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Rigorous convergence analysis of Jacobi spectral Galerkin methods for Volterra integral equations with noncompact kernels. (English) Zbl 1436.45008

The following weakly singular Volterra integral equation with noncompact kernel in the form \[ u(\xi)= \int^\xi_0\frac {\tau^{\alpha-1}} {\xi^\alpha} R(\xi,\tau)u(\tau)d\tau+\hat f(\xi),\ \ \xi\in[0,L], \] \((\alpha>0\), \(R(\xi,\tau)\), \(\hat f(\xi)\) are smooth functions) is considered in this paper. The authors apply Jacobi-Galerkin methods (spectral and nonspectral) to solve this given equation. Gauss quadrature rules are used for approximation of the integral and convergence analysis in \(L_\infty\) and \(L_2\) weighted norms is studied. The developed theory is verified by several numerical examples.

MSC:

45L05 Theoretical approximation of solutions to integral equations
45D05 Volterra integral equations
65R20 Numerical methods for integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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