Numerical solution of integral-algebraic equations with a weak boundary singularity by $k$-step methods

The article presents the construction of $k$-step methods for solving systems of Volterra integral equations of the first and the second kind with a weak power-law singularity of the kernels in the lower limit of integration. The matrix-vector representation of such systems has the form of an abstract equation with a degenerate coefficient matrix at the nonintegral terms, which is called an integral-algebraic equation. The methods proposed are based on extrapolation formulas for the principal part, Adams-type multistep methods, and a product integration formula for the integral term. The weights of the quadrature formulas constructed are obtained explicitly. A theorem on the convergence of the methods developed is proved. The theoretical results are illustrated by numerical calculations of test examples.