  Paul Chernoff in 1968 proposed his approach to approximations of
one-parameter operator semigroups while trying to give a rigorous mathematical
meaning to Feynman's path integral formulation of quantum mechanics. In early
2000's Oleg Smolyanov noticed that Chernoff's theorem may be used to obtain
approximations to solutions of initial-value problems for linear partial
differential equations (LPDEs) of evolution type with variable coefficients,
including parabolic equations, Schr\"odinger equation, and some other. Chernoff
expressions are explicit formulas containing variable coefficients of LPDE and
the initial condition, hence they can be used as a numerical method for solving
LPDEs. However, the speed of convergence of such approximations at the present
time is understudied which makes it risky to employ this class of numerical
methods. In the present paper we take two equations with known solutions (heat
equation and transport equation) and study both analytically and numerically
the speed of decay of the norm of the difference between Chernoff
approximations and exact solutions. We also provide graphical illustrations of
convergence and its rate. These model examples, being relatively simple, allow
to demonstrate general properties of Chernoff approximations. The observations
obtained build a base for the future employment of the approach based on
Chernoff's theorem to the problem of construction of new numerical methods for
solving initial-value problem for parabolic LPDEs with variable coefficients.
