Total variation regularization analysis for the inverse source problem of Crocco-type parabolic equation in thermal energy management

Purpose The purpose of this paper is to investigate the inverse problem of recovering two-dimensional spatial heat sources in thermal energy management based on the Crocco-type degenerate parabolic equation. Design/methodology/approach Unlike classical parabolic equations, the mathematical model discussed in the paper belongs to the Crocco-type equation, characterized by the absence of the second-order derivative with respect to x. Moreover, to better capture the discontinuity of the source term coefficients, this study employs the total variation regularization method to address the problem. Findings By transforming the original problem into an optimal control problem, a necessary condition of the control functional is derived. Combined with the a priori estimate of the forward problem, the global uniqueness and stability of the minimizer are obtained. For numerical verification, the Gauss–Jacobi iterative scheme is used to deal with the nonlinear term and the Euler equation is also derived. Numerical experiments indicate that the proposed algorithm is stable and effective, achieving successful reconstruction of the heat source distribution within the building. Practical implications This study enhances thermal energy management by precisely reconstructing heat source distributions, thereby optimizing energy efficiency and thermal comfort, with particular emphasis on irregular heat source configurations. Originality/value The paper first applies total variation regularization to Crocco-type parabolic equations to address inverse source problems in thermal energy management, optimizing energy utilization and enhancing thermal comfort.