We study the problem of finding time-dependent convection coefficient and source on the base of interior point or integral observations for an one-dimensional (1D) magnetohydrodynamic (MHD) flow, whose dynamics is modeled by a coupled parabolic equations. To find the numerical solution of nonlinear coefficient identification inverse problem, we construct linearized approximation in time after space difference discretizations with respect to space. The numerical method uses special decomposition transition to new time layer by solving ODEs. Then we perform decomposition with respect to the unknown coefficient/source or both of the solution of the obtained linear problem. Computational results confirm the efficiency of the numerical algorithms.