Modulating Functions-Based Method for Parameters and Source Estimation in One-Dimensional Partial Differential Equations

TL;DR

This paper introduces a modulating functions-based approach for estimating unknown parameters and sources in one-dimensional PDEs, simplifying the problem into linear algebraic equations and demonstrating effectiveness through wave and KdV equations.

Contribution

It presents a novel modulating functions-based method that simplifies parameter and source estimation in PDEs into linear algebraic systems, with proven well-posedness.

Findings

Effective in noise-free and noisy scenarios

Successfully applied to wave and KdV equations

Proven well-posedness of the method

Abstract

In this paper, modulating functions-based method is proposed for estimating space-time dependent unknowns in one-dimensional partial differential equations. The proposed method simplified the problem into a system of algebraic equations linear in unknown parameters. The well-posedness of modulating functions-based solution is proven. The wave and the fifth order KdV equations are used as examples to show the effectiveness of the proposed method in both noise-free and noisy cases.