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.

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.

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 An Improved Moving Least Squares Method for Curve and Surface Fitting

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Lei Zhang ◽  
Tianqi Gu ◽  
Ji Zhao ◽  
Shijun Ji ◽  
Ming Hu ◽  
...  
Keyword(s):  
Least Squares ◽  
Random Error ◽  
Least Squares Method ◽  
Surface Fitting ◽  
Measured Data ◽  
Moving Least Squares ◽  
Generate Curve ◽  
Moving Least Squares Method ◽  
Curve And Surface Fitting ◽  
Mls Method

The moving least squares (MLS) method has been developed for the fitting of measured data contaminated with random error. The local approximants of MLS method only take the error of dependent variable into account, whereas the independent variable of measured data always contains random error. Considering the errors of all variables, this paper presents an improved moving least squares (IMLS) method to generate curve and surface for the measured data. In IMLS method, total least squares (TLS) with a parameterλbased on singular value decomposition is introduced to the local approximants. A procedure is developed to determine the parameterλ. Numerical examples for curve and surface fitting are given to prove the performance of IMLS method.

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.

A Simplified Interpolating Moving Least-Squares Method and its Error Estimates

2011 ◽  
Vol 101-102 ◽  
pp. 271-274
Author(s):  
Ju Feng Wang
Keyword(s):  
Error Estimate ◽  
Least Squares ◽  
Shape Function ◽  
Least Squares Method ◽  
Delta Function ◽  
Moving Least Squares ◽  
Approximation Function ◽  
Kronecker Delta ◽  
Moving Least Squares Method ◽  
Mls Approximation

A disadvantage of the MLS approximation is that the shape function of this method does not satisfy the property of Kronecker Delta function. Thus developing an interpolating MLS approximation is very important. In this paper, the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas is discussed in detail and a simplified expression of the approximation function of the IMLS method is given. The simpler expression makes it more convenient to use this method. The error estimate of the approximation function also is discussed. And a numerical example is given to confirm the results.

scholarly journals A Finite-Difference Solution of Solute Transport through a Membrane Bioreactor

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
B. Godongwana ◽  
D. Solomons ◽  
M. S. Sheldon
Keyword(s):  
Finite Difference ◽  
Analytical Solution ◽  
Membrane Bioreactor ◽  
Elliptic Partial Differential Equation ◽  
Reaction Rates ◽  
Variable Coefficients ◽  
Concentration Profiles ◽  
Algebraic Equations ◽  
Regular Perturbation ◽  
Sink Term

The current paper presents a theoretical analysis of the transport of solutes through a fixed-film membrane bioreactor (MBR), immobilised with an active biocatalyst. The dimensionless convection-diffusion equation with variable coefficients was solved analytically and numerically for concentration profiles of the solutes through the MBR. The analytical solution makes use of regular perturbation and accounts for radial convective flow as well as axial diffusion of the substrate species. The Michaelis-Menten (or Monod) rate equation was assumed for the sink term, and the perturbation was extended up to second-order. In the analytical solution only the first-order limit of the Michaelis-Menten equation was considered; hence the linearized equation was solved. In the numerical solution, however, this restriction was lifted. The solution of the nonlinear, elliptic, partial differential equation was based on an implicit finite-difference method (FDM). An upwind scheme was employed for numerical stability. The resulting algebraic equations were solved simultaneously using the multivariate Newton-Raphson iteration method. The solution allows for the evaluation of the effect on the concentration profiles of (i) the radial and axial convective velocity, (ii) the convective mass transfer rates, (iii) the reaction rates, (iv) the fraction retentate, and (v) the aspect ratio.

scholarly journals An Improved Interpolating Element-Free Galerkin Method Based on Nonsingular Weight Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
F. X. Sun ◽  
C. Liu ◽  
Y. M. Cheng
Keyword(s):  
Least Squares ◽  
Shape Function ◽  
Weak Form ◽  
Moving Least Squares ◽  
Weight Functions ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Node Distribution ◽  
Mls Approximation ◽  
Element Free

Based on the moving least-squares (MLS) approximation, an improved interpolating moving least-squares (IIMLS) method based on nonsingular weight functions is presented in this paper. Then combining the IIMLS method and the Galerkin weak form, an improved interpolating element-free Galerkin (IIEFG) method is presented for two-dimensional potential problems. In the IIMLS method, the shape function of the IIMLS method satisfies the property of Kroneckerδfunction, and there is no difficulty caused by singularity of the weight function. Then in the IIEFG method presented in this paper, the essential boundary conditions are applied naturally and directly. Moreover, the number of unknown coefficients in the trial function of the IIMLS method is less than that of the MLS approximation; then under the same node distribution, the IIEFG method has higher computational precision than element-free Galerkin (EFG) method and interpolating element-free Galerkin (IEFG) method. Four selected numerical examples are presented to show the advantages of the IIMLS and IIEFG methods.

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