Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations

Solving of the coefficient inverse problems for a nonlinear singularly perturbed reaction-diffusion-advection equation with the final time data

Asymptotic-numerical approach to solving the coefficient inverse problem for a nonlinear singularly perturbed two-dimensional reaction–diffusion equation by knowing the location of moving front data is proposed. Asymptotic analysis of the direct problem allows to reduce the original two-dimensional parabolic problem to a series of more simple equations with lower dimension for the determination of moving front parameters. It enables to associate the observed location of the moving front to the parameters which have to be identified. Numerical examples show the effectiveness of the proposed method.

Solving of the coefficient inverse problem for a nonlinear singularly perturbed two-dimensional reaction–diffusion equation with the location of moving front data

A singularly perturbed boundary value problem for a second-order ordinary differential equation known in applications as a stationary reaction–diffusion equation is studied. A new class of problems is considered, namely, problems with nonlinearity having discontinuities localized in some domains, which leads to the formation of sharp transition layers in these domains. The existence of solutions with an internal transition layer is proved, and their asymptotic expansion is constructed.

Internal layers in the one-dimensional reaction–diffusion equation with a discontinuous reactive term

Abstract In this paper, a new asymptotic-numerical approach to solving an inverse boundary value problem for a nonlinear singularly perturbed parabolic equation with time-periodic coefficients is proposed. An unknown boundary condition is reconstructed by using known additional information about the location of a moving front. An asymptotic analysis of the direct problem allows us to reduce the original inverse problem to that with a simpler numerical solution. Numerical examples demonstrate the efficiency of the method.

Asymptotic analysis of solving an inverse boundary value problem for a nonlinear singularly perturbed time-periodic reaction-diffusion-advection equation

The asymptotical method of differential inequalities is developed for a new class of periodic problems of reaction-diffusion type. The problem of the existence and Lyapunov stability of periodic solutions with internal transient layers in the case of balanced nonlinearity is studied.

Development of the asymptotic method of differential inequalities for investigation of periodic contrast structures in reaction-diffusion equations

We propose an effective asymptotic-numerical approach to the problem of moving front type solutions in nonlinear reaction-diffusion-advection equations. The dimension of spatial variables for the location of a moving front is lower per unit then the original problem. This fact gives the possibility to save computing resources in numerical experiments and speed up the process of constructing approximate solutions with a suitable accuracy.

V. T. Volkov

Papers

Solving of the coefficient inverse problems for a nonlinear singularly perturbed reaction-diffusion-advection equation with the final time data

Solving of the coefficient inverse problem for a nonlinear singularly perturbed two-dimensional reaction–diffusion equation with the location of moving front data

Asymptotic analysis of solving an inverse boundary value problem for a nonlinear singularly perturbed time-periodic reaction-diffusion-advection equation

Development of the asymptotic method of differential inequalities for investigation of periodic contrast structures in reaction-diffusion equations

Asymptotic-numerical Investigation of Generation and Motion of Fronts in Phase Transition Models