This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations.Introduction. Since the middle of the XX century it is a well known fact [1,2] that the solution of a well-posed Cauchy problem for a linear evolution partial differential equation (examples: Schödinger equation, parabolic equations) is given by a strongly continuous semigroup of linear bounded operators whose infinitesimal generator is a (usually unbounded) linear operator from the right-hand side of the evolution equation. Let us explain this in more details and introduce some notation which will be useful for the main text.Let X be an infinite set, and F be a Banach space of (not necessarily all) number-valued functions on X, and let L be a closed linear operator L : Dom(L) → F with the domain Dom(L) ⊂ F dense in F . We consider the Cauchy problem for the evolution equation
This paper is devoted to a new method for constructing approximations to the solution of a parabolic partial differential equation. The Cauchy problem for the heat equation on a straight line with a variable heat conduction coefficient is considered. In this paper, a sequence of functions is constructed that converges to the solution of the Cauchy problem uniformly in the spatial variable and locally uniformly in time. The functions that make up the sequence are explicitly expressed in terms of the initial condition and the thermal conductivity coefficient, i.e. through functions that play the role of parameters. When constructing functions that converge to the solution, ideas and methods of functional analysis are used, namely, Chernoff's theorem on approximation of operator semigroups, which is why the constructed functions are called Chernoff approximations. In most previously published papers, the error (i. e., the norm of the difference between the exact solution and the Chernoff approximation with number n) does not exceed const/n. Therefore, approximations, when using which the error decreases to zero faster than const/n, we call fast convergent. This is exactly what the approximations constructed in this work are, as follows from the recently proved Galkin-Remizov theorem. Key formulas, explicit forms of constructed approximations, and proof schemes are given in the paper. The results obtained in this paper point the way to the construction of fast converging Chernoff approximations for a wider class of equations.
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