Integral transforms for explicit source estimation in non-linear advection-diffusion problems

In many engineering problems non-linear mathematical models are needed to accurately describe the physical phenomena involved. In such cases, the inverse problems related to those models bring additional challenges. In this scenario, this work provides a novel general regularized methodology based on integral transforms for obtaining explicit solutions to inverse problems related to source term estimation in non-linear advection-diffusion models. Numerical examples demonstrate the application of the methodology for some cases of the one- and two-dimensional versions of the non-linear Burgers' equation. An uncertainty analysis for the proposed inverse problem is also conducted using the Monte Carlo Method, in order to illustrate the reliability of the estimates. The results reveal accurate estimates for different functional forms of the sought source term and varying noise levels, for both diffusion-dominated and advection-dominated scenarios.