Abstract
The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.
Highlights
Nonlinear singularly perturbed reaction–diffusion–advection equations arise when solving the problems in gas dynamics [1], combustion theory [2], chemical kinetics [3,4,5,6,7,8,9], nonlinear wave theory [10], biophysics [11,12,13,14,15], medicine [16,17,18,19], ecology [20,21,22], finance [23], and other areas of science [24]
The formulation of inverse problems for the equation of this type with additional information on the dynamics of the reaction front has become relevant because the reaction front is an experimentally distinguishable contrast structure
Questions were considered about the possibility of applying the methods of asymptotic analysis [3,42,43,44] to recover some coefficients in inverse problems for nonlinear singularly perturbed reaction–diffusion– advection equations with data on the position of a reaction front
Summary
Nonlinear singularly perturbed reaction–diffusion–advection equations arise when solving the problems in gas dynamics [1], combustion theory [2], chemical kinetics [3,4,5,6,7,8,9], nonlinear wave theory [10], biophysics [11,12,13,14,15], medicine [16,17,18,19], ecology [20,21,22], finance [23], and other areas of science [24]. It was shown that in the case of sufficiently small values of the singular parameter included in the original partial differential equation, the reduced formulation provides an opportunity to obtain an approximate solution close to the solution of the inverse problem in the full statement. The method of [57] is close to it, which diagnoses the blow-up by the growth of the residual of a numerical solution when the grid parameter tends to zero (for a smooth solution, the residual must decrease) The application of these approaches makes it possible to construct a numerical algorithm for solving the inverse problem in a reduced formulation. Taking into account possible significant errors in the experimental determination of the values of the function u(x, t) lying on the reaction front, this method of restoring the function q(t) is incorrect
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