Geodesic
A high order numerical method for the integral Volterra equations with weak singularity
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 6 (2003) no. 2, pp. 181-195

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A new method of numerical solution of the linear integral Volterra equations with high accuracy, based on approximation of integrals by quadratures independent of the kernel values is proposed. This approach does allow to numerically solve special integral equations, for example, the equations with slightly singular kernels. The main idea of the method is to expand a sought for function by the Taylor formula and to use the kernel moments at the subgrid points to find the matrix of the quadrature coefficients. The dependence of the approximation constant on the number of the subgrid points is analyzed. It is shown that the constant exponentially decreases. An estimate of the solution error for a problem with perturbations of the kernel and the right-hand side is found. The theorem of convergence for the second kind Volterra equations is proved. 
@article{SJVM_2003_6_2_a7,
     author = {A. O. Savchenko},
     title = {A high order numerical method for the integral {Volterra} equations with weak singularity},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {181--195},
     publisher = {mathdoc},
     volume = {6},
     number = {2},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2003_6_2_a7/}
}
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A. O. Savchenko. A high order numerical method for the integral Volterra equations with weak singularity. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 6 (2003) no. 2, pp. 181-195. http://geodesic.mathdoc.fr/item/SJVM_2003_6_2_a7/