Approximations to the solution of Cauchy problem for a linear evolution equation via the space shift operator (second-order equation example)

TL;DR

This paper introduces a novel method using the Chernoff approximation and a space shift operator to solve linear parabolic PDEs with variable coefficients, ensuring uniform convergence to the exact solution.

Contribution

The paper develops a new approximation technique for linear evolution equations using a shift operator and proves its convergence, extending the applicability of Chernoff approximations.

Findings

Method converges uniformly to the exact solution.

Applicable to equations with derivatives of orders two, one, and zero.

Demonstrated on a second-order linear PDE example.

Abstract

We present a general method of solving the Cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two, one and zero. The method is based on the Chernoff approximation procedure applied to a specially constructed shift operator. It is proven that approximations converge uniformly to the exact solution.