scholarly journals PRODUCT INTEGRATION FOR WEAKLY SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS

2011 ◽  
Vol 16 (1) ◽  
pp. 153-172 ◽  
Author(s):  
Arvet Pedas ◽  
Enn Tamme
Keyword(s):  
Differential Equations ◽  
Convergence Rates ◽  
Discrete Version ◽  
Product Integration ◽  
Weakly Singular ◽  
Integration Techniques ◽  
Singular Kernels ◽  
Convergence Estimates ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

On the basis of product integration techniques a discrete version of a piecewise polynomial collocation method for the numerical solution of initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels is constructed. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms.

scholarly journals NUMERICAL SOLUTION OF VOLTERRA INTEGRAL EQUATIONS WITH WEAKLY SINGULAR KERNELS WHICH MAY HAVE A BOUNDARY SINGULARITY

2009 ◽  
Vol 14 (1) ◽  
pp. 79-89 ◽  
Author(s):  
Marek Kolk ◽  
Arvet Pedas
Keyword(s):  
Integral Equations ◽  
Volterra Integral Equations ◽  
Piecewise Polynomial ◽  
Boundary Singularity ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Convergence Estimates ◽  
Linear Volterra Integral Equations ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

We propose a piecewise polynomial collocation method for solving linear Volterra integral equations of the second kind with kernels which, in addition to a weak diagonal singularity, may have a weak boundary singularity. Global convergence estimates are derived and a collection of numerical results is given.

scholarly journals COLLOCATION APPROXIMATIONS FOR WEAKLY SINGULAR VOLTERRA INTEGRO‐DIFFERENTIAL EQUATIONS

2003 ◽  
Vol 8 (4) ◽  
pp. 315-328 ◽  
Author(s):  
I. Parts ◽  
A. Pedas
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Order Of Convergence ◽  
Piecewise Polynomial ◽  
Volterra Type ◽  
Weakly Singular ◽  
Uniform Grids ◽  
Attainable Order ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

A piecewise polynomial collocation method for solving linear weakly singular integro‐differential equations of Volterra type is constructed. The attainable order of convergence of collocation approximations on arbitrary and quasi‐uniform grids is studied theoretically and numerically.

Fully Discrete Galerkin Method For Fredholm Integro-Differential Equations With Weakly Singular Kernels

2008 ◽  
Vol 8 (3) ◽  
pp. 207-222 ◽  
Author(s):  
H. BRUNNER
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Optimal Order ◽  
Piecewise Polynomial ◽  
Fully Discrete ◽  
Proportional Delays ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Discrete Galerkin Method ◽  
Polynomial Collocation

AbstractWe analyze the optimal superconvergence properties of piecewise polynomial collocation solutions on uniform meshes for Volterra integral and integrodifferential equations with multiple (vanishing) proportional delays. It is shown that for delay integro-differential equations the recently obtained optimal order is also attainable. For integral equations with multiple vanishing delays this is no longer true.

A Piecewise Nonpolynomial Collocation Method for Fractional Differential Equations

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Shahrokh Esmaeili
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Differential Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Block Scheme ◽  
Poor Convergence ◽  
Fractional Differential ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

Since the solutions of the fractional differential equations (FDEs) have unbounded derivatives at zero, their numerical solutions by piecewise polynomial collocation method on uniform meshes will lead to poor convergence rates. This paper presents a piecewise nonpolynomial collocation method for solving such equations reflecting the singularity of the exact solution. The entire domain is divided into several small subdomains, and the nonpolynomial pieces are constructed using a block-by-block scheme on each subdomain. The method is applied to solve linear and nonlinear fractional differential equations. Numerical examples are given and discussed to illustrate the effectiveness of the proposed approach.

scholarly journals Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

2021 ◽  
Vol 5 (3) ◽  
pp. 70
Author(s):  
Esmail Bargamadi ◽  
Leila Torkzadeh ◽  
Kazem Nouri ◽  
Amin Jajarmi
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Algebraic System ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Chebyshev Wavelets ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Chebyshev Wavelet

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

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