Computing without a computer: a new approach for solving nonlinear differential equations

TL;DR

This paper introduces a novel analytical approach called the 'method of the computer analogy' for solving nonlinear differential equations by formalizing arithmetic operations on ordinary computing devices, bypassing traditional digital computation.

Contribution

It proposes a new method that formalizes arithmetic operations to solve nonlinear differential equations analytically, using a finite-difference scheme and probabilistic averaging.

Findings

Provides an explicit analytical solution representation.

Demonstrates the method with an example of nonlinear equations.

Shows the approach approximates finite-difference schemes effectively.

Abstract

The well-known Turing machine is an example of a theoretical digital computer, and it was the logical basis of constructing real electronic computers. In the present paper we propose an alternative, namely, by formalising arithmetic operations in the ordinary computing device, we attempt to go to the analytical procedure (for calculations). The method creates possibilities for solving nonlinear differential equations and systems. Our theoretical computer model requires retaining a finite number of terms to represent numbers, and utilizes digit carry procedure. The solution 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 schematic representation 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. All the stages of formalizing operations of the classical computer result in "the method of the computer analogy". The proposed method leads to an explicit analytical representation of the solution. We present the general features of the algorithm which are illustrated by an example of solutions for a system of nonlinear equations.