Numerical study of the rate of convergence of Chernoff approximations to solutions of the heat equation with full list of illustrations and Python source code (2301.05284v2)

Abstract: Chernoff approximations are a flexible and powerful tool of functional analysis, which can be used, in particular, to find numerically approximate solutions of some differential equations with variable coefficients. For many classes of equations such approximations have already been constructed since pioneering papers of Prof. O.G.Somlyanov in 2000, however, the speed of their convergence to the exact solution has not been properly studied. We select the heat equation (because its exact solutions are already known) as a simple yet informative model example for the study of the rate of convergence of Chernoff approximations. Examples illustrating the rate of convergence of Chernoff approximations to the solution of the Cauchy problem for the heat equation are constructed in the paper. Numerically we show that for initial conditions that are smooth enough the order of approximation is equal to the order of Chernoff tangency of the Chernoff function used. We also consider not smooth enough initial conditions and show how H\"older class of initial condition is related to the rate of convergence. This method of study in the future can be applied to general second order parabolic equation with variable coefficients by a slight modification of our Python 3 code. This arXiv version of the text is a supplementary material for our journal article. Here we include all the written text from the article and additionally all illustrations (Appendix A) and full text of the Python 3 code (Appendix B).