# A random regularized approximate solution of the inverse problem for the Burgers' equation

## TL;DR

This paper develops a regularized method combining quasi-reversibility, truncated expansion, and nonparametric regression to approximate solutions for the ill-posed inverse Burgers' equation problem, analyzing convergence rates.

## Contribution

It introduces a novel regularization approach for the inverse Burgers' equation using random perturbations and nonparametric regression, enhancing solution stability.

## Key findings

- Proposes a regularized solution method for the inverse Burgers' equation.
- Analyzes the convergence rate of the regularized solution.
- Demonstrates improved stability in the inverse problem solution.

## Abstract

In this paper, we find a regularized approximate solution for an inverse problem for the Burgers' equation. The solution of the inverse problem for the Burgers' equation is ill-posed, i.e., the solution does not depend continuously on the data. The approximate solution is the solution of a regularized equation with randomly perturbed coefficients and randomly perturbed final value and source functions. To find the regularized solution, we use the modified quasi-reversibility method associated with the truncated expansion method with nonparametric regression. We also investigate the convergence rate.