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Working paper

Rapidly Converging Chernoff Approximations to Solution of Parabolic Differential Equation on the Real Line

Abstract. The method of Chernoff approximation was discovered by Paul Chernoff in 1968 and now is a powerful and flexible tool of contemporary  functional analysis. This method is different from grid-based approach and helps to solve numerically the Cauchy problem for evolution equations, e.g., for heat equation and for more general parabolic second-order partial differential equations with variable coefficients. The case of heat equation is well-studied and does not require any approximations because Green's function is known in that case; but for more general equations exact formulas for solutions are unknown, so numerical approximations of solutions are often requested by engineers and other researchers dealing with PDEs. Chernoff approximations are functions defined by explicit expressions that contain variable  coefficients of the equation and initial condition as parameters. Traditionally researchers construct Chernoff approximations with the use of integral operators which lead to Feynman formulas that are connected with representation of solution in terms of Feynman path integral and Feynman-Kac formulas. This traditional approach has two features that restrict its practical use. First, one needs to integrate over the Cartesian product of $n$ copies of the real line hence $n$-tuple integrals are improper (for better quality of approximation one needs to take larger values of $n$, and this leads to improper integrals of high multiplicity which are difficult to handle in practice). Second, the speed of convergence usually is not very high (not better than $1/n$).  In the present paper we construct Chernoff approximations of a new kind which are free from these disadvantages: all integrals in our formulas are over the segment $[-1,1]$, and the speed of convergence is higher than $1/n$ for  initial conditions that are smooth enough. Our approximations provide solution in a form of quasi-Feynman formula which is not clearly connected with Feynman integral but is much better for practical purposes.