Barycentric Lagrange Interpolation Matrix–Vector Form Polynomial for Solving Volterra Integral Equations of the Second Kind

2021 ◽  
pp. 151-161
Author(s):  
E. S. Shoukralla ◽  
B. M. Ahmed
Keyword(s):  
Integral Equations ◽  
Lagrange Interpolation ◽  
Volterra Integral Equations ◽  
Vector Form ◽  
Interpolation Matrix ◽  
Matrix Vector

Numerical Treatment of Nonlinear Stochastic Itô–Volterra Integral Equations by Piecewise Spectral-Collocation Method

2019 ◽  
Vol 14 (3) ◽  
Author(s):  
Fakhrodin Mohammadi
Keyword(s):  
Approximate Solution ◽  
Finite Number ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Lagrange Interpolation ◽  
Interpolation Method ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Treatment ◽  
Nonlinear Integral Equations

This paper deals with the approximate solution of nonlinear stochastic Itô–Volterra integral equations (NSIVIE). First, the solution domain of these nonlinear integral equations is divided into a finite number of subintervals. Then, the Chebyshev–Gauss–Radau points along with the Lagrange interpolation method are employed to get approximate solution of NSIVIE in each subinterval. The method enjoys the advantage of providing the approximate solutions in the entire domain accurately. The convergence analysis of the numerical method is also provided. Some illustrative examples are given to elucidate the efficiency and applicability of the proposed method.

scholarly journals Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Qiumei Huang ◽  
Min Wang
Keyword(s):  
Integral Equations ◽  
Numerical Experiments ◽  
Volterra Integral Equations ◽  
Least Square ◽  
Convergence Order ◽  
Collocation Points ◽  
Weakly Singular ◽  
Interpolation Postprocessing ◽  
Collocation Solution

AbstractIn this paper, we discuss the superconvergence of the “interpolated” collocation solutions for weakly singular Volterra integral equations of the second kind. Based on the collocation solution $$u_h$$ u h , two different interpolation postprocessing approximations of higher accuracy: $$I_{2h}^{2m-1}u_h$$ I 2 h 2 m - 1 u h based on the collocation points and $$I_{2h}^{m}u_h$$ I 2 h m u h based on the least square scheme are constructed, whose convergence order are the same as that of the iterated collocation solution. Such interpolation postprocessing methods are much simpler in computation. We further apply this interpolation postprocessing technique to hybrid collocation solutions and similar results are obtained. Numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing methods.

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