A high order numerical method for the Volterra integral equations with weak singularity. (Russian. English summary) Zbl 1032.65148

Savchenko, A. O.

Sib. Zh. Vychisl. Mat. 6, No. 2, 181-195 (2003).

Summary: A new method for the numerical solution of a linear Volterra integral equations with high accuracy, based on the approximation of the integrals by quadratures independent of the kernel values is proposed. This approach does allow to numerically solve special integral equations, for example, equations with weakly singular kernels. The main idea of the method is to expand the unknown 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. A theorem on the convergence for second kind Volterra equations is proved.

MSC:

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

Keywords:

Gauss quadrature formula; Legendre polynomial; perturbation of the kernel; discrete variable method; error estimate; linear volterra integral equations; weakly singular kernels; Taylor formula; convergence