## 1 Introduction  
Viscous MHD duct flows present significant challenges and are of great interest in both physics and engineering due to their substantial theoretical and practical importance. They have widespread applications in various fields such as astrophysics, geology, biology, in the design of cooling systems using liquid metals for nuclear fission or fusion reactors, power generation, MHD generators, and electromagnetic pumps. Therefore, it is not surprising that numerous theoretical and experimental studies have been conducted on these issues over the past decades. Solving the equations governing MHD flows is quite challenging due to the coupling between fluid mechanics and electrodynamics equations. Consequently, analytical solutions are only available for specific cases. Therefore, in general, MHD flow problems must be solved numerically using various numerical techniques [1; 2; 3; 4; 5]. Biological effect of magnetic field is discovered during the last decades [6]. From a physical point of view, one evident phenomenon is the magnetic alignment observed in biological systems [7]. Application of MHD problems and their numerical modelling in biological systems are discussed in [8; 9]. Hydrodynamic model simulations are very important in forecasting the environmental fate and potential environmental consequences of chemical contaminants, see e.g., [10]. The authors created a hydrodynamic model to predict and assess the risk of chemical contaminants in Xiamen Bay. The book by [11] is devoted to the mathematical theory of climate and atmospheric pollution and application. In particular, ref. [12] numerically studied a general hydrodynamical model of air circulation, including magnetic fields. The laser-induced wave propagation and reflection phenomenon in a functionally graded porous medium subjected to an electro-magnetic field was investigated in [13], utilizing FEM. In [14] the authors used FEM to develope and numericaly solve new nonlinear coupled finite-strain electro-magneto-thermo-hyperelasticity models. The governing equations were derived in line with a nonlinear version of the Helmholtz free energy. While steady MHD duct flows have been extensively studied, there are less papers focusing on _unsteady_, two-dimensional, incompressible, viscous MHD flow in channels. Studies on time-dependent MHD flow systems, focusing on flow control and MHD stability, have addressed various issues. These control and stability problems for MHD flows are all model-based approaches, where the model parameters or initial conditions are assumed to be well-defined. However, when these model parameters (i.e., empirical parameters and initial conditions) are not precisely known or are uncertain, it typically leads to biased simulation results and inconsistencies between the output of the established control system and the actual physical system. Therefore, it is crucial to identify these unknown or uncertain parameters in the direct system model to ensure that the mathematical model accurately represents the real physical system under consideration. These challenges are commonly known as parameter estimation problems or inverse problems in the field of data assimilation. In this paper, we address a reconstruction problem for a simplified MHD Hartmann flow characterized by incompressible and Newtonian (constant viscosity) properties in a 1D space. The mathematical model for the 1D MHD Hartmann flow can be effectively constructed under certain suitable assumptions, tightly coupling the flow velocity and electromagnetic fields [2; 3; 4; 5]. Control and stability issues for MHD flows have been investigated over the past few decades, see, e.g., [1; 2; 3; 4; 5]. Identifying various unknown or uncertain parameters in the differential problem is crucial for ensuring that the mathematical model accurately represents the physical phenomena under study. This lack of information must be addressed by solving inverse problems.