The paper considers a model problem of a combined thermal and diffusion process in silicon. The mathematical model of this process is an initial-boundary value problem for a system of linear partial differential equations of parabolic type. In this system, one equation describes the process of heat propagation in silicon, and the other describes the process of impurity diffusion in it. In this case, the equations are not independent in the same way that the diffusion coefficient depends on temperature. For each equation in this system, the corresponding initial-boundary conditions are set. An implicit difference scheme and the classical sweep method are used to find an approximate solution to the problem that has arisen. The paper presents a description of a numerical algorithm and exact calculation formulas for solving a discretized parabolic problem.

Keywords: model of thermal diffusion process, numerical simulation, sweep method, implicit difference scheme

The paper considers a mathematical model of the ion-beam etching process. A nonlinear differential equation of first-order ion-beam etching is considered. It has been established that the model equation for ion-beam etching can be reduced to a homogeneous Monge-Ampere equation. Some classes of exact solutions are presented for this equation. A power-law solution is obtained by the method of functional separation of variables, which depends only on a set of constants and does not contain arbitrary functions. Solutions are also found that linearly depend on arbitrary functions of the coordinate variable and the time variable. Assumptions and explicit conditions are formulated on how to select solutions from the families of solutions of the Monge-Ampere equation that correspond to the model process under consideration. A class of nonlinear equations in partial derivatives of the first order is indicated, which can also be reduced to the Monge-Ampere equation. Limitations on the etching rate are established, which allow the ion-beam etching equation to be reduced to a second-order linear hyperbolic equation, for which, by separation of variables, it is possible to obtain a solution in the form of a Fourier series.

Keywords: ion-beam etching equation, Monge-Ampere equation, model solutions, exact solutions