Geodesic
Fast converging Chernoff approximations to the solution of heat equation with variable coefficient of thermal conductivity
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 24 (2022) no. 3, pp. 280-288.

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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.
Keywords: Cauchy problem for heat equation with variable coefficient of thermal conductivity, approximation of solution, rate of convergence to the solution, one-parameter semigroup of operators, Chernoff product formula. 
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A. V. Vedenin. Fast converging Chernoff approximations to the solution of heat equation with variable coefficient of thermal conductivity. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 24 (2022) no. 3, pp. 280-288. http://geodesic.mathdoc.fr/item/SVMO_2022_24_3_a1/

[1] Numerical methods for partial differential equationsed. . by G. Evans, J. Blackledge, P. Yardley, Springer, 2000, 304 pp. | MR

[2] V. Ruas., Numerical methods for partial differential equations: an introduction, Wiley, 2016., 376 pp. | MR

[3] Numerical methods for PDEs. : state of the art techniquesed. . by D. A. Di Pietro, A. Ern, L. Formaggia, Springer, Cham, Switzerland, 2018., 330 pp. | MR

[4] K.-J. Engel, R. Nagel., One-parameter semigroups for linear evolution equations, Springer, New York, 2000, 589 pp. | MR

[5] P. R. Chernoff., “Note on product formulas for operator semigroups”, J. Functional Analysis, 2:2 (1968), 238-–242 | DOI | MR

[6] Ya. A. Butko, “The method of Chernoff approximation”, Springer Proceedings in Mathematics and Statistics, 325 (2020), 19-46 | MR

[7] I. D. Remizov, “Solution-giving formula to Cauchy problem for multidimensional parabolic equation with variable coefficients”, Journal of Mathematical Physics, 60:7 (2019) | DOI | MR

[8] I. D. Remizov, “Quasi-Feynman formulas a method of obtaining the evolution operator for the Schrödinger equation”, J. Funct. Anal, 270:12 (2016), 4540-4557 | DOI | MR

[9] A.Gomilko, S.Kosowicz, Yu.Tomilov, “A general approach to approximation theory of operator semigroups”, Journal de Mathématiques Pures et Appliquées, 127 (2019), 216–267 | DOI | MR

[10] Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Rate of convergence of Feynman approximations of semigroups generated by the oscillator Hamiltonian”, Theoretical and Mathematical Physics, 172 (2012), 987–1000 | DOI | MR

[11] A. Gomilko, Yu. Tomilov, “On convergence rates in approximation theory for operator semigroups”, Journal of Functional Analysis, 266:5 (2014), 3040–3082 | DOI | MR

[12] I. D. Remizov, “On estimation of error in approximations provided by chernoff’s product formula”, International Conference ’ShilnikovWorkshop-2018’ dedicated to the memory of outstanding Russian mathematician Leonid Pavlovich Shilnikov (1934-2011), book of abstracts, 2018, 38–41 | MR

[13] A. V. Vedenin, V. S. Voevodkin, V. D. Galkin, E. Yu. Karatetskaya, I. D. Remizov, “Speed of convergence of Chernoff approximations to solutions of evolution equations”, Mathematical Notes, 108:3 (2020), 451–456 | DOI | MR

[14] O. E. Galkin, I. D. Remizov, “Rate of convergence of Chernoff approximations to $C_0$-semigroups of operators”, Mathematical Notes, 111:2 (2022), 305–307 | DOI | MR

[15] O. E. Galkin, I. D. Remizov, Upper and lower estimates for rate of convergence in the Chernoff product formula for semigroups of operators, 2022, 33 pp. | MR

[16] P. S. Prudnikov, Speed of convergence of Chernoff approximations for two model examples: heat equation and transport equation, 2012, 27 pp.