LEAST-SQUARES MESHFREE COLLOCATION METHOD

2012 ◽  
Vol 09 (02) ◽  
pp. 1240031 ◽  
Author(s):  
BO-NAN JIANG
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Least Squares ◽  
Collocation Method ◽  
Present Method ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
Dual Variables

A least-squares meshfree collocation method is presented. The method is based on the first-order differential equations in order to result in a better conditioned linear algebraic equations, and to obtain the primary variables (displacements) and the dual variables (stresses) simultaneously with the same accuracy. The moving least-squares approximation is employed to construct the shape functions. The sum of squared residuals of both differential equations and boundary conditions at nodal points is minimized. The present method does not require any background mesh and additional evaluation points, and thus is a truly meshfree method. Unlike other collocation methods, the present method does not involve derivative boundary conditions, therefore no stabilization terms are needed, and the resulting stiffness matrix is symmetric positive definite. Numerical examples show that the proposed method possesses an optimal rate of convergence for both primary and dual variables, if the nodes are uniformly distributed. However, the present method is sensitive to the choice of the influence length. Numerical examples include one-dimensional diffusion and convection-diffusion problems, two-dimensional Poisson equation and linear elasticity problems.

scholarly journals Solving Some Differential Equations Arising in Electric Engineering Using Hermite Polynomials

2020 ◽  
Vol 12 (4) ◽  
pp. 517-523
Author(s):  
G. Singh ◽  
I. Singh
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Hermite Polynomials ◽  
Electric Circuit ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Electric Engineering ◽  
Collocation Points ◽  
Linear Algebraic Equations

In this paper, a collocation method based on Hermite polynomials is presented for the numerical solution of the electric circuit equations arising in many branches of sciences and engineering. By using collocation points and Hermite polynomials, electric circuit equations are transformed into a system of linear algebraic equations with unknown Hermite coefficients. These unknown Hermite coefficients have been computed by solving such algebraic equations. To illustrate the accuracy of the proposed method some numerical examples are presented.

scholarly journals Least-Squares Chebyshev Method for Solving Fractional Integro-diffrential Equations

2021 ◽  
Vol 06 (07) ◽  
Author(s):  
Oyedepo Taiye ◽  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Least Squares ◽  
Chebyshev Polynomials ◽  
Exact Results ◽  
Algebraic Equations ◽  
Graphical Solution ◽  
Numerical Examples ◽  
Chebyshev Method ◽  
Linear Algebraic Equations

The main purpose of this study gears towards finding numerical solution to fractional integro-differential equations. The technique involves the application of caputo properties and Chebyshev polynomials to reduce the problem to system of linear algebraic equations and then solved using MAPLE 18. To demonstrate the accuracy and applicability of the presented method some numerical examples are given. Numerical results show that the method is easy to implement and compares favorably with the exact results. The graphical solution of the method is displayed.

scholarly journals An Efficient Numerical Method for Solving Volterra-Fredholm Integro-Differential Equations of Fractional Order by Using Shifted Jacobi-Spectral Collocation Method

2018 ◽  
Vol 15 (3) ◽  
pp. 344-351
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Jacobi Polynomial ◽  
Fractional Derivatives ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
Jacobi Spectral Collocation Method ◽  
Polynomial Collocation

The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique.

scholarly journals Sinc Collocation Method for Finding Numerical Solution of Integrodifferential Model Arisen in Continuous Mixed Strategy

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
F. Hosseini Shekarabi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Mixed Strategy ◽  
Evolutionary Game ◽  
Collocation Methods ◽  
Mixed Strategies ◽  
Algebraic Equations ◽  
New Techniques ◽  
Numerical Tool

One of the new techniques is used to solve numerical problems involving integral equations and ordinary differential equations known as Sinc collocation methods. This method has been shown to be an efficient numerical tool for finding solution. The construction mixed strategies evolutionary game can be transformed to an integrodifferential problem. Properties of the sinc procedure are utilized to reduce the computation of this integrodifferential to some algebraic equations. The method is applied to a few test examples to illustrate the accuracy and implementation of the method.

scholarly journals A New Scheme for Solving Multiorder Fractional Differential Equations Based on Müntz–Legendre Wavelets

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Pseudospectral Method ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Derivative Operator ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
Fractional Differential

In this study, we apply the pseudospectral method based on Müntz–Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative. Using the operational matrix for the Caputo derivative operator and applying the Chebyshev and Legendre zeros, the problem is reduced to a system of linear algebraic equations. We illustrate the reliability, efficiency, and accuracy of the method by some numerical examples. We also compare the proposed method with others and show that the proposed method gives better results.

Solving systems of high-order linear ordinary differential equations with variable coefficients by means of exponential Chebyshev collocation method

2017 ◽  
Vol 8 (1-2) ◽  
pp. 40 ◽  
Author(s):  
Mohamed Ramadan ◽  
Kamal Raslan ◽  
Talaat El Danaf ◽  
Mohamed A. Abd Elsalam
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Variable Coefficients ◽  
High Order ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Linear Algebraic Equations

The purpose of this paper is to investigate the use of exponential Chebyshev (EC) collocation method for solving systems of high-order linear ordinary differential equations with variable coefficients with new scheme, using the EC collocation method in unbounded domains. The EC functions approach deals directly with infinite boundaries without singularities. The method transforms the system of differential equations and the given conditions to block matrix equations with unknown EC coefficients. By means of the obtained matrix equations, a new system of equations which corresponds to the system of linear algebraic equations is gained. Numerical examples are given to illustrative the validity and applicability of the method.

scholarly journals A numerical method for solving systems of higher order linear functional differential equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 15-25 ◽  
Author(s):  
Suayip Yüzbasi ◽  
Emrah Gök ◽  
Mehmet Sezer
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Legendre Polynomials ◽  
Linear Functional ◽  
Functional Differential Equations ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Linear Algebraic Equations ◽  
Functional Differential

AbstractFunctional differential equations have importance in many areas of science such as mathematical physics. These systems are difficult to solve analytically.In this paper we consider the systems of linear functional differential equations [1-9] including the term y(αx + β) and advance-delay in derivatives of y .To obtain the approximate solutions of those systems, we present a matrix-collocation method by using Müntz-Legendre polynomials and the collocation points. For this purpose, to obtain the approximate solutions of those systems, we present a matrix-collocation method by using Müntz-Legendre polynomials and the collocation points. This method transform the problem into a system of linear algebraic equations. The solutions of last system determine unknown co-efficients of original problem. Also, an error estimation technique is presented and the approximate solutions are improved by using it. The program of method is written in Matlab and the approximate solutions can be obtained easily. Also some examples are given to illustrate the validity of the method.

scholarly journals Perturbed Collocation Method For Solving Singular Multi-order Fractional Differential Equations of Lane-Emden Type

2020 ◽  
pp. 141-148
Author(s):  
O. A. Uwaheren ◽  
A. F. Adebisi ◽  
O. A. Taiwo
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Collocation Method ◽  
Fractional Differential Equations ◽  
General Class ◽  
Computer Software ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential ◽  
Interior Points

In this work, a general class of multi-order fractional differential equations of Lane-Emden type is considered. Here, an assumed approximate solution is substituted into a slightly perturbed form of the general class and the resulting equation is collocated at equally spaced interior points to give a system of linear algebraic equations which are then solved by suitable computer software; Maple 18.

scholarly journals Laguerre Collocation Method for Solving Fredholm Integro-Differential Equations with Functional Arguments

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Burcu Gürbüz ◽  
Mehmet Sezer ◽  
Coşkun Güler
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Matrix Equation ◽  
Numerical Experiments ◽  
Approximate Solutions ◽  
Correction Method ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Residual Correction

Laguerre collocation method is applied for solving a class of the Fredholm integro-differential equations with functional arguments. This method transforms the considered problem to a matrix equation which corresponds to a system of linear algebraic equations. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Also, the approximate solutions are corrected by using the residual correction method.

scholarly journals Application of the Least Squares Method in Axisymmetric Biharmonic Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Vasyl Chekurin ◽  
Lesya Postolaki
Keyword(s):  
Boundary Conditions ◽  
Least Squares ◽  
Lateral Surface ◽  
Least Squares Method ◽  
Neumann Boundary ◽  
Cylindrical Domain ◽  
Algebraic Equations ◽  
Homogeneous Solutions ◽  
Linear Algebraic Equations ◽  
Biharmonic Problems

An approach for solving of the axisymmetric biharmonic boundary value problems for semi-infinite cylindrical domain was developed in the paper. On the lateral surface of the domain homogeneous Neumann boundary conditions are prescribed. On the remaining part of the domain’s boundary four different biharmonic boundary pieces of data are considered. To solve the formulated biharmonic problems the method of least squares on the boundary combined with the method of homogeneous solutions was used. That enabled reducing the problems to infinite systems of linear algebraic equations which can be solved with the use of reduction method. Convergence of the solution obtained with developed approach was studied numerically on some characteristic examples. The developed approach can be used particularly to solve axisymmetric elasticity problems for cylindrical bodies, the heights of which are equal to or exceed their diameters, when on their lateral surface normal and tangential tractions are prescribed and on the cylinder’s end faces various types of boundary conditions in stresses in displacements or mixed ones are given.

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