## 1 Introduction  
The research concerning the construction and analysis of numerical methods for partial differential equations (PDEs) is more developed and actual than analytical methods. It is due to the necessity of solving many problems of physics, mechanics, heat-mass transfer, finance, biology, etc., see e.g., [1; 2; 3; 4; 5; 6; 7; 8], for which it is difficult to construct analytical solutions. The main interest in efficient numerical solving of the PDEs in applied science and engineering is concerned with construction of fast computational _algorithms_. The inverse problem is formulated for a particular process described by a PDE when one or some important data such as a coefficients, boundary data, and source data, are provided in many publications [1; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. In the current paper, we study a convection coefficient and a source inverse problem of a parabolic system from MHD. In recent years, the viscous MHD duct flows, as a challenging and problem arising in the fields of physics and engineering, have considerable theoretical and practical importance owing to their important and widespread applications in astrophysics, geology, designing cooling systems with liquid metals for a nuclear fission or fusion reactor, power generation, MHD generators, and electromagnetic pumps. A number of theoretical and experimental studies have been performed for the aforementioned. Because of the coupling of the equations of fluid mechanics and electrodynamics, the solution of the equations governing MHD flows is quite difficult, so analytical solutions are available only in some special cases. For this reason, in the general situation, the MHD flow problems can be only numerically solved by taking advantage of some numerical methods [4; 15]. While steady MHD duct flows have been studied extensively, some papers have appeared on _unsteady_ 2D incompressible, viscous MHD flow in channels. Works on _time-dependent_ MHD flow systems, concerning the areas of the flow control and the MHD stability, have investigated different problems. These related control and stability problems for the MHD flows are all model-based approaches, in which the model parameters or the initial conditions have to be identified. However, when model parameters (i.e., empirical parameters and initial conditions) are unknown or uncertain, it often leads to biased simulation results and inconsistencies between the output of the established control system and the real physical system. Thus, it is fundamentally important to recover these various unknown or uncertain parameters in the direct system model so that the mathematical model can best fit the real physical system under considerations. Such problems are commonly referred to as parameter estimation problems or inverse problems in the areas of data assimilation. In the paper, we consider a recovering problem for a simplified MHD Hartman flow with characteristics of incompressible and Newtonian (constant viscosity) in 1-D space. The mathematical model of the 1-D MHD Hartman flow can be successfully built under some suitable assumptions and tightly coupled by the flow velocity and electromagnetic fields [17; 21]. Least squares cost functionals are often used for parameter estimation of the objective functions. However, such estimation procedures (gradient-like search) are computationally time-consuming [11; 22]. Also, through each step in the iteration, the solutions of forward problems of PDEs have to be solved numerically. In this paper, we propose a direct system method to avoid these drawbacks. This approach leads to the technique of the loaded equation method, first introduced for the solution of the source heat problem in [6]. In [23], a Galerkin finite element method was used for solving an inverse problem, similar to IP1, described in Section 2.2 of the present paper. This paper is structured as follows. In the next section, the direct and the inverse problems are introduced.