The best parameterization of initial value problem for mixed difference-differential equation

We develop an approach to the numerical integration of initial value problem for mixed difference-differential equations that are differential with respect to one argument and difference with respect to others. Preliminary reduction of the problem to a set of Cauchy problems for systems of ordinary differential equations depending on a parameter affords to state it as the problem of the continuing the solution with respect to the best continuation parameter, namely, the integral curve length. This statement has numerous advantages over the usual statement. Namely, the right-hand sides of the transformed system remain bounded even if right-hand sides of the original system become infinite at some points.