Abstract
A boundary value problem for a second-order parabolic equation with a non-self-adjoint operator is considered. Such problems are mathematicalmodels for a number of problems, describing convective-diffusion processes of matter transfer, breakdown mechanisms of laser activity in plasma, etc. While studying the physics of breakdown, one should take into account the avalanche-like increase in the number of free electrons due to multiphoton ionization processes under the influence of optical pulses. This requires the inclusion of related boundary conditions in the problem formulation. An important circumstance that must be taken into account when developing a method for solving the problem is fulfillment of a certain conservation law for its solution. To solve the boundary value problem an approach based on the finite difference method is proposed. The approximation of the equation and boundary conditions is constructed so that the difference scheme is completely conservative. It approximates the original problem with the second order in the spatial variable and in time, and it has the second order of convergence. To effectively solve a system of linear algebraic equations at each time layer, the sweep method for complex systems in combination with the non-monotonic sweep method for systems with a tridiagonal matrix is used. Software based on computer mathematics MATLAB is developed to perform numerical calculations. It is obtained an approximate solution of an applied problem for different instants of time, as well as values of an absorption coefficient, the change in sign of which determines the transition of the plasma in a laser-active state.
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