On the convergence of a cellular automaton-based solution to the solution of the original Cauchy problem

The paper presents a rationale for using cellular automata as mathematical models for solving Cauchy problems for ordinary first-order differential equations. Reasons are given for using cellular automata for modeling socio-economic processes and solving related applied problems, including those usually solved with the help of ordinary first order differential equations. A method for replacing a first-order ordinary differential equation with a cellular automaton is presented. A theorem is given on the convergence in probability of the solution obtained by means of a cellular automaton to the solution of the original Cauchy problem as the number of cell states tends to infinity. This makes cellular automata another method like finite-difference one for obtaining approximate solutions to differential equations. The obtained result is discussed, and a corollary to the theorem is given, considering the convergence of the cellular automaton solution to the solution of the original problem when the number of computational experiments tend to the infinity. This means that there is no need to significantly increase the number of states of the cellular automaton cell to increase the accuracy of the solution. The use of cellular automata for solving socio-economic problems is discussed, including expanding the applicability of mathematical models in cases where existing problems cannot be solved with the help of differential equations. An example of such use of the cellular automata approach is given.