The author considers the problem of numerical solving the initial problem for the equation
subject to , where are invertible square matrices of the order and denotes the -th order derivative of the unknown -dimensional function . This problem is deep theoretically investigated, and the local existence theorem 1 presented and proved in the paper is presented (with non-essential simplification ) in the text-book by A. B. Vasilieva and A. N. Tikhonov [Integralnye Uravnenia (Russian). Izdat. Moskovskogo Universiteta, Moscow (1986)]. The author presents and discusses a few variants of iterative algorithms of Picard kind, and a series method of solving the initial problem. The latter is the presentation of an approximation of the required solution as a finite functional sum. The first summand is the given initial value , and the each following summand is an integral iteration of one or two predecessors. Convergence of the sum to the required solution is proved.