Numerical methods for solving the second boundary value problem for a multidimensional Sobolev type equation

Abstract

The second boundary value problem is investigated for a multidimensional Sobolev-type differential equation with variable coe cients. The considered equation is reduced to an integro-differential equation of parabolic type with a small parameter. For an approximate solution of the obtained problem, a locally one-dimensional dierence scheme is constructed. Using the method of energy inequalities, an a priori estimate is obtained for the solution of a locally one-dimensional dierence scheme, which implies its stability and convergence. For a two-dimensional problem, an algorithm is constructed for the numerical solution of the second boundary value problem for a partial differential equation of Sobolev type.