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Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. (English) Zbl 1207.65157
The authors consider Volterra integral equations of the second kind with a weakly singular kernel of the form $$y(t) = g(t) + \int_0^t (t-s)^{-\mu} K(t,s)y(s)\,ds,\ 0<\mu<1,\quad 0\le t\le T,$$ and develop a Jacobi-collocation spectral method for the above integral equation. The main aim is to use Jacobi-collocation method to numerically solve Volterra integral equation on the interval $[-1,1]$. They obtain higher-order accuracy for the numerical approximation using a Jacobi spectral quadrature rule for the integral term. Finally, numerical results are given by tables and figures. Numerical and exact solutions are compared by graphics in the $L^{\infty}$-norm and $L_{w}^{\infty}$-norm.

65R20Integral equations (numerical methods)
45D05Volterra integral equations
45E10Integral equations of the convolution type
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