Applied Mathematics and Computation., 127 (2002), 195–206.

[26] S. A. Yousefi, A. Lotfi, and M. Dehghan, Hes variational iteration method for solving nonlinear mixed Volterra Fredholm integral equations, Computers & Mathematics with Applications., 58 (2009), 2172–2176.

[8] K. Maleknejad, H. Almasieh, and M. Roodaki, Triangular functions (TF) method for the solution of nonlinear Volterra Fredholm integral equations, Communications in Nonlinear Science and Numerical Simulation., 15 (2010), 3293–3298.

[9] D. Mirzaei and M. Dehghan, A meshless based method for solution of integral equations, Applied Numerical Mathematics., 60 (2010), 245–262.

[10] D. Mirzaei, R. Schaback, and M. Dehghan, On generalized moving least squares and diffuse derivatives, IMA Journal of Numerical Analysis., 32 (2012), 983–1000.

[11] D. Mirzaei, Analysis of moving least squares approximation revisited, Journal of Computational and Applied Mathematics., 282 (2015), 237–250.

[12] M. M. Mustafa and I. N. Ghanim, Numerical solution of linear Volterra-Fredholm integral equations using Lagrange polynomials, Mathematical Theory and Modeling., 4 (2014), 137–146.

[13] H. Netuzhylov, Enforcement of boundary conditions in meshfree methods using interpolating moving least squares, Engineering analysis with boundary elements., 32 (2008), 512–516.

[14] H. P. Ren and Z. Wu, An improved boundary element-free method (IBEFM) for two-dimensional potential problems, Chinese Physics B., 18 (2009), 4065–4073.

[15] H. P. Ren, Y. M. Cheng, and W. Zhang, An interpolating boundary element-free method (IBEFM) for elasticity problems, Science China Physics, Mechanics and Astronomy., 53 (2010), 758–766.

[16] H. P. Ren and Y. M. Cheng, The interpolating element-free Galerkin (IEFG) method for twodimensional elasticity problems, International Journal of Applied Mechanics., 3 (2011), 735–758.

[17] H. Ren and Y. M. Cheng, The interpolating element-free Galerkin (IEFG) method for twodimensional potential problems, Engineering Analysis with Boundary Elements., 36 (2012), 873–880.

[18] H. P. Ren, J. Cheng, and A. Huang, The complex variable interpolating moving least-squares method, Applied mathematics and computation., 219 (2012), 1724–1736.

[19] H. P. Ren and Y. M. Cheng, A new element-free Galerkin method based on improved complex variable moving least squares approximation for elasticity, International Journal of Computational Materials Science and Engineering., 1 (2012), 1250011.

[20] F. Sun, J. Wang, and Y. Cheng, An improved interpolating element-free Galerkin method for elasticity, Chinese Physics B., 22 (2013), 120203.

[21] F. Sun, C. Liu, and Y.Cheng, An improved interpolating element-free Galerkin method based on nonsingular weight functions, Mathematical Problems in Engineering., 2014 (2014), 1–13.

[22] J. Wang, F. Sun, and Y. Cheng, An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems, Chinese Physics B., 21 (2012), 090204.

[23] J. Wang, J. Wang, F. Sun, and Y. Cheng, An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems, International Journal of Computational Methods., 10 (2013), 1350043.

[24] J. Wang, F. Sun, Y. Cheng, and A. Huang, Error estimates for the interpolating moving leastsquares method, Applied Mathematics and Computation., 245 (2014), 321–342.

[25] S. Yal¸cinba¸s, Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equations, Applied Mathematics and Computation., 127 (2002), 195–206.

[26] S. A. Yousefi, A. Lotfi, and M. Dehghan, Hes variational iteration method for solving nonlinear mixed Volterra Fredholm integral equations, Computers & Mathematics with Applications., 58 (2009), 2172–2176.

July 2021

Pages 830-845

**Receive Date:**25 January 2019**Revise Date:**15 May 2020**Accept Date:**15 May 2020**First Publish Date:**01 July 2021