An inverse problem for a steady-state hydrodynamic diffusion-advection-reaction model is considered. The desired quantity in the inverse problem is the thermal diffusivity of the model in a certain region of the model’s operation inaccessible to direct measurements. To find this desired quantity, it is proposed to use the trace of the normal derivative of the model’s state on an accessible part of the boundary of its implementation domain. The solution to the inverse problem is reduced to an extremal (variational) problem of minimizing some suitable residual functional and minimizing it over a certain set of possible desired coefficients. The paper focuses on algorithmic issues and numerical modeling of the inverse problem. The problem is considered under conditions where the trace of the normal derivative of the model’s state is not an ordinary regular function, but a generalized function (functional). To solve the minimization problem, it is proposed to use the gradient projection method. The gradient of the residual functional is found analytically from the solution of the optimality system, which consists of the direct and adjoint problems. For the practical implementation of gradient method, various acceptable options for choosing the descent step are proposed. The results of numerical simulations of several model examples are presented, demonstrating the viability of the proposed approach for solving the inverse problem. The approach is highly sensitive to the choice of the initial approximation in the iterative process.
Keywords: diffusion-advection-reaction equation, thermal diffusivity coefficient, inverse problem, residual functional, extremal problem, minimum point, gradient projection method
Received April 28, 2026
Revised May 26, 2026
Accepted June 1, 2026
Alexander Illarionovich Korotkii, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: korotkii@imm.uran.ru
Igor Anatolievich Tsepelev, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: tsepelev@imm.uran.ru
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Cite this article as: A.I. Korotkii, I.A. Tsepelev. Numerical simulation of the problem of recovering the leading coefficient of a diffusion-advection-reaction in a closed zone using boundary data. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 3, pp. 110–121.
