Difference methods for solving some classes of multidimensional loaded parabolic equations with boundary conditions of the first kind

Abstract

Background. In the literature, loaded differential equations are usually called equations containing functions of the solution on manifolds of smaller dimension than the dimension of the domain of definition of the sought function. The purpose of the work is to study a difference scheme of second order accuracy in terms of mesh parameters for solving the first boundary value problem for loaded parabolic equations in a multidimensional domain with variable coefficients. Two different types of equations are considered. Problems of this type arise when studying the movement of groundwater, in problems of managing the quality of water resources, when a pollutant of a certain intensity enters a reservoir from n sources, when constructing a mathematical model of the transfer of dispersed pollutants in the atmospheric boundary layer when describing the mass distribution function of drops and ice particles, taking into account microphysical condensation processes, coagulation (combination of small drops into large aggregates), fragmentation and freezing of drops in convective clouds, as well as in the study of natural processes and phenomena that take into account the memory effect. Materials and methods. The finite difference method and the method of energy inequalities are used to obtain a priori estimates for the solution of difference schemes. Results. For each problem, a difference scheme with the order of approximation O( h 2 + τmσ ) is constructed, where mσ = 1, if σ ≠ 0,5 and mσ = 2 , if σ = 0,5; an a priori estimate was obtained using the method of energy inequalities to solve the difference problem. From the obtained estimates it follows that the solution is unique and stable with respect to the right-hand side and the initial data, as well as the convergence of the solution of the difference problem to the solution of the corresponding initial differential problem with a speed O( h 2 + τmσ ) at σ = 0,5. Conclusions. New numerical schemes of second order approximation have been developed to solve the problems posed

Keywords

• first initial-boundary value problem
• loaded equation
• a priori estimate
• difference scheme
• parabolic equation