Product integration for Volterra integral equations of the second kind with weakly singular kernels
Mathematics of Computation
1896–1996: one hundred years of Volterra integral equations of the first kind
Selected papers of the second international conference on Numerical solution of Volterra and delay equations : Volterra centennial: Volterra centennial
A new approach to the numerical solution of weakly singular Volterra integral equations
Journal of Computational and Applied Mathematics
Quadrature rule for Abel's equations: Uniformly approximating fractional derivatives
Journal of Computational and Applied Mathematics
Uniform approximation to fractional derivatives of functions of algebraic singularity
Journal of Computational and Applied Mathematics
A Finite Element Splitting Extrapolation for Second Order Hyperbolic Equations
SIAM Journal on Scientific Computing
An algorithm using the finite volume element method and its splitting extrapolation
Journal of Computational and Applied Mathematics
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This paper presents high accuracy mechanical quadrature methods for solving first kind Abel integral equations. To avoid the ill-posedness of problem, the first kind Abel integral equation is transformed to the second kind Volterra integral equation with a continuous kernel and a smooth right-hand side term expressed by weakly singular integrals. By using periodization method and modified trapezoidal integration rule, not only high accuracy approximation of the kernel and the right-hand side term can be easily computed, but also two quadrature algorithms for solving first kind Abel integral equations are proposed, which have the high accuracy O(h^2) and asymptotic expansion of the errors. Then by means of Richardson extrapolation, an approximation with higher accuracy order O(h^3) is obtained. Moreover, an a posteriori error estimate for the algorithms is derived. Some numerical results show the efficiency of our methods.