Difference schemes for the numerical solution of the heat conduction equation with aftereffect

A family of grid methods is constructed for the numerical solution of the heat conduction equation in the general form with delay; the methods are based on the idea of separating the current state and the prehistory function. A theorem is obtained on the order of convergence of the methods, which uses the technique of proving similar statements for functional differential equations and methods from the general theory of difference schemes. Results of calculating test examples with constant and variable delay are presented.