Solution of nonlinear weakly singular Volterra integral equations using the fractional-order Legendre functions and pseudospectral method

2015 ◽  
Vol 39 (12) ◽  
pp. 3411-3425 ◽  
Author(s):  
Jafar Eshaghi ◽  
Hojatollah Adibi ◽  
Saeed Kazem
Keyword(s):  
Integral Equations ◽  
Fractional Order ◽  
Pseudospectral Method ◽  
Volterra Integral Equations ◽  
Legendre Functions ◽  
Weakly Singular

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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