A method of formalizing computer operations for solving nonlinear differential equations

A new mathematical formalization of the computation process in a classical computer is proposed as a tool for solving nonlinear differential equations. This model includes retaining a finite number of ranks and using the rank transfer procedure. A method for solving nonlinear differential equations based on this model is suggested, in which the solution of a differential equation is represented in the form of a segment of a series in the powers of the step size of the independent variable in the finite-difference scheme. The algorithm generates a scheme that approximates the convergent finite difference scheme, which, in turn, approximates the equation under consideration. The use of probabilistic methods allows us to average the recurrent calculations and exclude intermediate levels of computation in our numerical scheme. The proposed method results in an explicit representation of the solution. Examples of solutions for nonlinear equations and systems of nonlinear equations are given.