A meshless method based on the moving least squares (MLS) approximation for the numerical solution of two-dimensional nonlinear integral equations of the second kind on non-rectangular domains

2013 ◽  
Vol 67 (2) ◽  
pp. 423-455 ◽  
Author(s):  
Pouria Assari ◽  
Hojatollah Adibi ◽  
Mehdi Dehghan
Keyword(s):  
Numerical Solution ◽  
Least Squares ◽  
Integral Equations ◽  
Meshless Method ◽  
Moving Least Squares ◽  
Two Dimensional ◽  
Nonlinear Integral Equations ◽  
Mls Approximation

THE INTERPOLATING ELEMENT-FREE GALERKIN (IEFG) METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS

2011 ◽  
Vol 03 (04) ◽  
pp. 735-758 ◽  
Author(s):  
HONGPING REN ◽  
YUMIN CHENG
Keyword(s):  
Least Squares ◽  
Shape Function ◽  
Weak Form ◽  
New Method ◽  
Moving Least Squares ◽  
Two Dimensional ◽  
Element Free Galerkin ◽  
Elasticity Problems ◽  
Mls Approximation ◽  
Element Free

In this paper, a new method for deriving the moving least-squares (MLS) approximation is presented, and the interpolating moving least-squares (IMLS) method proposed by Lancaster is improved. Compared with the IMLS method proposed by Lancaster, a simpler formula of the shape function is given in the improved IMLS method in this paper so that the new method has higher computing efficiency. Combining the shape function constructed by the improved IMLS method with Galerkin weak form of the elasticity problems, the interpolating element-free Galerkin (IEFG) method for the two-dimensional elasticity problems is presented, and the corresponding formulae are obtained. In the IEFG method, the boundary conditions can be applied directly which makes the computing efficiency higher than the conventional EFG method. Some numerical examples are presented to demonstrate the validity of the method.

Application of moving least squares algorithm for solving systems of Volterra integral equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mashallah Matinfar ◽  
Elham Taghizadeh ◽  
Masoumeh Pourabd
Keyword(s):  
Least Squares ◽  
Integral Equations ◽  
Approximation Method ◽  
Variable Coefficients ◽  
Volterra Integral Equations ◽  
Moving Least Squares ◽  
Numerical Examples ◽  
Acceptable Accuracy ◽  
Mls Approximation ◽  
Mls Method

Abstract 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.

THE COMPLEX VARIABLE ELEMENT-FREE GALERKIN (CVEFG) METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS

2009 ◽  
Vol 01 (02) ◽  
pp. 367-385 ◽  
Author(s):  
MIAOJUAN PENG ◽  
PEI LIU ◽  
YUMIN CHENG
Keyword(s):  
Complex Variable ◽  
Trial Function ◽  
Meshless Method ◽  
Moving Least Squares ◽  
Two Dimensional ◽  
Element Free Galerkin ◽  
Variable Element ◽  
Elasticity Problems ◽  
Mls Approximation ◽  
Element Free

Based on element-free Galerkin (EFG) method and the complex variable moving least-squares (CVMLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems is presented in this paper. With the CVMLS approximation, the trial function of a two-dimensional problem is formed with a one-dimensional basis function. The number of unknown coefficients in the trial function of the CVMLS approximation is less than in the trial function of moving least-squares (MLS) approximation, and we can thus select fewer nodes in the meshless method that is formed from the CVMLS approximation than are required in the meshless method of the MLS approximation with no loss of precision. The formulae of the CVEFG method for two-dimensional elasticity problems is obtained. Compared with the conventional meshless method, the CVEFG method has a greater precision and computational efficiency. For the purposes of demonstration, some selected numerical examples are solved using the CVEFG method.

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