scholarly journals Convergence of the collocation method and the mechanical quadrature method for systems of singular integro-differential equations in Lebesgue spaces

2012 ◽  
Vol 236 (15) ◽  
pp. 3796-3804 ◽  
Author(s):  
Iurie Caraus ◽  
Feras M. Al Faqih
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Quadrature Method ◽  
Lebesgue Spaces ◽  
Mechanical Quadrature ◽  
Mechanical Quadrature Method

A High-Accuracy Mechanical Quadrature Method for Solving the Axisymmetric Poisson's Equation

2017 ◽  
Vol 9 (2) ◽  
pp. 393-406 ◽  
Author(s):  
Hu Li ◽  
Jin Huang
Keyword(s):  
Three Dimensional ◽  
High Accuracy ◽  
Poisson’S Equation ◽  
Two Dimensional ◽  
Quadrature Method ◽  
Poisson's Equation ◽  
Mechanical Quadrature ◽  
Asymptotic Error ◽  
Accuracy Order ◽  
Mechanical Quadrature Method

AbstractIn this article, we consider the numerical solution for Poisson's equation in axisymmetric geometry. When the boundary condition and source term are axisymmetric, the problem reduces to solving Poisson's equation in cylindrical coordinates in the two-dimensional (r,z) region of the original three-dimensional domain S. Hence, the original boundary value problem is reduced to a two-dimensional one. To make use of the Mechanical quadrature method (MQM), it is necessary to calculate a particular solution, which can be subtracted off, so that MQM can be used to solve the resulting Laplace problem, which possesses high accuracy order and low computing complexities. Moreover, the multivariate asymptotic error expansion of MQM accompanied with for all mesh widths hi is got. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at least by the splitting extrapolation algorithm (SEA). Meanwhile, a posteriori asymptotic error estimate is derived, which can be used to construct self-adaptive algorithms. The numerical examples support our theoretical analysis.

scholarly journals Numerical Solution of Weakly Singular Integrodifferential Equations on Closed Smooth Contour in Lebesgue Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Feras M. Al Faqih
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Error Estimates ◽  
Closed Contour ◽  
Lebesgue Spaces ◽  
Solvability Conditions ◽  
Mechanical Quadrature ◽  
Weakly Singular ◽  
Quadrature Methods ◽  
Singular Integrodifferential Equations

The present paper deals with the justification of solvability conditions and properties of solutions for weakly singular integro-differential equations by collocation and mechanical quadrature methods. The equations are defined on an arbitrary smooth closed contour of the complex plane. Error estimates and convergence for the investigated methods are established in Lebesgue spaces.

scholarly journals A Mechanical Quadrature Method for Solving Delay Volterra Integral Equation with Weakly Singular Kernels

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Li Zhang ◽  
Jin Huang ◽  
Yubin Pan ◽  
Xiaoxia Wen
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Richardson Extrapolation ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Proportional Delays ◽  
Weakly Singular ◽  
Uniqueness Of The Solution ◽  
Singular Kernels ◽  
Mechanical Quadrature Method

In this work, a mechanical quadrature method based on modified trapezoid formula is used for solving weakly singular Volterra integral equation with proportional delays. An improved Gronwall inequality is testified and adopted to prove the existence and uniqueness of the solution of the original equation. Then, we study the convergence and the error estimation of the mechanical quadrature method. Moreover, Richardson extrapolation based on the asymptotic expansion of error not only possesses a high accuracy but also has the posterior error estimate which can be used to design self-adaptive algorithm. Numerical experiments demonstrate the efficiency and applicability of the proposed method.

scholarly journals Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Hu Li ◽  
Yanying Ma
Keyword(s):  
Integral Equation ◽  
Helmholtz Equation ◽  
Boundary Integral Equation ◽  
Dirichlet Boundary ◽  
Boundary Integral ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Splitting Extrapolation ◽  
Accuracy Order ◽  
Mechanical Quadrature Method

We study the numerical solution of Helmholtz equation with Dirichlet boundary condition. Based on the potential theory, the problem can be converted into a boundary integral equation. We propose the mechanical quadrature method (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote byhmthe mesh width of a curved edgeΓm  (m=1,…,d)of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied withO(hm3)for all mesh widthshmis obtained. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at leastO(hmax⁡5)by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

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