Abstract:
We study the Dirichlet problem for a multidimensional differential equation of Sobolev type with variable coefficients. 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 difference scheme is constructed. Using the method of energy inequalities, an a priori estimate is obtained for the solution of a locally one-dimensional difference scheme, which implies its stability and convergence. For a two-dimensional problem, an algorithm for the numerical solution of the problem posed was constructed, numerical experiments were carried out on test examples, illustrating the theoretical results obtained in this work.
Keywords:boundary value problems, a priori estimate, multidimensional Sobolev-type equation, Dirichlet problem, locally one-dimensional scheme, stability, convergence.
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