ИСТИНА

We consider two approaches based on the asymptotic analysis which are able to solve ill-posed problems for nonlinear singularly perturbed equations with internal and boundary layers. The first approach is based on the idea that the asymptotic analysis allows to extract a priory information about interior layer (moving front), which appears in the direct problem, and boundary layers, which appear in the conjugate problem (in the case of using some gradient method for numerical solving of the considered problem). In this case we are able to construct so called dynamically adapted mesh based on this a priory information. The dynamically adapted mesh significantly reduces the complexity of the numerical calculations and improve the numerical stability in comparison with the usual approaches. The effectiveness of this approach are shown on the example of coefficient inverse problem for a nonlinear singularly perturbed reaction-diffusion-advection equation with the final time observation data. The second approach is based on the idea that in particular cases the asymptotic analysis allows to reformulate the initial ill-posed problem to the problem that is well-posed. The effectiveness of this approach are shown on the example of boundary inverse problem for a nonlinear singularly perturbed reaction-diffusion-advection equation with the observation data based on the position of interior layer.