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.

Moving Least Squares (MLS) Method for the Nonlinear Hyperbolic Telegraph Equation with Variable Coefficients

2017 ◽  
Vol 14 (03) ◽  
pp. 1750026 ◽  
Author(s):  
A. Mardani ◽  
M. R. Hooshmandasl ◽  
M. M. Hosseini ◽  
M. H. Heydari
Keyword(s):  
Finite Difference ◽  
Least Squares ◽  
Telegraph Equation ◽  
Variable Coefficients ◽  
Moving Least Squares ◽  
Algebraic Equations ◽  
Mesh Structure ◽  
Background Mesh ◽  
Mls Approximation ◽  
Mls Method

Telegraph equation which widely used for modeling many engineering and physical phenomena has considered by some researchers in recent years. In this paper, a numerical scheme based on the moving least squares (MLS) approximation and finite difference method (FDM) is proposed for solving a class of the nonlinear hyperbolic telegraph equation with variable coefficients. In the new developed scheme, we use collocation points and approximate solution of the problem under study by using MLS approximation. The MLS method is a meshless approach and does not need any background mesh structure. A time stepping approach is employed for the first- and second-order time derivatives. The proposed method provides a semi-discrete solutions for the problems under study. In space domain, the MLS approximation and in time domain, the finite difference technique are employed. This method after discretization leads to a linear system of algebraic equations. Some numerical results are given and compared with analytical solutions to demonstrate the validity and efficiency of the proposed technique.

Coupled Finite Element-Moving Least Squares Technique for Stochastic Structural Response of Cracked Structures

2006 ◽  
Author(s):  
K. Aravinda Priyadrashini ◽  
B. N. Rao
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Approximation Method ◽  
Reproducing Kernel ◽  
Computational Cost ◽  
Structural Response ◽  
Moving Least Squares ◽  
Shape Functions ◽  
Cracked Structures ◽  
Mls Approximation

In recent years, a class of Galerkin-based meshfree or meshless methods have been developed that do not require a structured mesh to discretize the problem, such as the element-free Galerkin method, and the reproducing kernel particle method. These methods employ a moving least-squares (MLS) approximation method that allows resultant shape functions to be constructed entirely in terms of arbitrarily placed nodes. These methods employ a moving least-squares approximation method that allows resultant shape functions to be constructed entirely in terms of arbitrarily placed nodes. Meshless discretization presents significant advantages for modeling fracture propagation. By sidestepping remeshing requirements, crack-propagation analysis can be dramatically simplified. Since no element connectivity data are needed, the burdensome remeshing required by the FEM is avoided. A growing crack can be modeled by simply extending the free surfaces, which correspond to the crack. In addition, stochastic meshless method (SMM) facilitates to have different discretizations for capturing the spatial variability of the material properties and the structural response, without much difficulty in mapping between the two discretizations. However, the computational cost of a SMM typically exceeds the cost of a SFEM. Hence, it is advantageous to adopt MLS approximation for material property discretization and FEM for structural response computation. This paper presents a new coupled finite element-moving least squares technique for predicting probabilistic structural response and reliability of cracked structures.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

scholarly journals Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation

2019 ◽  
Vol 24 (4) ◽  
pp. 101
Author(s):  
A. Karami ◽  
Saeid Abbasbandy ◽  
E. Shivanian
Keyword(s):  
Free Boundary ◽  
Least Squares ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Moving Least Squares ◽  
Heaviside Step Function ◽  
Mlpg Method ◽  
Mls Approximation

In this paper, we study the meshless local Petrov–Galerkin (MLPG) method based on the moving least squares (MLS) approximation for finding a numerical solution to the Stefan free boundary problem. Approximation of this problem, due to the moving boundary, is difficult. To overcome this difficulty, the problem is converted to a fixed boundary problem in which it consists of an inverse and nonlinear problem. In other words, the aim is to determine the temperature distribution and free boundary. The MLPG method using the MLS approximation is formulated to produce the shape functions. The MLS approximation plays an important role in the convergence and stability of the method. Heaviside step function is used as the test function in each local quadrature. For the interior nodes, a meshless Galerkin weak form is used while the meshless collocation method is applied to the the boundary nodes. Since MLPG is a truly meshless method, it does not require any background integration cells. In fact, all integrations are performed locally over small sub-domains (local quadrature domains) of regular shapes, such as intervals in one dimension, circles or squares in two dimensions and spheres or cubes in three dimensions. A two-step time discretization method is used to deal with the time derivatives. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

scholarly journals A Modification of the Moving Least-Squares Approximation in the Element-Free Galerkin Method

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Yang Cao ◽  
Jun-Liang Dong ◽  
Lin-Quan Yao
Keyword(s):  
Least Squares ◽  
Singular System ◽  
Basis Functions ◽  
Moving Least Squares ◽  
Shape Functions ◽  
Monomial Basis ◽  
Element Free Galerkin ◽  
Small Matrix ◽  
Mls Approximation ◽  
Element Free

The element-free Galerkin (EFG) method is one of the widely used meshfree methods for solving partial differential equations. In the EFG method, shape functions are derived from a moving least-squares (MLS) approximation, which involves the inversion of a small matrix for every point of interest. To avoid the calculation of matrix inversion in the formulation of the shape functions, an improved MLS approximation is presented, where an orthogonal function system with a weight function is used. However, it can also lead to ill-conditioned or even singular system of equations. In this paper, aspects of the IMLS approximation are analyzed in detail. The reason why singularity problem occurs is studied. A novel technique based on matrix triangular process is proposed to solve this problem. It is shown that the EFG method with present technique is very effective in constructing shape functions. Numerical examples are illustrated to show the efficiency and accuracy of the proposed method. Although our study relies on monomial basis functions, it is more general than existing methods and can be extended to any basis functions.

scholarly journals A linear approximation for solutions of Abel–Volterra integral equations

2020 ◽  
Vol 14 (2) ◽  
pp. 596-604
Author(s):  
Mostefa Nadir
Keyword(s):  
Integral Equations ◽  
Linear Approximation ◽  
Singular Integral ◽  
Volterra Integral Equations ◽  
New Technique ◽  
Numerical Examples ◽  
Weakly Singular

Abstract In this work, we present a modified linear approximation for solving the first and the second kind Abel–Volterra integral equations. This approximation was used by the author to approximate a weakly singular integral on the curve. Noting that this new technique gives a good approximation of these solutions compared with several methods in several numerical examples.

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