A global method for solving second-kind Volterra–Fredholm integral equations

Abstract The paper presents a Nyström-type method to approximate the solution of second-kind Volterra–Fredholm integral equations. Two forms are considered, that is the disjoint form, in which the Volterra and Fredholm operators are additive integrals; and the mixed one, in which the two integrals appear in a single term through composition. In both situations, the right-hand side and the kernel functions may have algebraic singularities at $$\pm 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and hence equations are treated in suitable weighted spaces equipped with the uniform norm. The proposed methods, based on product and Gauss rules, are stable and convergent. The error is of the order of the best polynomial approximation of the given functions. Numerical examples are presented to illustrate the accuracy of the method.