On the numerical methods with optimal rates of convergence for cordial Volterra integral equations of a specific class

An important type of cordial Volterra integral equations is considered. The equation of this type appears in heat conduction problems with mixed boundary conditions. We propose some high order methods for approximating the unique solution of the equation. The common approach in these algorithms is to consider a representation of the solution and design appropriate composite quadrature rules for its approximation. Then we can reach an optimal rate of convergence and, comparing to the previous methods, enhance the accuracy of the approximations considerably.